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TK-YUPANA - Kunturweb

1. Table 33 133x97 Further summations Continuing the process and gradually adding all the balls we get the following result Table 34 133x97 Result The result is a blue ball 10 000 two yellow balls 2 000 nine red balls 900 no white ball 0 a black ball 1 12901 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 47 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 7 Kak s Theory 2014 Positional non uniform representation 144 with powers progression stepping by 2 Vertical 5 rows x 4 columns 1 6 24 72 for the first row 125 125 125 125 12 Subhash Kak proposes for the Poma de Ayala s Yupana a non uniform representation of numbers with positional notation in base 144 with progression 129 122 12 12 128 and set the yupana in a vertical position long side vertical ISKAKI In his work Kak suppose that the abacus were used as a tool to perform astronomical calculus by the Incas He gives his own explanation of the position of the seeds drawn by Poma de Ayala and finds a connection with the sinodic sub periods of some planets Kak explains also how the abacus may be used to represent add and subtract numbers 2 7 1 Representation of a number Kak based his theory on a non uniform representation system of numbers T
2. Table 6 Operations for the addition 3 595 515 3 471 7 581 in the Yupana by C Radicati di Primeglio As an example we add to the number represented in the previous paragraph seeds blacks first column the number 515 red seeds second column and the number 3471 blue seeds third column The first step of the procedure consists in representing said numbers in the first three columns see Table 6 Part A columns 1 2 and 3 We start here from the units row adding up all the blacks red and blue seeds and moving them to the last column Table 6 Part A column 4 Since the total sum of the seeds is 11 greater than 10 ten seeds black red seeds can be replaced with a single seed in the top row Table 6 Part B line 2 column 4 seed orange and in the box 1 x 4 will remain onlya blue seed Similarly in row 2 the black seeds and the red seeds sum to give 10 tens and can then be replaced by one seed orange seed in the hundreds row Table 6 Part B line 3 column 4 seed orange while the blue seeds are moved in column 4 Even in row 3 the seeds red and blacks result in 10 and then may be replaced with a single seed of row 4 Table 6 Part B row 4 column 4 orange seed and the blue seeds are moved to column 4 In row 5 you have only to move the blue and black seeds to column 4 because the number 10 is not reached You can read the result vertically from top to bottom 7581 Author Kunturweb Web http kunturweb al
3. 0 ax 2x 0 Illustration 60 Yupana by Florio with the default decomposition of the multiplier 3 2 x Note that at the base of the column two second column from the right is shows the caption 2 x indicating the multiplication factor of the first term while at the bottom of column three the third column from the right is shown the caption 3 x corresponding to the multiplication factor of the second addend The default configuration shown is 3 2 and can be changed later Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 100 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Entering the multiplicand T To insert a multiplying proceed as in paragraph 3 11 1 For example to enter the number 111 1 x 107 1 x 10 1 x 10 we select the red seed x 100 and click the 3 box from the top in the right column then we select the white colour x 10 and click on the first box from the top of the right column and finally we select the black colour x 1 and click on the second box from the top of one column Illustration 61 Note that we enter the tens 1n the 1 row just to show that the numeral system is additive and does not depend on the position of the seeds in the yupana Once you have entered at least one seed in a box of column 1 then the sec
4. USER MANUAL and corrections are necessary Frankly this theory 1s weak and is widely open to criticism 2 7 3 Addition TO DO 2 7 4 Multiplication TO DO Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 53 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 8 Which theory should we choose The debate about the numeral system that best fit the Incan Abacus shown in Nueva Coronica is far from exhausted The purpose of Tk yupana is to provide an overview of all the possible solutions and not to make judgements on the solutions adopted however some aspects that can support a certain theory and exclude others can be considered e the purpose of the abacus is to assist the user simplifying complex calculations and should be used in an intuitive and simple way Are therefore to be preferred simple methods of calculation to complex ones see for example the complexity of the calculations of the multiplication in Radicati s theory For the same reason are to be preferred simple numeral systems to complex ones see for example the complexity of numbers representation in the theories of Kak or De Pasquale The importance of the need on the part of the Khipu Camayuq to keep under visual control the displacements of the seeds on the yupana is advocated both by Cinzia Floriol FLRI and by supporters of the Glynn theory LES This vi
5. The instrument used by these teachers was something like a series of trays made of wood stone or clay with different separations in which they put stones of different shapes and colours even angular shapes Information collected The instrument singular was composed of several parts series of trays each of them having different separations e Used coloured stones with different shapes Assuming that the three journalists were referring to the same object and the same method of calculation after all why the accountants belonging to the same elite and educated in the same way would have had to use different methods and tools to perform a specific calculation we can make further assumptions that are at the basis of some theories developed over the years on the Incan Abacus portrayed by Poma de Ayala e the black circles of the drawing by Poma de Ayala may be used as the equivalent of the grains in place of a certain amount e the number of holes or seeds present in the boxes could correspond to 1 the value to be attached to the seeds present in a certain box 2 the number of elements corresponding to a given value determined by the row column of the box Still we do not have any certainty about the method used by accountants to perform arithmetic 9 The fact that the Incas were using a decimal system is also confirmed in a previous step of Nueva Coronica when it comes to the division of the empire into classes SPOM El ca
6. The vertical progression is instead 1 1 1 M that is every seed planted in any box of the first three rows of the table starting from bottom has the value 1 and has a value of M memory 10 x 10 with x equal to the number of the column starting from right to left when located in the fourth row from bottom The representation of a number is very simple we begin to fill the table from bottom to top and from right to left bearing in mind that each digit corresponds to a column positional system in the first three rows starting from low fit the seeds each one has value 1 See the example in Table 9 QQ QQ QQ QQ Q Q Q Q QQ QQ eo OO O9 QO 6 o0 OO QQQ QQ 660 eee QQ Table 9 Representation of number 1 971 The number O is represented by not putting any seed into the yupana empty table The maximum number representable in the yupana by Glynn is 100 000 or 222 220 if also the memory Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 23 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL holes are filled Note that with the theory of Glynn is possible to represent all numbers from 1 to 99 999 this representation however is not univoca The choice to use the number one as memory as well as providing an aid in the calculations allows the numeral system to become purely decimal 2 3 2 Addition To add t
7. USER MANUAL Tk yupana by Kunturweb OOOOO QQQ QQ Q OOOOO QQQ QQ Q OOOOO QQQ QQ Q 00000 66 60 o0 o 00000 eee 6 o Table 27 32X5 Summation of columns 2 and 3 and result in column 4 5 Result a You can already read the result in column 4 fifteen white balls ten black balls namely 5 x 3 101 5x2 150 102 160 b However wanting to get a more natural reading starting from the seeds of lower weight let us group ten units and convert the corresponding ten seeds in a seed of higher weight c We get two different weights red for the hundreds and white for the tens with their associated values respectively 1 and 6 d start from the higher power and read the result 100 60 160 Table 28 O O OOOOOO QQQ QQ Q QQQ OO O 66 60 o0 o 66 60 o0 e Table 28 32x5 Grouping of the seeds of the same colour and reading of the result product in the 5 column Example 2 133x97 Let us decompose the multiplicand 133 2 3 3 x 101 1 x 102 Assign the various powers of ten to the following colours e Black unit 10 x 1 e White tens 10 2 x 10 e Red hundred 10 100 x e Yellow thousand 10 1000 x e Blue tens of thousands 104 x 10000 Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 44 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL
8. when it represents the number using up to five seeds for the numbers 1 9 for each decimal place see an example in Table 17 4 Concrete representation simplified form where represents the number using the rules of step three with the addition that the boxes of the sixth huachu all have equal value equal to the average of the three seeds see an example in Table 18 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 33 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 1 Coupled representation when the number is represented by only pitu circles see an example in Table 15 2 Uncoupled representation when the number is represented both by pitu circles and chulla circles see an example in Table 16 3 Concrete representation when the number is represented by only five seeds for numbers from to 9 for each decimal position see an example in Table 17 4 Simplified concrete representation when the number is represented by the rules of step three with the addition that the circles of the sixth huachu all have equal value that is the average value of the three seeds see an example in Table 18 O 110 O 60 O 80 O 30 100 O 50 O 10 O 70 e gt O 90 Q 40 O 1l O 6 O8 3 O 10 O 5 O 1 O7 gt O 9 O 4 Table 15 Coupled representation of number 23 pitu O 110 O 60 O 80 O 30 O 100 O 50 O 10 O 70 O 20
9. 5 elements in the boxes in row 1 3 elements in the cells of row 2 and 2 elements in the cells of row 3 1 element in the cells of row 1 The line 4 the highest has a different connotation than the other three below According to the theory of Glynn in fact it was used as a memory in arithmetic operations to reduce the possibility of error during substitutions see for example the addition operation Each seed will have unitary value this means that in the interpretation of Glynn as in that of Wassen the circles of the yupana drawn by Poma de Ayala will have value 0 if empty and when filled with a seed The horizontal progression is based on ten decimal and each column is associated with a power of ten Starting from the right and growing to left columns correspond to unit 109 tens 105 hundreds 107 thousands 10 tens of thousands 10 Each seed will therefore have Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 22 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL different value 1 x 10 depending on the column in which it is located see the diagram in Table 8 O9 O9 QQ QQ QQ Every circle 9 O O O O if filled has 00 OO 00 OO OO d multiplied by OO OO OO OO QO the value of QQQ QQQ QQQ QQQ QOO the column Table 6 Scheme of the numeral system by W Glynn positional notation in base 10
10. Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 69 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Sum of the addends IKO By clicking on the Calculate button the sum of the three numbers you entered previously is made If the speed 1s set to 0 the calculation 1s done instantly and the result appears in the fourth column of the yupana If the speed 1s set to a value greater than zero the filling of the table 1s done gradually and with increasing speed according to the number entered in field speed Illustration 15 Somma degli addendi 1268 3 6 3 Subtraction DELL 8 NENTENG The window allows you to subtract two numbers together The numbers minuend and subtracting are inserted through the panel on the left you can also select the speed with which it 1s filled with the yupana please see Illustration 24 Illustration 16 Yupana di Radicati sottrazione Entering the minuend X You can enter the minuend using the arrows in the related field or by entering the number directly using the keyboard Once you leave the field the number is represented in yupana with KEH SN white seeds according to the rules described in F F section2 2 3 and the subtrahend input field is enabled otherwise disabled Please see CO Tr Illustration 21 Illustration 17 Inserimento del primo addendo 6789 Au
11. Then put a red ball three white and three black balls in the first column of the yupana for a total of 7 lines corresponding to the sum of the digits of the multiplicand Table 29 Table 29 133 x 97 Entering the multiplicand The second step is to decide the decomposition of the multiplier The number 97 is decomposable in the sum 50 40 4 3 Table 30 but theoretically I could also choose other decompositions I need therefore 4 intermediate columns and one for the result for a total of 6 columns S0X 40X 4X 3X eee O0 0 O0 Table 30 133497 Decomposition of the multiplier 97 in 50 40 4 3 For each row multiply the value of the seed present in column by the values of the columns 2 and 3 and distribute an adequate number of seeds in said columns Put attention because when multiplying a row of the multiplicand for the values of the columns 4 and 5 the result increases to higher power from the units to the tens from tens to hundreds and so on You must then use a ball of the appropriate colour according to the scale of conventions adopted above Table 31 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 45 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL OO QQ OO O QQQ Q OO QQ OO OO OO OO Q QQQ Q 900 OO OO OO O
12. http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 17 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL The representation of a number in the yupana has a vertical progression of base 10 decimal This means that the each number can be represented by placing the units in the a column starting from the bottom units to the top tens of thousands or vice versa It was possible to put up to nine 9 seeds all having value 1 in each box see Table 4 As an example for the representation of a number choose 3 595 Let s begin filling the column 1 of the table from the top to the bottom bearing in mind that each digit corresponds to a row positional system We start with the thousands 4 row three seeds black circles in the first box then continue with the hundreds 3 4 row five seeds in the first box of the row then the tens 2 4 row as the seeds are nine the box will be completely filled Finally units 1 row 5 seeds in the 1 t box See Table 5 Note that you can represent all the numbers from 1 to 99999 O O O O CO OQ oO o O OC Oo 0 O O o o o O O C 0 C00 O OO CO OC 0 O 0 O O CO 0 CO O OO O OC 0 O 0 O O O 0 C00 O 0 0 CO OC O O O O O C 0 0 0 O O O CO OC O 0 0 O O O 0 CO O CO O 0 OF O O 0 O O O 0 0 0 O c O ccc O 0 0 O O O 0 CO O CO O 0 0 O 0 0 O O C 0 09 0 O COO CO OC 0 0 0 O O O 0 0 0 O 0 O O cc O 0 0 Table 5 R
13. http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 28 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 12800000 7680000 5120000 2560000 O O O O O O O O 320000 192000 12800 64000 O O O O O O O O 8000 4800 3200 1600 O O O O O O 200 120 80 40 O O O O O O O O 5 3 2 1 O O O O O O Q Table 11 Scheme of the Yupana by De Pasquale Positional notation in base 40 To represent a number you begin to fill the table from bottom to top and from right to left the first row may contain a minimum value equal to zero yupana empty and a maximum equal to 1 x 1 x 40 2 x2 x 40 3 x3 x 40 4 5 x5 x 40 1 4 9 425 39 The number 40 corresponds to a yupana filled solely with a seed positioned in the box to the right of the second row Even in this case there are numbers that can have multiple representations as shown in Table 12 e Table 13 for number 100 Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 20 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 200 120 80 40 Q Q Q Q Q Q Q Q Q 5 3 2 1 Q e o o O e O e Table 12 First representation of number 100 2x1x40 5x2x40 3 x3 x40 1 x 1 x 40 804 10 9 1 100 Or 200 120 80 40 Q Q Q Q Q Q Q Q Q 5 3 2 1 Q Q e O e O o Q Table 13 Alternative rep
14. kunturweb altervista org e mail kunturweb altervista org The window allows you to add two numbers The numbers are inserted through the panel on the left which involves the insertion of the first and second addend as well as the ability to select the speed with which the yupana will be filled see Illustration 35 You can enter the first term using the arrows in the field or by entering the number directly using the keypad Once you leave the field the second term is enabled otherwise disabled and the first number 1s represented in yupana with white seeds according to the rules described in the previous paragraph Once you have entered the first term and only then you can enter the second term using the arrows or by directly entering the field with the keyboard When you leave the field is 3 enabled the Calculate button Doc Rev EN 0 7 1 07 02 14 79 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Addition of the terms By clicking on the Calculate button the addition of the two numbers previously entered begins If the speed is set to 0 the calculation is done instantly and the result displayed on the yupana The seeds for the second addend are black to distinguish them from those of the first addend see Illustration 37 If the speed is set with a value greater than zero the filling of the table is performed step by step and with increasing speed according to the number entered in
15. oc es INC PC SETRA ION oct saa iosa tans aa aaa Bana ki vat a naa a lim eec Bag Ka aa aa an a AN ada pie aie pana aba 88 9 9 D ICME TUNIC BIO Sara sana E as adana na aaa aa aaa naa an a uenak aaa daana Feu a ah 91 Sloe Yupanrdi Kak C2 OU Jasari sagen a an adan aaa na eaaa an aaa aa aa banjang aaa naga daa ag 92 DO A MENSES SIEMENS T 02 Dia VIC MUL TAREN DN aana aka aaa E a a E OE 94 SoM eod Te aa 2 O06 a a AA ENE NT RN NE Ke TAN ANE TKR A TAN NANA AA KI 95 Sal TUE ING OC SC a DUE eec AN AG KAEN TAG EA NGA OE ma Rap OE AU EM TEN 95 SW N INPUNE E a P PANE EET 98 Farm OS TEK asasi aaa anaa aaa ai Ban aaa a Lang ag aaa a aaa ada al aa aan ana ban aaa na ai ae aaa aba 98 Earn The 2 HO CIN kasatan a apaa naa ae aana aana eae ga a a aka aa aa na a aaa La Pa naba a Dia anaa aaa UR 98 DANA OT TS BUE aa sasanga aaa iga aa bra a aaa deve T 99 AL Man ND CAO oes aa ane naa aga E TENAR Pa NANG E NANA GE aaa eom a a Sak aga aan A 99 Paterne thie THUTUD HC AG asasaran iga aE bagia KAE gena gana ae nega Nang iga aba aaa ai e iga gena Fed ik 101 Entering the THOU UDLTIGE essri NA BAN KERA TEN GA e BENG DENE E AN KA a Ba aa a SPAN SANA 101 The options Speed and Step bDy Step sscccssssseeececccccceceeeeeaeeeeeessseeeeeeeeeceeeeeees 102 Calculation oL prod Cis sos0stncacsaaaceectmetncicoamegnsconvesecansaadusanteeshonoemuges EN EEEa 103 SIEEN AU ECHOING QOO LO E E E E A S E N Ee 104 EE e e sen a age aga aaa E E E A E E A A Ng E A 106 SEN PO M
16. 1 and replace them with a single seed on the second column of each row see Table 3 Part B red circles Finally grouping together the seeds that have filled the boxes in column 2 and will replace it with a single seed of box 1 of the next row see Table 3 Part C blue circles 2 1 3 Multiplication TO DO Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 16 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 2 Theory by Carlos Radicati di Primeglio 1979 Positional 10 Vertical 5 rows x 4 columns 1 1 1 1 or 9 9 9 9 1 10 100 1000 10000 2 2 1 Representation of a number Radicati di Primeglio settled his theory on a positional numeration system Unlike Wassen He did not think that the white and black circles drawn by Poma de Ayala were empty or full gaps but seeds placed in the different boxes O O 0 0 CO O O CO CO 0 0 0 O O O O OO 0 0 Occ cc 0c O 0 0 O O 0 O 0 0 O O C O O0 O O O O O O O CO O O CO 0 O 0C O O O O O 0 O 0 0 O O C O 0 QC 0 O O O O O O 0 0 O O CO 0 0 Q0 O O O O O O O CO O O C O O0 O O O O O OO C 0 Occ ccc O 0 O O O OO C 0 Occ O 0 0c O 0 0 O O OO C 0 Occ cc 0 O 0 0 O O O O 0 0 O O C 0 O0 O O O O O OO 0 0 Occ O 0 0 O 0 O Table 4 Scheme of the Yupana by C Radicati positional notation in base 10 with progression 1 1 1 1 Author Kunturweb Web
17. Ayala according to Kak s theory Weaknesses The weaknesses of the Kak s theory are more than one and highlighted by the author himself in his article see notes 19 20 and 21 However other objections can be raised nota number is directly associated with a synodic period of a planet only one number is perfectly associable with a synodic period of a planet the synodic period of Mercury is directly proportional to the number 29 e if all the numbers had a uniform and direct correspondence with the synodic periods of the planets this would provide a strong basis for his theory but unfortunately multiplications 19 8 is not a divisor of 399 nor of 378 Kak writes This interpretation rests on special significance being given to 8 cycles of the 819 day period Alternatively the Inca may have held to the theory that 56 codes the synodic periods of Jupiter and Saturn because its factors 8 and 7 almost exactly divide 399 Jupiter and 378 Saturn respectively n d A 20 The author writes that this discrepancy of 1 in the product is a weakness of his theory but that there might be astronomical reasons for that value 2 The author points out that this may be just a coincidence and that it cannot be a significant number but used by the Incas to get the 287 as a sum Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 52 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb
18. O 90 O 40 O 1 O 6 e 6 O 10 e 0 O7 6 gt O 9 4 Table 16 Uncoupled representation of number 23 chulla O 110 O 60 O 80 O 30 O 100 O 50 e io O 70 O 20 O 90 O 40 in O 6 O8 a O 5 O 1 O7 2 OP O 4 Table 17 Concrete representation of number 23 Note that unlike the previous the number is represented with Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 34 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL maximum five seeds in the circles from I to 9 O 100 O 60 O 80 O 30 O 100 O 50 O 10 O70 O 20 100 O 40 1 O 6 O8 3 B d5 O 5 O 1 07 Se a O 2 0 io Table 18 Simplified concrete representation of number 23 Here the values of the circles belonging to the sixth huacho are all equal to the average value of the three seeds 2 5 2 Addition TO DO 2 5 3 Multiplication Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 35 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 6 Cinzia Florio s Theory 2008 2009 In 2008 2009 Cinzia Florio proposed a completely different approach from the traditional positional point of view Instead of formulating a theory of computation with a yupana five rows by four columns and the fixed set 1 2 3 5 Florio looked for a plaus
19. Q 31 Q Q Q e Q e Q e e e 92 Q e O Q Table 37 Representation of the five numbers of the drawing by Poma de Ayala according to Kak s theory Kak proposes the existence of a strong relationship between the numbers shown in the table above and astronomy also referring to some numbers found in Maya s astronomy Refer to in this regard Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 51 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL A first explanation is related to the Mayan calendar since 56 x 117 819 x 8 and then 8 cycles of the Mayan calendar A second explanation links the number to the synodic periods of the gaseous planets since 56 7 x 8 which are almost perfect divisors of 399 synodic period of Jupiter and 378 synodic period of Saturn 79 x 12 948 This value decreased by 365 days in the calendar year gives 583 which differs by 1 from the synodic period of Venus 56 Jupiter and Saturn 79 Venus 29 x 4 116 which is the synodic period of 29 Mercury Mercury 31 is the number of days present in some a Mont months of the calendar year Number of days between the autumn equinox and the winter solstice 287 365 78 with 78 which should be 1 10 of the synodic period of Mars 92 1 4 of a year sum 287 Mars Table 38 Explanation of the numbers represented by Poma de
20. X number of the column e A number of seeds that can be inserted in column X e 10 symbol for exponentiation of the number 10 S power of number 10 weight to be assigned to column X Note that by increasing the variable S from column 2 progressively decrease the possibility of selecting high values of the seeds see Illustration 63 in which some seeds are disabled This has been introduced to avoid errors of the program due to a breakthrough of the upper limit of 10 The rule is that the greatest exponent S for column 2 depends on the maximum value of the power J introduced in the multiplicand and vice versa The sum of S and J must not exceed the upper limit of 7 By now you can not change the exponent of column 3 In the future I will introduce also this feature The options Speed and Step by step Near the group of fields relating to the multiplier are present Achek box with the label Step by Step Acheck box with the label Speed By checking the box Step by step calculation will be performed a step at a time first the multiplication of column 1 to column 2 then the multiplication of column 1 to column 3 then the summations and simultaneously displaying the result If the box is not selected the calculation is performed without pause The speed selector is used to change the speed at which the boxes are filled 10 corresponds to instantaneous while values below 10 correspond to values of the speed gradu
21. an Ian aaa tena Jaa Sega ag aaa 68 Os OO OMG e PAE AKEN PAN KATANG PR LAT PA AR KA TTNG PRA LA PN PAN A TAN PAE 69 Peh the TGS dde Tia coi pa aana r AG E ABA NG AA D atio KDA ED E GRA MENU na ag 69 Entering the second and third addend cc eeecccccccccccceeseseeccececeeeaeeseseeceeeeeseaaenseeeeeees 69 DOL ANG 6 HOS ne ee ere ee ee 70 2 10 Dan SANAN AC MO Bg aan ana EN a a A anan IA OT Disco sS eC KE Na KANG ana EN aa a a a KIT E En aga aaa aaa 70 EDA IA TS TIS erus a aaa aaa a a aa B NGI aa daana a a aaa aaa aa a eaaa aii US OU 70 Entcuns Me SUDO IG aaa ata nandana aga ag ayake a aaa DG Ta aaa a aga bag ante 71 DHE AC IO aan dana ak kana aan kna TTE TO TET 71 OF NUT ID DUC AUN Uh sarana E agan taga anaa nia ak banga aaa E a daga a DA aaa AB gada aba aaa agak 71 PALON the TERUTUED HEC ATIN aE cen SEG BE E IB a GE BE ING BN aaa SEBA DNE 72 Enter be TAN TIO E ooo ooo E E a BER o PHI RAT IB Ce ORAE HERO an R 12 SL SIE D parta DEOCUEDIS ud eua etc maveed eati peso edr aee EE rd onmi DIUI E Sud MSAN EIE Th 72 ZING SECO SOMMA ONS asana teste um dest anie ng ii aaa aapa ia a a aia tanga kaa aaa abani reU M UE 73 DEO SEED PEOOUGE asana aaa a aaa aa aaa a ga a Ka A ES E ee aa a da ee 73 Pe Oe NA SK TANG NIU TET Waaa raaa aaa aaa EO OO D ai aana ag aaa aaa 73 2r DADA Dy Gs asana abaan si ah aa aaa a ak baa a E an Ka a a aaa a 74 S Ale EDEN GINAIB a a a NAN Bag ene ne KG arak aa Aa LASS A rer E aaa Ga aan er a 74 LENG a IA DK aaa caress anak ana
22. base 10 with progression 1 11 3 9 1 Representation The numbering of the columns grows from right to left and that of the rows from bottom to top The little boxes seeds in the table have different weight depending on the row and the column in which they are placed please refer to paragraph 2 5 7 for details Note that row 1 corresponding to the lower row stands for decimals 29 Tk yupana has been tested on an Asus Eee PC 701 4G with Linux Xandros operating system N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 88 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Clicking with the left mouse button on a box you put a seed in the Yupana To enter a number you have to click with the left button of the mouse on any of the boxes When the check box is selected you will see a seed in it the corresponding value is added to the total and it is then displayed next to the table Since the numbering is in base 10 and it starts from decimals in the first row at the bottom you can enter any number from 0 row and empty table to 6 6 full row Each number does not have a unique representation as can be seen in Illustration 49 e Illustration 50 where are shown two representations of number 9 8 M w Yupana di Chirinos Rappresentazione v X _ Hk Yupana ud by Kunturweb ES l Eyal Totale 9 8 E s
23. field speed 3 7 3 Multiplication Yupana di Glynn Moltiplicazione The window allows you to multiply two numbers The numbers are inserted through the panel on the left which involves the insertion of a A rn E the multiplicand multiplier as well as the ability to select the speed with which the yupana is filled see Illustration 38 pauco c Illustration 38 Yupana by Glynn multiplication Entering the Multiplicand You can enter the multiplicand using the arrows in the field or by entering the number directly using the keypad Once you leave the field the multiplier is enabled otherwise disabled and the multiples 1x 2x 3x and 5x are calculated automatically and shown in the table below Illustration 39 Illustration 39 Entering the multiplicand Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 80 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Entering the multiplier Once inserted the multiplicand and only then you can enter the multiplier using the arrows or by directly entering the field with the keyboard When you leave the field the multiplier entered is broken down according to predefined rules and shown in the board below the Calculate button 1s then enabled Illustration 40 in oF o 18 Illustration 40 Entering the multiplicator Adding
24. in the Total field on the upper left of the window Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 86 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 8 2 Menu functions Clicking on the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F1 2 Help Displays a help file F12 Close Closes the window F Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 87 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 9 Yupana by Chirinos 2008 A e Ik Yupana Lud by Kunturweb Totale 0 0 E E Z a E El Es pes pes ips pe e p p ps e es s s s n EH Eh SES SES ES AU Eso ES nl E EE E Illustration 48 Empty yupana corresponding to number 0 zero When you open the window Yupana by Chirinos the program presents an empty table in an upright position corresponding to the number zero Illustration 48 The size of the table 1s different depending on the resolution of the screen and is able to operate even on a net book with resolution 800x4807 In this case have not yet been implemented algorithms but it is only possible to represent numbers in the positional numeral system in
25. positioned in the second box of the 1 row See Table 2 It should be noted that all numbers can be represented using only the first two columns 1 and 2 This obviously implies the uselessness of columns 3 and 4 and constitutes a limit of the theory of Wassen if we consider true what Poma de Ayala wrote cfr Chapter 1 1 upper limit of 100 000 By Wassens theory we can represent all the numbers up to the upper limit of 888 880 inclusion of all seeds or yupana completely filled The theory of Wassen based on a positional notation system was taken as a model by most Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 14 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL subsequent authors who while rejecting his validity never rejected the positional nature 10000 50000 50000 150000 150000 300000 300000 O O O O O O O O O O O 1000 5000 5000 15000 15000 30000 30000 e O O O O e O O O e O 100 500 500 1500 1500 3000 3000 e Q Q e O e O O e e O 10 50 50 150 150 300 300 e e O O Q e O O e e O 1 5 5 15 15 30 30 O O O O O O O O e Table 2 Representation of number 3 595 in the Yupana by H Wassen 2 1 2 Addition To add two numbers the first addend should be represented in the yupana see previous paragraph Subsequently starting from the column of units add as many seeds as there are units o
26. subtraction of 3595 146 3449 in the yupana by C Radicati di Primeglio As an example we subtract the number 146 red seeds second column to the number shown in the previous section black seeds first column The first step of the procedure consists in representing said numbers in the first two columns see Table 7 Part A columns and 2 We start here from the units row and and move the black seeds to the last column while subtracting the red seeds since the result would be negative we transform a black seed of the next row 2 row into ten seeds of the 1 row Table 7 part B row 1 column 1 half orange The result of the subtraction of row one are the blue seeds Table 7 Part B row 1 column 4 half blue Proceed with the tens 8 black seeds minus 4 red seeds and report the results in column 4 Table 7 Part B line 2 column 4 semi blue Similarly for the hundreds we have to subtract a red seed to 5 black seeds Table 7 Part B line 4 column 4 blue seeds Finally for the thousands is sufficient to report the three black seeds of column 1 to column 4 since there are no seeds to be subtracted 1n column 2 The result can be read vertically from top 3449 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 20 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 2 4 Multiplication TO DO Author Kunturweb Web http kunturweb altervi
27. tens In each cell is it possible to insert up to 9 seeds each of them with value one 1 To enter a number you must click with the left mouse button on a free box a green frame shows which of them are enabled a seed will appear right in the box and its value will be added to the total the total 1s shown under the corresponding column The color of the seeds is white for the odd coluns starting from left and black for the even this color difference is only to distinguish the represented numbers Illustration 11 Inserimento del 3046 m REM ue RM MU To enter number 3046 in column 1 for example you need to select some boxes in the square R2xC 1 it sums to 4 tens and six boxes in the square R1xC1 total of 6 units The total is shown under column C1 Please see Ilustration 11 It is possible to enter all numbers from 1 to 99999 While holding down the CTRL key and clicking with the right mouse button on a seed is removed from the Yupana I To remove a seed from the table and decrease the total of the corresponding quantity you must click with the right button of the mouse on the seed while holding the control key CTRL 25 Tk yupana stato testato su un Asus eee PC 701 4G con sistema operativo Linux Xandros n d A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 68 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana
28. the chronicler In any case the probability that Poma de Ayala has drawn circles at random and represented precisely the numbers 32 64 and 96 is practically negligible e The yupana designed by Poma de Ayala with five rows and four columns would be contingent on the specific calculation drawn and not all numbers can be represented using that table The author solves this impasse whereas Guaman Poma would have drawn the rows and columns needed for that particular multiplication but that yupanas had larger sizes She also points out that for multiplicands where many digits between 6 and 9 appears there would be needed a very large yupana or maybe more yupanas arranged side by side Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 36 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL e e Q e e O O e O O e O e e e O O O O e O e O e e e O e O O Table 19 The calculation of Poma de Ayala according to the numeral system proposed by Cinzia Florio 2 6 1 Representing a number Although the example in Nueva Coronica refers to a particular multiplication Cinzia Florio also proposes the use of yupana as abacus for addition and division op cited in note 25 It is important to keep in mind that the numeral system under consideration is additive by powers of 10 This means that during the representation of a number the Khip
29. the power of ten associated to the cipher to be entered just click with the left mouse button in any gap of any box of the first column a seed of the same power selected will appear in the gap and its value will be added to the total of what was introduced before and that will also be displayed at the bottom of the column itself Clicking with the left mouse button on a gap of one of the boxes in the first column you can insert a seed in the Yupana The value of the seed will depend on the selected colour to the left of the table NOTE The parts of the number that is inserted do not depend on the position in fact the numeral system 1s additive The order of filling should be done from top to bottom so as reported by Jos de Acosta see note 15 e As long as you do not enter at least one seed you can not change the value of the multiplier the fields at the top left of the window are disabled For example let us enter the number 1 291 1 x 105 2 x IO 9 x 10 1 x 10 We select the black seed x 1 unit and click any gap in the extreme right column last column let s begin from the top of the column so from the first row from the top then select a red seed x 100 32 In many modern dictionaries of the Quechua language the word Hunu is translated as one million following the translation made by Gonz lez Holgu nB5P U in 1608 In Tk yupana I preferred to follow the interpretation of the linguist Pila
30. total 21 See Illustration 28 While holding down the CTRL key and clicking with the right mouse button on a seed ts removed from the Yupana To clear a seed decrease the number of the corresponding quantity you must click on the corresponding seed box by holding down the Control key CTRL 26 Tk yupana has been tested on an Asus Eee PC 701 4G with Linux Xandros operating system N A 27 It is not possible in version 0 4 to transfer pairs or sets of seeds in the highest slots of the yupana This feature will be possibly included in a later version N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 19 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL You can enter any number from 1 one to 111 110 or up to 222 220 using also memories The result is displayed in the Total field present above the table Use of memory x we Tk Yupana rev 0 1 by Kunturweb Once a column is totally filled with seeds 10 seeds it 1s possible to promote all the seeds in the memory box in order to be able to perform arithmetic operations without having to worry about keeping in mind the seeds moved Suppose for example that you have set in Yupana the number 103 and therefore you have completed the column 4 corresponding to ten p tens or a hundred and filled three boxes of Illustration 29 Number entered 103 column 5 Illustration 2
31. upper limit of one hundred thousand and a lower limit to unity 4 The one who takes a census of the population of the Tawantinsuyo Inca empire the one who receives the taxes of the Inca the one who keeps the books of the empire A N 5 It must be said that previous authors translate as He counted with strings made of wool of Andean deer a clear reference to the khipu one of them is Radicati di Primeglio op cited above But the type of wool used for Khipu came from Llama or alpaca or cotton so my translation seems more realistic especially in light of the fact that emparejar means to match 6 The one that took into account the serious violations and shortages A N He took account of the violations and shortages of minor importance A N Contador mayor de todo este rreyno Condor Chaua hijo de apo A ste le llamauan Tawantin Suyo runa quipoc Yncap haziendan chasquicoc tezorero mayor Dize que este prencipal ten a grande auilidad para sauer su auilidad el Ynga mand contar y numirar ajustar con los yndios deste rreyno Con la lana del cierbo taruga enparexaua con la lana a los yndios y enparexaua con una comida llamado quinua gram nea de altura contaua la quinua y los yndios Fue muy grande su auilidad mejor fuera en papel y tinta Contador mayor hatun hucha quipoc contador menor huchuy hucha quipoc Cuentan en tablas numiran de cien mil y de dies mil y de ciento y de dies hasta llegar a una De todo lo que pasan en
32. yupana r0 7 Tk yupana by Kunturweb USER MANUAL Addition of terms By clicking on the Calculate button the operation of addition of the two numbers previously entered begins If the speed 1s set to 0 the calculation is done instantly and the result displayed on the yupana The seeds for the second addend are black to distinguish them from those of the first term see previous paragraph If the speed is set with a value greater than zero the filling of the table is done step by step and with increasing speed according to the number entered in field Illustration 6 Addition 3 5 3 Multiplication TO DO 3 5 4 Menu functions Clicking on the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F1 2 Help Displays a help file F12 3 Close Closes the window F8 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 66 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 6 Yupana by Radicati Rg rappresentazione base 10 progr 1111 F1 i Addizione F2 ul Sottrazione F3 F4 Pe Moltiplicazione Illustration 9 Radicati di Primeglio sub menu possible choice addition operation between two or three numbe
33. yupana will be filled please see Illustration 20 Illustration 20 Yupana di Radicati moltiplicazione Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 71 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Entering the multiplicand You can enter by multiplying by acting on the arrows on the field or by entering the number directly using the keyboard Once you leave the field the number is represented in the first column from the left with purple seeds and the multiplier field is enabled otherwise disabled Please see Illustration 21 You can only represent numbers between and 999 and such that the product does not exceed 99999 Illustration 21 Inserimento del moltiplicando 123 Entering the multiplier Once inserted the multiplicand and only mm ae IT ET then you can enter the multiplier using the TRO FRR FF arrows or by directly entering the field with the keyboard When you leave the field the number is represented with yellow seeds in the first row of the yupana columns 2 3 and 4 from the left Also in this case can only represent numbers between 1 and 999 and such that the product does not exceed 99999 The Calculate button is enabled See Illustration 22 Illustration 22 Inserimento del moltiplicando 456 1 step partial products X HE By clicking on the Calculate button the multiplication of the t
34. 0 1 2 3 9 and therefore the base is 10 and is said decimal The right most digit corresponds to the unit the previous one to the tens then hundreds and so on So the number can be written as a succession of previous figures provided you comply with the Convention of positions e g 5342 5 x 10 3 x 10 4x 10 2 x 109 1 2 1 Limitations of the sign value notation A sign value numeral system is fine as long as you are dealing with small numbers When you start to scale to tens of thousands or millions the number of symbols used to represent a certain value increases dramatically and we are forced to invent other symbols For example consider the number as above and let s write it with Roman numerals 5342 4 symbols MMMMMCCCXXX XII 14 symbols Note that the sign value numeral systems were used since ancient times even by primitive and prehistoric populations The use of a sign value numeral system however does not indicate a low degree of civilization think for example to the Romans or the Greeks However a system of positional type being more advantageous than sign value one is a winning choice over time and is intended to supplant it definitively Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 1 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 A century of theories Here we will analyse the theories that have been
35. 12 2 1 Henry Wassen s Theory 1931 cccccesescccccccccccceeeeeeaaeeessssseseeeeeeeececceeeeeesaeeeeeeeeeeeeeees 13 2 1 1 Representing NUMDEC eso a UE EE KA KAE Na AA AA E ANAA KENA 13 251 2 5 Add NO anaa a aan taa a a ga aaa Ta ag ad KAR TAN a Na AR RN TE aa NA E AA Ha Ka aaa a E Aan a Tag a a Na aa taa gan aka 15 ska IN AD TAG SITI D TIN saa E A LEOTE 16 2 2 Theory by Carlos Radicati di Primeglio 1979 eeccsseccceececeseesseeececeeeeesaaeeseeeeeees 17 2 25 e Ik CONSE MEAL ON OL a NUM DOT assises o tdt a eaaa ada aa aaa uut aaa aa var iunt aa aa ka RF aaa anala daa daa 17 2 25 NAN NT OM ror nan aga agahan anaa E aga na daa aana gana aa anaa ag aa aaa aana CUR a 18 2 2 rs UN AG CUO DL aana aa aaa aana a ga a a anga aa a ag an na ag aaa s a RAN agak aaa a io aaa an aaa a ah aaa aas Es 20 2 Dabs NAD EIS UELUT RN Em 2 2 3 William Glynn Burns Theory 1981 sss nnne nnn nnne 27 Di ler CONC SC MENS a mnb esos aaa gana ete vni eu E ete io uem urat ad PEE MEI Pete A PEDE 27 PS BP TA NT E IEEE TEC TEE T T Om 24 PASRSE OM IIS FTO TEE ET aa aa aa ga aa aaa aaa aaa Sahg T 24 2 4 De Pasquale s Theory A ZOUTJiaan panas aaa an aga aaa ag a a aa a nan aa gag a aaa ke aaa aa aaa gag aana a Na aaa 26 24 l Representine a UMD sasadan aaa aan aa aa gana paka aana nda aaa aaa ea pagaen anga gaun aaa gana aaa gagange at 26 2 70 2 NAJA TAN Pae aga aaa ete Bana a NG Ban NANA aaa E AN a UNDER EAE CIBUS aaa Nana a aaa aa dU 31 DA N
36. 2 3 Sa and so on Illustration 42 and following Ngan Nn Illustration 42 15 step additions Yupana di Glynn Moltiplicazione Yupana di Glynn Moltiplicazione Illustration 44 5 step result Once concluded the calculation the final result product is shown in the upper right corner of the window Illustration 44 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 82 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 7 4 Menu functions Clicking on the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F1 2 Help Displays a help file F12 Close Closes the window F Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 83 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 8 Yupana by De Pasquale uw z x Tk Yupana Pe by Kunturweb Totale 0 ps 4 P hut 4 A NM rw nl M i ah n 1 A i nh lj Po J LII y 2 A IN L Bm Hi Sr E ic ESI D EE ES ES EE EST E m peores e eae Su oe E Ei d Eom En E Ez B E t E Eun BAA EIER ESL ERSA ES E EE Es e ES E Illus
37. 9 If at this point you wanted to add 35 to the amount already entered would be sufficient to select 5 boxes of column 5 and 3 cells in column 4 The operation on column 5 does not present any problems and can be carried out immediately illustration 30 by clicking with the left mouse button on the five boxes above the three already filled ad Illustration 30 Adding 5 units Because the fourth column is filled Illustration 30 before performing the operation on the tens add three seeds you must free the necessary space replacing the ten seeds of column 4 with a seed memory equal to one hundred We thus replace the ten seeds with a seed of memory in column 4 Illustration 31 To do this 5g kala simply click on the Memory of column 4 all the Illustration 31 Click with the left button on the memory seeds below will be replaced with a single seed in POG ORIEL ORC COGN the green box The result is shown in Illustration 32 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 76 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL When a column has been filled with ten seeds you can click with the left mouse button on a memory box in order to replace the ten seeds with one in the memory Note that if you have not filled all ten boxes below the operation is not p
38. ALA A CAO ana aaa ab ai baa a Aa a EO EP 31 2 COS MMC Oy LAJU Janaka a gana ah a genea A aka Gana aa IN BAANG BEA Aa IA BAN BAANG E eh BE aa Ga 32 Zoe Rrepresentanon of a NUMDET uoo tons pana kaa aapa aaa ag Urin ga a aaa a due a a Aa an ag aa NAN 22 D DoF PO OMI OU RENE anaa aaa ata a na Danan an a agan an aia pa anana aana 35 D Map E O a mA 35 2 6 Cinzia Florio s Theory Z008 2009 cisssiececsscoussussicsasencdepasasasovarcesdooustnaseeaneaddonnsauncobancesdeeess 36 2 5 MIRC pre scinn OL NDE esre E E E R 37 DOP AO E E EE EE E E EE AE 38 PE REV Se ETE IR Ones e DOS 39 Example 1 32x5 drawing by Poma de Ayalla ccccccccccccccccesseeeeeeeeeeceeeaeeeseeeeeeeeeeeaas 42 LADEN m 44 DEEP CIR IVC OR O aaa aa NR PR E EEE E A AE RN NR ER KA E 48 2 JL edsepresentattOn Or AMUN CE aana wasn aaa a kaa a Ee Ee cis a esce Dd DO ENGGAN 48 2 7 2 The yupana as a computational tool for astronomy eeeeeeseseeeeeeee sl NGAKEN SEN bana ats neb aia bet ag eaaa ag ab aaa aga bangi utis ten aga aba ab oru UA deu aga abhang 52 Daha E NAJA BIO Tsang nanda aana aaa a b nai agan ad diadani a kada aaa aaa a aga aa aa naa gana aba gadi 39 PTS REO e OA TO DOTT EU 39 2 02 Which theory should we choose 7st ee seres GA e ENENG MR E KAN E GAR AN E RE aa NG AN 54 REMISIT TEE m 56 SL Boite re TIE Sie spect coca ce tou a ERA MUI IMMUNE UP EE UMP MU IINE E 56 Author Kunturweb Web http kunturweb altervista org
39. Doc Rev EN 0 7 1 07 02 14 3 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 2 Download install and running the program cceeeessecccccccceeeeeeeeeccceeeeeaeeeeseeeeeeeeeeaaas 56 SS EM A PURIS I ses to Bag a EE 58 5 4 c ONY CONS paraanake ah ana aaa ang ERAT aapa pah esu aan pat RUP P IURE RE EISE a ng kan aa aaa S 59 3 55 WANA VV ASS Cece sas pact te soe ease aaa aa ag a a Ba BAN ED E ecc uad Ba a lias sed paga ank a 62 Sco al Maral DEE SENI OT raaa aaa ea aaa aaa aa aa aa angga aa gi ee ee 62 Encre IMAM CE gcse cs scat sia aaa oe areola actinic ase are a ag an anaa occa aan ida eaaa aaa nodose saeka oreo 63 Operations of displacement of the S eds ccccccccssssssseeccceeeceeeeeseeecceeeeeeaeeseeeeeeeeeeeaas 64 2 9 A a i a Ak a KN PARAN NA ce RUF SURE detec teem UTE EE E E M E EN NN 65 Pe TSW o a a aaa tana bea ag ene eka BENE a Mak E T E E DD Ma ak Bag taa aaa 65 Pae a The 2 nad Ce NIG saga ngatah dan E E ga ak aa aa NA A AA E akad ga aja a AN GAN a Ka 65 PCIE ON OL TONG jasa nasak aaa E gana GE OO d a ana aan a GE 66 Br eI ENO ICAU RE see naga aa aan ean ba naa anaa TOO aa aaa dak in 66 2 9 4 VIE DUE BUCO aaa cc Pace sine dente Dass E NS a Pa aa Sede e HAN EDU 66 DEO WW Mel Dy AC NG RTT mM 67 OMe ING ie NGIRO n yo casera peed nai ga you aa igo aa Tana eng aga via tones aaa aa aga uuu UR FERRE aga 67 PTN ANC Biss ores aston aga ana ajaa aa Dena EGONG a ING E EA Kg Ng
40. ER MANUAL 4 While holding down the CTRL key and clicking with the right mouse Ctrl e e button on a seed it is removed from the Yupana You can enter all the numbers from 1 one to 61 917 364 223 sixty one billion nine hundred seventeen million three hundred and sixty four thousand two hundred and twenty three The result is displayed in the Total field on the left side of the table 3 10 2 Menu functions Clicking on the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F10 amp F1 2 Help Displays a help file F10 amp F12 3 Close Closes the window F10 amp F8 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 94 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 11 Yupana by Florio 2008 The sub menu Yupana Florio see Illustration 53 has three choices also selectable via the function keys Fl F2 and F3 which correspond to the following functions Illustration 53 Sub menu Florio 1 Representation is used to represent a number on the yupana 2 Addition allows to add two numbers 3 Multiplication allows to multiply two numbers 3 11 1 Representing a number To insert a number in the table You need to break it down into units tens hund
41. N 0 7 1 07 02 14 59 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL I4 Sign value notation Systems Sub menu about Sign value notation systems M Yupana by Florio 2008 Sub menu of yupana by C Florio powers of 10 Representation Representation of numbers in Florio s system F1 Addition Addition of numbers in Florio s system F2 Multiplication Multiplication of numbers in Florio s system F3 Positional notation Systems Sub menu about positional notation systems Wy Yupana by Wassen 1931 1 120 yupana by H Wassen base 10 progression Representation Representation of numbers in Wassen s system F1 Addition Addition of numbers in Wassen s system F2 Sub menu of the yupana by C Radicati di Primeglio By Yupana by Radicati 1979 base 10 progression 1 1 1 1 wy Representaiton Representaiton of numbers in Radicati s system F1 uy Addition Addition of numbers in Radicati s system F2 uy Subtraction Subtraction of numbers in Radicati s system F3 X Multiplication Multiplication of numbers in Radicati s system F4 Sub menu of yupana by Burns Glynn base 10 progression 1 1 1 M Yupana by Glynn 1980 aay Representation Representation of numbers in Glynn s system F1 dy Addition Addition of numbers in Glynn s system F2 wy Multiplication Multiplication of numbers in Glynn s system F3 Su
42. O 6 e e e Q O O 9 o oO 6 oO O O OQO 6 o o o Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 27 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 4 De Pasquale s Theory 2001 Positional 40 Vertical 5 rows x 4 columns 1 2 3 5 Fibonacci s sequence 1 40 1 600 64 000 2 560 000 Nicolino de Pasquale proposes for the Abacus by Poma de Ayala a numeral system with positional notation in base 40 and expects to have the yupana vertically long side vertical SPEI 2 4 1 Representing a number Each seed will have a value that depends on the row R and column C in which it 1s placed Each row R of yupana corresponds to a power of 40 according to the formula f R 40 Re 1 5 1 4 1 with R growing from bottom to top The horizontal progression of the values of the columns is 1 2 3 5 in correspondence to the number of circles in the drawing of Poma de Ayala which is also equal to the maximum number of seeds that can be inserted in a given cell R C In mathematical terms the column value 1s expressed g C C whereC e 1 4 U 4 5 1 4 2 with C growing from right to left The value of a seed placed in the box R C is then given by V R C f R g C C 40 whereC e 1 4 U 4 5 e Re 1 5 1 4 3 See also the scheme in Table 11 Author Kunturweb Web
43. O OO Q QQQ Q OO OO OO OO QQ OO Q QQQ Q OO QQ OO OO Q S i eee e OO OO Q C MM eee e OO OO Q saha eee e OO Table 31 133x97 Multiplication of column I by columns 2 3 4e 5 Finally collect the seeds of the same colour and put them in the last column taking care to replace ten seeds of the same colour with a seed of the colour of power higher Table 32 The procedure should be done step by step e We can now move the nine yellow balls in the box corresponding to thousands of row 1 in column 6 then we add the seeds belonging to column 5 which having as a coefficient a number that can be broken down into 5 x 10 will give a sum of 10 and then can be easily transformed into a ball of higher colour see the red and yellow areas in Table 32 Table 32 133x97 Summations We can see the result in Table 35 where also are shown the balls that will be added next and promoted to the higher power yellow area in which 10 red balls are added and then promoted to a Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 46 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL yellow ball red area where ten white balls are added and promoted to a red eee E nate oo o O od O OOO O E 20 ee eee ate be eee e S g ow
44. OST AP ee OE E E E E A E E E E RN E 107 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 5 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Introductory Note This software and related documentation are the result of personal interest of the author to pre Columbian cultures and are PURELY INFORMATIVE If You think you have found any mistake please write to kunturweb altervista org as soon as possible If You liked this software or it has been useful or if you have any comments or suggestions please drop kunturweb altervista org a line THE VERSION OF THIS DOCUMENT MAY BE OUTDATED PLEASE DOWNLOAD THE LATEST VERSION FROM EN http kunturweb altervista org pag en tk yupana html OTHER LANGUAGES IT http kunturweb altervista org pag it tk yupana html ps http kunturweb altervista org pag es tk yupana html THE ENGLISH TRANSLATION OF THIS MANUAL IS ONLY AT AN EARLY STAGE SO BE KIND AND FORGIVE ANY ERROR IF YOU WANT TO COLLABORATE AND IMPROVE THIS MANUAL PLEASE CONTACT ME AT kunturweb Q altervista org OR VISIT TK YUPANA PAGE http kunturweb altervista org pag en tk yupana html Thanks Kunturweb Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 6 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 1 Ihe Yupana With the t
45. Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL TK YUPANA A SIMPLE INCAN ABACUS EMULATOR By Kunturweb Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Kunturweb Kunturweb is a project born from my passion for everything 1s concerned with history and culture of pre Columbian civilizations Kuntur is the quechua name for the Andean Condor Vultur Gryphus Web is a synonymous of World Wide Web or WWW Kunturweb does not receive any external funding nor hosts advertisements from third party in his web pages Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 2 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Table of contents TAUT NING saa aa ne bcos nse ak a onc aaa naa a E bac Aa ng a pa DE AS PG 1 IM D DEN I saksapaa aaa aaa aaa Gana NA m 7 1 1 References to the Yupana by the Chroniclers of the Indies oeneeeaaaeneenaaaa anane e nenen 8 1 2 Abacus and numeralsySie DIS os esoers eres aad senceasiesscaivonaeoeeaeied nana Euge mae nane eaaa anana ne nee anaa nenen nenen 10 1 2 1 Limitations of the sign value notatlon eeeseeeeenaaa anane e ena anana naen nenen een 11 2 ANDE OL PAC OMICS asasi a ga At E A Ga a ang GA a E D a aa a gag nada gak di M gan aaa aa anaa
46. ally decreasing Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 102 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Calculation of product Once filled the multiplicand the multiplier and decided how to set the calculation parameters you can start the calculation by pressing the Calculate button If the option step by step is not selected the calculation is carried out without interruption and the result product shown right below column 4 Illustration 66 otherwise the button will change the word Calculate in First multiplication column 2 and waits after a button is pressed Pressing the button again the second column is filled according to the values selected and the words of the button changes to Second multiplication column 3 The program waits for a button to be pressed again Illustration 64 Illustration 64 111 x 4005 First multiplication multiply the first column by the second Pressing the button again the third column is filled according to the values selected and the words of the button changes to Addition of columns 2 and 3 The program waits for a button to be pressed again Illustration 65 Illustration 65 111 x 4005 Second multiplication multiply the first column by the third Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 103 e mail kunturw
47. are consecutive the theorem is not satisfied and the representation is not unique N A Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 24 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL ways 0 0x 10 1 1x 10 2 2x 10 3 3 x 10 4 2 x 10 2 x 10 or 1 x 10 3 x 109 5 5 x 10 6 5x 10 1 x 10 or 1 x 10 2 x 10 3x 109 7 5x 10 2 x 10 or 1 x 10 3 x 10 3 x 109 8 5 x 10 3 x 10 or 2 x 10 3 x 10 3 x 109 9 5 x 10 2 x 10 2 x 10 or 1 x 10 3 x 10 5 x 109 In general then a digit of the multiplier can be represented on the base 1 2 3 5 such as n a X10 b X10 c X10 d x10 e x10 tc a b c d e 0 1 2 3 5 A0nx4 Having decided on the decomposition of the multiplier you can proceed to the actual calculation multiplying each occurrence of non zero digits of the multiplier by the corresponding multiplication factors M1 M2 M3 M5 of the multiplicand and bringing in yupana the values gradually doing the sums For example suppose you want to perform the following multiplication 3 x 12 359 We find the 4 multiples of the multiplicand 3 6 9 15 Then decompose the multiplier 12359 we compose the following table to which we have added as the last line the multiples of the multiplicand that will help us later Table 10 Multiplication of 3 X 12 359 Author Kunturweb We
48. b http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 23 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL At this point we have all the elements to perform the calculation which will be reduced to simple sums Starting from the unit represent the multiplication factors M1 M2 M3 M5 by the number of occurrences b c dy ey 15X14 6x2 O OQ O 9 OQ 9 OQ 9 OQ 9 O O O O O OQ OQ QO OQ O 9 O 9 e e O OQ O 9 OQ OQ QO 9 ee O O OQ O O O O O 9 O e e eee Then the tens 15 x 1 O OQ O 9 OQ OQ OQ 9 J O Q Q Q Q Q O O O 9 O O e e e e Q O Q O O O 6 o0 O OQ OQO O OQ O O O e 660 eee Then hundreds 9 x 1 note that adding 9 to the already present 1 a ten of hundreds is reached so a thousand O OQ O 9 OQ OQ J Q O 9 Q Q Q Q Q O 9 O 9 QO O e e e Q O Q O Q Q 6 o0 O O O O O e O OQ O 660 660 Then thousands 6 x 1 O OQ OQ OQ OQ OQ OQ OQ OQ OQ Q Q Q Q Q Q O e QO O 6 e Q O eo QO O 6 6 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 26 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL O O O 660 O O O 660 660 Finally we add the tens of thousands 3 x 1 thereby obtaining the result 37 077 O O O O Q O O O Q 9 Q Q Q Q Q Q O e e QO
49. b menu of the Yupana by De Pasquale base 40 progression 1 2 3 5 y Yupana by De Pasquale 2001 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 60 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Representation Representation of numbers in De Pasquale s system F1 ar Yupana by Chirinos 2008 dici e ha Yupana by Chirinos base 10 Representation Representation of numbers in De Pasquale s system F1 o Author amp License Information about Kunturweb amp GPL 3 License F12 b Exit Exit the program ESC Table 40 Descriptions of Tk yupana Menu Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 61 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 5 Yupana by Wassen The sub menu Yupana by Uap Rappresentazione base 10 progr 1 5 15 30 FL Wassen see Illustration 3 has two choices also selectable via gb Addizione EZ the function keys F1 and F2 which correspond to the following functions Illustration 3 Sub menu of the Yupana by Wassen 1 Representation is used to represent a number on the yupana 2 Addition Allows you to add two numbers 3 5 1 Representation Y Yupana di Wassen Rappresentazione a X Tk Yupana ad by Kunturweb Tota
50. ble to put a maximum of one seed The first line from the bottom corresponds to units the second to tens the third to hundreds the fourth and the fifth to the thousands and tens of thousands Clicking with the left mouse button on a box you put a seed in the Yupana To enter a number you have to click with the left button of the mouse on one of the boxes Note that the order of insertion is important the system 1s positional and You must proceed from bottom to top and left to right starting with the units and rising gradually When the box is selected you will see a seed in it the corresponding value is added to the total and it is then displayed next to the table To enter the number 652 for example we J start from the units and insert two seeds in the box 1n the first row from the bottom and first column from left and then add the tens by placing five seeds in the box in the second row and first column and finally add the hundreds by inserting a seed in the box of the third row and second column value 5 and a seed in the box of the third row and first column value 1 total 5 1 6 See Illustration 5 Illustration 5 Entering number 652 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 63 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 4 While holding down the CTRL key and clicking with the right mouse button
51. by Kunturweb USER MANUAL The order of removal is opposite to the insertion direction so you can not remove the seeds that are not adjacent 3 6 2 Addizione The window allows you to add three numbers together The numbers are inserted through the panel on the left which involves the insertion of the first second and third addend besides the possibility to select the speed with which is filled the yupana please see Illustration 12 Entering the first addend You can enter the first term using the arrows in the related field or by entering the number directly using the keyboard Once you leave the field the second term is enabled otherwise disabled and the first number is represented in yupana with the white seeds according to the rules described in the previous paragraph A ae eee al Entering the second and third addend Once inserted the first term and only then you can enter the second term using the arrows or by directly filling the field with the keyboard When you leave the field the number is displayed in the second column by black seeds and the next field is enabled for entering the third summand Once you have entered the third summand when you leave the respective field the number is represented in the third column Illustration 14 Inserimento del secondo 456 e del terzo ith yellow seeds and the Calculate button is AGOSTO HOHEM enabled which allows you to compute the sum
52. corresponds to write it as a sum of N terms 1 BREAKDOWN AND INCLUSION OF MULTIPLICAND a first thing is to break down the multiplicand M in base 10 This corresponds to write it as a sum of N terms M x0 t c j 0 1 2 N b you assign to each power of ten a coloured seed N different types of seeds or different colours will be necessary one for each power Later we will call these seeds seeds j or powers j Their type or colour will then account the relative power of ten c start drawing a column of R rows where R corresponds to the sum of the digits of Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 39 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL the multiplicand R x t c 0 1 2 N d each rows of column 1 has value 1 e let us insert in column 1 first column from the right x seeds each for every type of power j the number of seeds of the same colour therefore corresponds to the coefficients of the summations 2 BREAKDOWN AND INCLUSION OF THE MULTIPLIER a The numeral system by Florio 1s based on the distributive property of multiplication with respect to addition therefore we must factor the multiplier in the sum of two or more terms not necessarily grouped by the same power of ten m a 10 t c k 0 1 2 3 Kes 0 1 2 3 K The number K and the value a of the terms is at the discretion of the Khipu Camay
53. cos del gt 2 4y jdLi Quechua y el Aimara Direcci n Nacional de Educaci n Biling e Ministerio de Educaci n Lima Junio 2005 SPOM POMA DE AYALA Felipe Guaman 1615 El Primer Nueva 9 9 Cor nica y Buen Gobierno Royal Library di Copenhagen SRDP RADICATI DI PRIMEGLIO Carlos El sistema contable de los Incas SSAT SANTO TOMAS Domingo De 1560 Lexicon o Vocabulario de la Rttp to 1y jal lengua general del Peru SVEL VELASCO Juan De 1841 44 Historia del Reino de Quito en la 5z Am rica Meridional Quito Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 108 e mail kunturweb altervista org
54. design that appears in the manuscript El Primer Nueva Coronica y Buen Gobierno by Felipe Guaman Poma de Ayala written in 1615 but found relatively recently in the library of Copenhagenb MI This drawing depicts an accountant of the Inca empire and at its foot a board consisting of five rows and four columns which boxes contain whites and blacks or empty and full circles The two classes of objects are 1n my opinion be treated separately because the artwork and archaeological finds are absolutely inconsistent Furthermore while the description of yupana of Poma de Ayala certifies its use as a table of calculation there are not enough elements to establish with certainty the function of the yupana a casetta Literally Khipu knot meaning rope to which were made some knots to record accounting events and Camayuq teacher teacher of the knots A N 2 Some author translated it as Digits of numbers or seeds corresponding to numbers to make knots on the Khipus BL U and others with letter or numbers 5 9 N A 3 Some think it could be tables for gambling others models of fortresses cfr RDP chap 3 page 11 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 7 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL In this paper we will study the yupana of Poma de Ayala and the yupana a casetta will henceforth ignored 1 1 Reference
55. e product of the two factors is carried out and the number is shown in column 6 with red seeds The product is also numerically shown at the bottom left of the window Please see Illustration 25 A further press of the Calculate button allows you to perform another operation Illustration 25 Primo passo dell operazione 3 6 5 Menu functions Clicking on the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F1 2 Help Displays a help file F12 Close Closes the window F8 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 73 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 7 Yupana by Glynn The sub menu Yupana by Upp Rappresentazione base 10 progr 111M F1 Glynn see Illustration 26 has two choices also selectable via the po function keys Fl and F2 which correspond to the following pa functions Illustration 26 Sub menu of Yupana by Glynn 1 Representation is used to represent a number on the yupana 2 Addition Allows you to add two numbers 3 Multiplication allows you to multiply two numbers 3 7 1 Representation A Tk Yupana rev 0 1 by Kunturweb wi X i Tk Yu Da na Totale 0 pm by Kunturweb EPERU Gp up mp pr
56. e yupana by Poma de Ayala 1s shown in a vertical position with five rows and four columns the two central columns correspond to a multiplier divided into 3 5 see Illustration 60 The size of the table is different depending on the resolution of the screen and is able to operate even on a net book with resolution 800x480 4 33 It is recalled that according to the interpretation of the author the yupana represented by Poma de Ayala is a drawing of the calculation carried out in particular 32 x 5 and then the table would be used as a multiplying tool in which the column one the right one was used to represent the multiplicand while the columns two and three the central ones were used to represent the multiplier decomposed into two addends The fourth and last column the left one was used to derive the result N A 34 Tk yupana has been tested on an Asus Eee PC 701 4G with Linux Xandros operating system N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 99 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL In this window you can perform the following actions access the drop down menu in the upper right corner icon of condor select a different value for the seed to be inserted in the table e puta seed in the table a K NU Daha m by Kunturweb OS chunke QR pachaq LI J waranqa hunu C 10 5 cg io s EN 1077
57. eb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL A last press of the button allows you to fill in column 4 the sum of columns 2 and 3 and to display the product Illustration 66 Illustration 66 111 x 4 005 sum of columns 2 and 3 and display the result You can also remove an entire row by pressing the CTRL key and clicking with the right button of the mouse on the seed of column 1 The total is reset and the calculation must be repeated Illustration 67 Illustration 67 Deleting an entire row by removing the seed in column 1 3 11 4 Menu functions Clicking on the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F1 2 Help Displays a help file F12 3 Close Closes the window F8 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 104 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 105 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 4 License Tk yupana is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Fou
58. em is additive and not positional the arrangement of the balls can be random and the result will always be the same However as a matter of order and as written by Jos de Acosta we suggest inserting the multiplicand from the top of the yupana and proceeding downwards N A 18 Theoretically I could also choose 1 4 or 2 2 1 and so on N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 42 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL and 2 Table 26 Table 25 32X5 Decomposition of the multiplicator 3 Multiplications a for each row of yupana multiply the values in column 1 by the multiplication factor in column 2 and then add 2 seeds of the same power in box n x 2 Then for each row multiply the values in column 1 by the multiplication factor in column 5 and add 3 seeds of the same power in box n x 3 Table 26 QQQ QQ Q QQQ QQ Q QQQ QQ Q 66 60 o0 o 660660 o0 o Table 26 32X5 Multiplications of column I by columns 2 and 3 4 Summations a create column 4 where will be written the sum of the previous columns b For each row of the yupana sum the seeds k of the columns 2 and 3 and put as many seeds in column 4 Table 18 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 43 e mail kunturweb altervista org Tk yupana r0 7
59. epresentation of number 3 595 in the Yupana by C Radicati di Primeglio 2 2 2 Addition To add two numbers the two addends must be represented in the yupana see previous Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 18 e mail kunturweb altervista org Tk yupana by Kunturweb Tk yupana r0 7 USER MANUAL paragraph each of them in a corresponding column Subsequently starting from the column of units move all the seeds of the row in the last column column 4 keeping in mind that every time you reach the number 10 you can replace all the 10 seeds with a seed in the upper row Obviously the substitution principle also applies to the ciphers rows above O O O O QO QOQ O 0 0 O0 2 9 O O O OQO OO O0 OO OO 9 O O O O QO QO O 0 0 O0 O0 9 O O O OQO OO O0 OO OO 9 O O O O OQO QO O 0 0 O0 O0 9 O O O OQO OO O0 OO OO 9 e O O O O Oe O O e e 6 O O O OO e O 9 e e O 6 O O O O Oe O O e e e O O O OO e O 9 e O e O O O O 90 6 O QOQ e e O O O OO e O 9 e e O QO e OQ O 9 e o O 9 e O 9 O 9 O eeodiee O 0 e e o e e e O e e O e e O e e O eo e e O e O e O e e O e O O 9 ee 9000 e e O e e O O O e e Oo e e O e O O QOQ 8 O ee eee O OO e 9 e 6 9 e oe O O e e e oe e o e O O e e e e eo Q O O O O 0 eo o0 e O O e O 9 O OO OO 9 e O 6 6 O O 0 0 ee o e O e 6 O O O QOQ Q 9 e O le 60 OO eee o e O e e O e O O eO O
60. erm Yupana we mean an abacus used at the time of the Incas by the accountants and treasurers of the empire called Khipu Camayud The name comes from the Quechua word Yupay count while on paternity has arisen over the years a lot of confusion not only among scholars but also among researchers and professors alcuni5M9V attribute it to the engineer William Burns Glynn otherFPFl at the time of the Incas as the word is mentioned in an ancient Quechua language vocabulary compiled by Diego Gonzalez Holgu n5P9 in 1608 but Holguin translates the term Yupana with Letra los numeros de guarismos which we can translate Numbers while Yupana qqellca o qquipu as Las cuentas con nudos o por escrito meaning Calculations with knots or in writing Those who translates Table to count with Yupana o quippo is Domingo de Santo Tom s in his Lexicon o Vocabulario de la lengua general del Peru SAT dated 1560 So the word has Quechua origin and was used to indicate the instrument used for counting There are two classes of objects to which we refer when we speak of yupana The first class consists of a series of archaeological finds similar to boards of wood or stone whose boxes have different shapes and sizes are thought but is not certain that could be used as schedules Henceforth we will refer to them with the name of yupane a casetta The second class is actually made up of a single element a
61. este rreyno lo acienta y fiestas y domingos y meses y afios Y en cada ciudad y uilla y pueblos de yndios au a estos dichos contadores y tesoreros en este rreyno Y contaua desta manera comensando de uno dos y tres Suc uno yscay dos quinza tres taua cuatro pichica 5 zocta 6 canchis 7 puzac 8 yscon 9 chunga 10 yscay chunga 20 quinza chunga 30 taua chonga 40 pisca chunga 50 zocta chunga 60 canchis chunga 70 pozac chunga 80 yscon chunga 90 pachaca 100 uaranga 1000 SPOM page 361 363 translated by the Author oo N Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 8 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL The numeral system used was decimal A second chronicler who wrote a few words about the counting system of the Incas is the Jesuit Jos de Acosta who in his work entitled Historia Natural y Moral de las Indias described even approximately how to use the abacus These Indians took the grain and put one here three there eight in another location move a grain from one mailbox to another exchange three other grains in order to obtain the perfect result error free Information collected e They used seeds e They moved and exchanged seeds from one box to another They were very precise The third chronicler that I take 1nto consideration 1s Juan de Velasco who wrote
62. f the second addend If you complete the first column five seeds they are removed and replaced with a seed in the second column When number ten is reached all the seeds are removed and a seed is added in the first column of the second row Obviously the substitution principle also applies to the ciphers rows above Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 15 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 10000 50000 50000 150000 10000 50000 50000 150000 10000 50000 50000 150000 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1000 5000 5000 15000 1000 5000 5000 15000 1000 5000 5000 15000 e QO e QO e Q Q Q Q e Q e e e Q Q Q e QO e QO e Q 100 500 500 1500 100 500 500 1500 100 500 500 1500 e Q Q e Q Q e e Q Q Q Q Q e e e Q e Q 10 50 50 150 10 50 50 150 10 50 50 150 e e O e Q Q e e Q Q Q Q Q e e Q Q e e Q 1 5 5 15 1 5 5 15 1 5 5 15 Q Q o Q Q Q e Q Q Q Q Q Q e Q e Q Q Table 3 Operations for the addition 3 595 515 4 110 in the Yupana by H Wassen As an example we add the number 515 to the number represented in the previous paragraph The first step of the procedure is to add the ciphers of units tens and hundreds to the Yupana see Table 3 Part A red circles The second step is to group the seeds that have filled the boxes of column
63. f occurrences of holes a b Z is possible to assign the least significant value to the first gap a and then determine the values v v of the subsequent holes using the formula li a v l 1 JXv iza j lt i This ensures the uniquness of the representation 42998 1696 2579890176 10319560704 30958682112 O O O O O O O O O O O 2985984 17915904 71663616 214990848 O O O O O O O O O O O 20736 124416 497664 1492992 O O O O O O Q Q Q Q Q 144 864 3456 10368 Q Q Q Q Q Q Q Q Q Q 1 6 24 72 O O Q Q Q Q Q Q Q Q Table 35 Kak s Yupana Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 49 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Also to determine the value of a single hole on the first box of the next line line 2 we proceed in the same way considering previous boxes all the boxes of the previous row row 1 then 1x5 6x3 24x2 7241 144 122 Filling the second row will get the starting value of the third 124 and so on up to the fifth line We notice that the vertical progression of the power is not continuous but occurs in steps of two i e the system is positional base 144 with progression X such that n 0 2 4 6 8 The results are shown in Table 35 To represent a number you start to fill the table from the bottom upwards and from left to right the fi
64. grouping the seeds of minor power whenever ten seeds of the same colour are reached these are replaced by 1 seed of higher colour power c After having done so up to the maximum weight You will get R seeds 5 l c 0x s x9 d Starting by the weights of greater power You can read the result as p s 10 tc i 0 1 2 3 P Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 41 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 1 x seed 1 x seed 2 x seed N Table 23 Yupana N rows x K 1 columns Product column Example 1 32x5 drawing by Poma de Ayala As a first example we want to execute the multiplication drawn by Poma de Ayala 32 x 5 l Breakdown of 32 a 3222x100 3x 101 b let us assign the black colour to the units power 0 and the white colour to tens power 1 c then draw a yupana of 5 rows which are sufficient to contain the multiplicand in fact 5 is given by the sum of the digits of the multiplicand 3 2 d then three white balls and two black are placed in the first column of the yupana Table 24 Table 24 32 x 5 Entering the multiplicand 2 Decomposition and integration of the multiplier a decompose the multiplier into the sum of two terms 5 3 255 a we will enter in the table other two columns of equal power 0 and distinct values 3 17 Note that since the numeral syst
65. he author attributes to blacks circles designed by Poma de Ayala the concept of full holes while to the white ones the concept of empty holes So in each box there will be a different number of gaps seeds 5 in the first box 3 in the second 2 in the third 1 in the fourth He starts to fill the yupana from left to right and from bottom to top Then we number the boxes of the first row below as 1 2 3 and 4 In the 1 box you can enter 5 seeds with a value of 1 unit minimum value least significant for a total value TOT 1 gt From TOT 1 he determines the value of the individual gaps in the next box or TOT 1 1 6 In this box there are 3 holes for a total value TOT 22 TOT_1 x3 6x3 18 From the sum of TOT 1 TOT 2 he can determine the value of the individual gaps in the next box namely TOT TOT_2 1 24 In this box there are 2 gaps for a total value TOT 3 TOT 3x2 24 x2 48 Finally the value of a single gap in the next box is TOT 4 TOT3 TOT 2 TOT_1 1 2 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 48 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Each hole seed of column 1 has value 1 Each hole seed of column 2 has value 6 ossia 5 x 1 1 Each hole seed of column 3 has value 24 5 x 1 2 6 x3 1 Each hole seed of column 4 has value 72 5 x 1 6x3 24x 2 1 In general given a set o
66. ible solution to the exact arrangement of the circles in the design by Poma de Ayala and takes it as starting point for considering the non random distribution The result of his treatment 1s surprising when you consider the white circles as tens and the black circles as units the first column of yupana provides sum 32 second 64 third 96 and the fourth 151 Table 19 As long as you admit that Poma de Ayala made a typing error and has drawn a black dot instead of a white dot the sum for the fourth column would become 160 or 96 64 This observation led Florio to the conclusion that the Fibonacci sequence shown in the figure was just a case related to the example and the yupana was used in that example as a multiplication board The multiplication represented would be 32 x 5 where 5 is broken down as 2 3 and then as a result of the application of the distributive property of multiplication with respect to addition this would lead to the result 32 x 5 32 x 2 3 32 x 2 32 x 3 64 96 160 In addition to the clear strengths of being the only numeral system of calculation designed to have a hit in the same drawing by Poma de Ayala the theory proposed by Cinzia Florio has in my opinion to its advantage the simplicity of use The points against him are two e You have to admit the error of Poma de Ayala to support the theory However as shown by Florio in her article cited in note 25 it is unlikely an error on the part of
67. l kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Sum of its parts By clicking on the Calculate button the addition of the two numbers previously entered begins If the speed is set to 0 the calculation is done instantly and the result displayed on the yupana Illustration 59 If the speed is set with a value greater than zero the filling of the table is done step by step and with increasing speed according to the number entered in field speed Illustration 59 Addition of the terms 3 11 3 Multiplication The implementation of the multiplication by Cinzia Florio performed in Tk yupana does not allow you to enter all the numbers nor the multiplicand nor the multiplier This is not due to a program error or by the limitations of the numeral system but is simply due to a programmer choice that wanted to represent the table drawn by Poma de Ayala With regards to the multiplicand see note 16 With regard to the multiplier the limits depend on both the number of columns only 2 and the number of values 0 9 attributable to the same To understand the functioning of the Florio s multiplication algorithm and use tk yupana properly it must be quite clear that this is a multiplier table then an aid in the calculation of a multiplication of two terms M x m where M is the multiplicand and m the multiplier see paragraph 2 6 3 When activated the multiplication window th
68. le 0 NEN E EAA p EEN KA E mw il EX EET E ES ES BAN S ES E Ed Een EE ESO E mE Be E E E E EA KAN EE WA Exo d ERU HU EIS E D LEE KAN E Ed Illustration 4 Yupana by Wassen empty table number zero The program presents an empty table in a vertical position corresponding to the number zero Illustration 4 The size of the table is different depending on the resolution of the screen and is able to operate even on a net book with resolution 800x480 24 Tk yupana has been tested on an Asus Eee PC 701 4G with Linux Xandros operating system N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 62 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL This window provides for representation of numbers in the positional system developed by H Wassen with vertical progression of powers of 10 base 10 and horizontal progression of the weights 1 5 15 30 Entering a number The boxes in the table have different weights depending on the column in which they are found and different power of 10 depending on the row in which they are located The first column from the left has weight 1 and it 1s possible to put into it a maximum of 5 seeds the second column has weight 5 and it 1s possible to put a maximum of 3 seeds the third column has weight 15 and it is possible to put a maximum of 2 seeds and the last column has weight 30 and it 1s possi
69. mbers 1 2 3 4 5 but not 6 because it would have required an extra row while it is still possible to enter the numbers 10 11 12 13 and 14 but not on 15 and so on Generalizing the multiplicands that can be inserted in N rows are all M that 1 10 M x N j 107 j20 N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 40 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 1 x seed 1 Value ax Value a 1 x seed 2 Power Sx Power S 1 x seed N Table 22 Yupana N rows x K 1 columns 3 MULTIPLICATIONS column 1 x column k a for each row of the yupana multiply column 1 by column k and add a weights in the box n x k taking care to respect the following rule if the column has power s s 20 you must enter in the boxnxk 4 seeds equal to power s power b The procedure is repeated for K columns inserted 4 ADDITION column 2 column 3 column K 1 a Create a column to the left of the column K 1 In this column which we will call column K 2 or Product column will be written the sums of the previous columns Table 23 b For each row of the yupana sum the number of seeds of the same power k of every column k with 2 lt k K 1 and put an equal number of seeds in column K 2 5 PRODUCT a The sum of all the weights of column K 2 is the result of the multiplication or product b Start by
70. na aa ag a BI A jana d toxin aa E E ak kanga a aan aa 75 eOr O TETTE TIC E TT 76 Ins ersc o ic E E AE E E E EA 78 o O E E aia aaa anaa naa E EE E T 79 iie De IS WCE E D saa E E E E A E E E a Nga 79 Enternne the Se COMO TETI sas anan agak a aa aga nang Aa E Aga nandan gag aaa EEEE kaa arakan 79 ZUOCHUOBOT T6 1G IS aaa agak aa aaa aa aaa SN e aaa arana aa aaa anaa saa Na nana anaa kala a ag andaga kN 80 Soa oce MIUTEEDPICOL O eessen NEE EEEN UN E EOT 80 Entering the Multiplicand eeeeeeeeiessessseeeeeeeeeee nennen nennen enne nnn nnne nes 80 Entering the MUM IDOL sasa aaa Pape gg AA Aga GEN P Edo KA ENG EEE KET GE A AANE Na aa a Ka Ke 81 AAN TING a 0 ff Oe ene EO a aa en ad ene ae aia ana Ee aa an a ga aa ae TO E 81 Dee VICI EUG LI ONG eae daane ace cesar pasion aaa aa aa aan aaa E anga aaa aaa aaa aa ana aaa aaa anan 83 205 rupana OY MIC PaSdudle asana testes genes ae aaa BE daa DIEI Im aana E Da ANAK d an aaa UN ang a Ba adi 84 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 4 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Dl KG PROSE MUALION aana aa ag Pa a ste aaa ian Gi aa a an ana o eaaa baba ga aaa ne den Pi ga aana ena ah aa shoes 84 Deyo IVC IN LUCE ONS aan saa a gana na naa aa aapa E kaanan naa aapa bana pasa aana aaa aaa anapa 87 S Y Wala Dy Chirinos 2008 sasapan pan again ang aaa R aa aa E aa agen aaa pa Kaanan aa 88
71. nd Tk yupana Positional 10 Vertical 5 rows x 4 columns 1 5 15 30 or 5 15 30 30 1 10 100 1000 10000 Henry Wassen was the first to provide an interpretation of the drawing of the Yupana by Poma de Ayala The hypothesis that is the basis of the theory of Wassen is that the white circles were holes where to deposit the seeds while the black circles were such holes filled with a seed 2 1 1 Representing a number The representation of a number in the yupana has a vertical progression of base 10 decimal This means that the number can be represented by placing the units in the first row starting from the bottom the tens digit in the second and so on The value that each seed can assume instead depends on the column in which it is located following the horizontal progression based on the principle that 1n the first column it 1s possible to place a maximum number equal to 5 seeds having a value of 1 for a total of 5 and equal the value of a single seed placed in column 2 in the second column a maximum number of three seeds having a value of 5 each for a total of 15 and equal to the value of a single seed placed in column 3 in the third column a maximum of two seeds having the value of 15 each for a total of 30 and equal to the value of a single seed placed in column 4 See Table 1 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 13 e mail ku
72. ndation either version 3 of the License or at your option any later version Tk yupana is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with Tk yupana If not see http www gnu org licenses Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 106 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 9 Bibliography These are the articles sites from which all information about the functions of the program implemented have been retrieved SACO ACOSTA Jos De Historia Natural y Moral de las Indias 1590 Retp r to 1y duo Libro VI cap XVIII De los memoriales y cuentas que usaron los Indios del Per SCHI CHIRINOS RIVERA Andres 2008 Quipus del Tahuantinsuyo SDEP DE PASQUALE Nicolino 2001 Il volo del condor Pescara htP 7 t9 2y 3875 Informa ottobre 2001 SFLO FLORIO Cinzia 2010 Ruote di Acosta elementi decorativi strumenti mnemonici o mezzi di scrittura in Eliana Guagliano curatrice Letterature Americane ed atre Arti Atti del XXXI Convegno Internazionale di Americanistica Salerno O dipus Editore Salerno cinziaflorio tiscali it SFLR FLORIO Cinzia 2009 Incontri e disincon
73. ns When a seed is in memory you can also redistribute its value in the boxes of the column below if it they are all free To accomplish this just click on the seed into memory with the right mouse button the seed disappear from the memory box and ten boxes below will be filled A seed in the memory box can be redistribute in the boxes below where they are all free by clicking with the right button on the memory box Similarly the last seed entered in column X can be moved in the memory of the previous column by clicking on it with the right mouse button The seed disappears from the box to appear in the memory of the previous column Since this 1s only possible on the highest suit last entered if you click on one of the other seeds nothing happens S Any seed of a given column X can be moved to the memory of the previous column as long as the corresponding box is free Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 78 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 7 2 Addition Illustration 35 Yupana by Glynn addition Entering the first term AETERNE 3 ia mimm ELI IEEE HE Illustration 36 Yupana by Glynn entering number 1 254 Entering the second term Illustration 37 Yupana by Glynn entering 2 addend 5 675 and addition Author Kunturweb Web http
74. nturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 10000 50000 50000 150000 150000 300000 300000 O O O O O O O O O O O 1000 5000 5000 15000 15000 30000 30000 O O O O O O O O O O O 100 500 500 1500 1500 3000 3000 O O O O O Q Q Q Q Q Q 10 50 50 150 150 300 300 Q Q Q Q Q Q Q Q Q Q Q 1 5 5 15 15 30 30 Q Q Q Q Q Q Q Q Q Q Table 1 Scheme of the Yupana by H Wassen positional notation in base 10 with progression 1 5 15 30 As an example for the representation of a number choose 3 595 It fills the table from top to bottom and from left to right bearing in mind that each digit corresponds to a row positional system We start with the thousands 4 row three seeds circles blacks in the first section of the row covering three holes white circles then continue with the hundreds 3 row five seeds in the first cell then the tens 2 row since 9 is greater than 5 when five holes are filled adding 5 seeds in the first box of the row we can group the five seeds and replace them with one in the second box of the 2 row then proceed by adding the remaining four seeds in the first box of the 2 row Finally units 1 row these also are five then we can operate in the same way of the hundreds or in order to show that a number can be represented in different ways one might group the five seeds and replace them with one
75. o Linux Xandros n d A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 02 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Clicking with the left mouse button on a box you put a seed in the Yupana To enter a number you have to click with the left button of the mouse on any of the boxes When the box is selected you will see a seed inside it and the corresponding value is added to the total and the total is then displayed next to the table The first row at the bottom corresponds to the power 12 and the values of the boxes varies according to the progression 1 6 24 72 so in this row it 1s possible to enter all the numbers from 0 bare table and row to number 143 full line Each number has a unique representation Oo Yupana by Kak Repres _ _ Tk Yupana E ad by Kunturweb jmd Total 223 Ni EHEN EE LIN ee E E KN E EEA 4 EEEN E E Image 52 Representation of the number 223 144 2nd row 1 6 72 Ist row To clear a box decrease the total of the corresponding quantity you have to click with the right button of the mouse on the corresponding seed box while holding down the Control key Ctrl Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 93 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb US
76. o m p ele eae Hi j Babu iui el b is n s e n n p n sn n pn e e pn d JEN dU NR CES ERA E ER ERI Er E ST dies ge n rr TA Co Illustration 49 A representation of number 9 8 given from 6 3 0 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 89 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL X G Yupana di Chirinos Rappresentazione v X Tk Yupana d by Kunturweb ES Wil E Totale 9 8 DE Ep ESI EX n 5i e b ELENEN ELE JAM DEAE E ET EA EX 2E ELE ESL ESI Em d Bl HIE ERE ijui Sele Let S EUER Ve RN Illustration 50 Another representation of number 9 given from I 2 5 second row 1 0 8 first row To clear a seed decrease the number of the corresponding amount you have to click with the right button of the mouse on the corresponding seed box by holding down the Control key CTRL pi While holding down the CTRL key and clicking with the right mouse Ctri e e button on a seed it is removed from the Yupana You can enter any number from one to 73332 6 The result 1s displayed in the Total field on the upper left of the window Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 90 e mail kunturweb altervista org Tk yupana by Kunturweb Tk yupana r0 7 USER MANUAL 3 9 2 Menu functions Clicking on
77. olumn of Table 21 2 Then we add a second column in which we represent the number 2 847 as 2 x 1000 8 x 100 4 x 10 7 For this number we will need 4 rows see the second column of Table 21 3 Add the seeds of the same colour and returns the result in the next column see the third column of Table 21 4 Finally where possible regroup ten seeds of the same colour and replace it with one of the colour power of the ten higher see the fourth column of Table 21 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 38 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL hehehe 6051 6051 2847 3204 Table 21 Yupana by Florio representation of numbers 3 204 column 1 and 2 847 column 2 and their sum columns 3 and 4 2 6 3 Multiplication Let us now see the operation principle of the yupana used as multiplication table according to the theory by Cinzia Florio In this case the abacus was used as an aid to perform multiplications and what appears to be drawn in Nueva Coronica by Poma de Ayala 1s to be considered on the calculation carried out 32 x 5 Let us then consider the multiplication of two terms M X m where we denote by M the multiplicand and by m the multiplier The process executed by the Khipu Camayug is as follows 1 a first thing is to break down the multiplicand M in base 10 This
78. on a cell a seed is removed from the Yupana To clear a check box decrease the number of the corresponding quantity you must click on the corresponding seed box by holding down the Control key CTRL You can enter all the numbers from 1 to 99 999 using the first two columns or up to 888 880 using all the boxes and of course the zero empty table The result is displayed in the Total field present on the table Operations of displacement of the seeds Once you have filled a box R x C row R column C you can promote all the seeds of that box the next box R x C 1 or turn all the seeds in a single seed of the box of next column to do so simply click on any of a seed present in the full box holding down the CTRL key The operation is only possible 1f there is at least one free space in the next box Z While holding down the CTRL key and clicking with the left mouse E button on a box full of seeds all the seeds are transformed into a v single seed belonging to the box of the same row and next column Once you have filled a box R x C row R column C you can promote all the seeds or part them of that box to the box of next power R41 x 1 or turn all seeds in one or more seeds of the box into one of the next line column 1 to do so simply click on any of seed of the full box holding down the SHIFT key The operation is only possible if there is at least one free space in the first box of the ne
79. ossible Illustration 32 Replace ten seeds of column 4 with a seed in the memory in the same column Then to complete the operation You can add three seeds in the now free boxes of column 4 Illustration 33 which corresponds as we have said to the three tens of the number 35 Illustration 33 Adding tens 3 seeds to column 4 A seed in memory can be moved to the first empty space of the next column as long as there is one free by clicking on the seed itself with the left mouse button Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 77 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL To conclude You can move the seed from the memory of column 3 to the first free box in this case the first box in the lower right of column four To accomplish this simply click with the left mouse button on the seed in memory of column 4 the seed is removed and placed automatically in the first free box of column 3 free box in this case the first box in the lower right Illustration 34 Illustration 34 Moving the seed stored in the first free If the operation is not possible e g when cell to the next column the next column is completely filled the seed will remain in the memory box To move the seed You must promote all the seeds of the next column in its memory Inverse operatio
80. p pers Illustration 27 Empty yupana number zero Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 74 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL The window shows an empty table in a horizontal position corresponding to the number zero Illustration 27 The size of the table is different depending on the resolution of the screen and is able to operate even on a net book with resolution 800x480 The program allows to represent numbers in base 10 on positional system Entering a number Each box of the table has a unit weight the first column counting from the right corresponds to the column of the unit the second of tens the third of hundreds the fourth and the fifth of thousands and tens of thousands To enter a number you have to click with the left button of the mouse on one of the boxes Note that the order of insertion is important and must proceed from bottom to top and from right to left When the check box is selected you will see a seed in it the corresponding value is added to the total and it is then displayed above the table To enter the number 21 for example will need HDI XX to click on the box at the bottom right of the fourth Illustration 28 Entering number 21 column Total 10 then click on the box immediately to its left Total 20 then click on the box at the bottom right of the fifth column
81. pa the number 8 kinsakallqu 5 and 3 kinsa and 9 llatunka corresponds to 10 subtracted 1 23 It should be noted that while supporting the thesis of Pilares Casas and therefore a clearly additive numeral system the two authors support also the theory positional by Glynn e Also according to Shakiban and Leonard the infamous Fibonacci series of which the inventors of the yupana were certainly unaware turns out to be see work cited in note 25 the real value of the abacus Inca but its presence on the yupana should not be sought in something transcendental In fact thanks to the division of yupana in groups of units it was 22 It is Guaman Poma who wrote that the Camayuq counted in thousands hundreds tens and units compare the text cited in note 9 Below in the same book the author describes how the Inca empire was divided and once again numbering based on 10 returns N A 23 This number refers to a probable proto Aymara from which with some contamination is derived the Aymara spoken today Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 54 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL easier for the accountant from a visual point of view to move the seeds on the table in order to perform the basic operations The first group of seeds belonging to the first column total value 5 is in fact clearly separable into three two namely the t
82. pitulo de la visita general o censo pages 195 236 10 Y pondr n uno aqu tres all ocho no s donde pasar n un grano de aqu trocar n tres de all y en efecto ellos salen con su cuenta hecha puntualisimamente sin errar un tilde SACO book VI chap XVIII De los memoriales y cuentas que usaron los Indios del Per translated by the author 11 SVEL from pages 1841 44 T II cap 7 Translated by the author Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 9 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL operations with the yupana of Poma de Ayala and interpretations continue to proliferate To understand the topics below it is important that the reader focuses its attention on some essential concepts and that will hopefully deepen their own 1 2 Abacus and numeral systems An abacus is an instrument invented and used by people of various cultures as an aid in mathematical calculations The method of calculation used in the schedule is based on a specific numeral system By numeral system we refer to a way of representing the numbers using a series of symbols The numeral systems are divided into two main categories sign value numeral systems and positional number systems In sign value systems the value of the number represented is the sum of the values attributed to a limited series of basic symbols the numbering of the Romans a
83. proposed by various authors in nearly 100 years after the discovery of a copy of Nueva Coronica of Poma de Ayala Pay special attention to the following aspects which will be highlighted and listed in a similar table at the beginning of each paragraph Sign value or positional 10 12 20 40 60 Powers of 10 Vertical 5 rows x 4 columns Horizontal 4 rows x 5 columns Progression of numeral values assigned to each row Progression of numeral values assigned to each column Consider also the following questions regarding the solution adopted at the end of each chapter we will try to give an answer that will ultimately serve to draw conclusions e Is it possible to represent all the numbers from one to a hundred thousand with only a 5x4 table and the numeral system adopted e Are the operations to use the abacus simple e Is there any problem or inconsistency Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 12 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 1 Henry Wassen s Theory 1931 Unfortunately I still have not found the original writings of Henry Wassen therefore the arguments in this section are based on certain articles of other authors 3 0 e SVOVI who describe briefly the method by Wassen In the future I hope to confirm what by now I just deduced and to make any changes to this document a
84. reater versions For example with Debian or Ubuntu open a terminal and type sudo apt get install tcl tk e MINDOMS Install tcl tk from this page http www activestate com activetcl downloads 3 2 Download install and running the program PAY ATTENTION Before running the program you must install tcl and tk ad described in the previous chapter LINUX debian ubuntu 1 Download the file tfkyupana rev all deb from the web page http kunturweb altervista org pag it tk yupana html 2 Open a terminal enter the directory where the file was downloaded and type sudo dpkg i tkyupana rev all deb 3 In order to run the program type tkyupana 4 For the man page type man tkyupana Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 56 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL LINUX other distros 1 Download the file thyupana_ lt rev gt tar gz from the web page http kunturweb altervista org pag it tk yupana html Open a terminal enter the directory where the file was downloaded and untar the file tar xzf tkyupana rev tar gz A directory tfkyupana rev will be created It contains the tcl script and all the files No enter the directory tkyupana rev cd tkyupana rev and type tkyupana tcl For the manual type man tkyupana 1 e MINDOMS 1 Download the file tkyupana_ l
85. reds etc and represent it 1n the first column from the right as a sum of powers of ten To change the power of ten You must select the corresponding seed from the list of seeds next to the table Illustration 54 Some names are written in Quechua and their meanings are shown in Table 41 The power of 10 blue without seed 1s an upper limit not due to Florio s algorithm but to reasons of programming Compared to the writings by Cinzia Florio that are mainly about the algorithm of multiplication which is limited and relevant to the particular multiplication portrayed by Poma de Ayala in Tk yupana you can enter all the numbers from 1 to 10000 limit of five rows as the program allows you to enter the digits from 1 to 9 in each cell of the first column See in this regard the warnings about the limitations in the representation of the multiplicand always in paragraph 2 6 3 Illustration 54 List of seeds 3 See section 2 6 3 for details Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 95 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Huk Units 10 1 Chunka Tens 10 10 Pachaq Hundreds 10 100 Waranqa Thousands 10 1 000 Hunu Tens of Thousands 104 10 000 105 100 000 10 1 000 000 107 10 000 000 Table 41 Correspondence between numbers and names in Quechua Once you have selected
86. res Casas which seems more probable and the value that he attributes is ten thousand even more after the reading of the vocabulary of Domingo de Santo Tomas55 which translates to Huno or chunga Guaranga with ten thousand in number N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 96 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL hundreds we have deliberately missed the tens that will be added later to show the additivity of the numeral system and click on two gaps in the second row of the last column then select the white seed x 10 tens and click on nine boxes of the third row from the top of the same column then select the yellow seed lt 1000 thousands and click on a square of the fourth row from the top of the last column and you are done Illustration 55 z Yupana di Florio Rappresentazione a X 3444 3444 3424 HH Tk Yupana l by Kunturweb ES EM I I BERE EE ee Er E F e e 444 gp ESI a mimm pachak ec FLEE a of of ei EE ee P una ists 10 5 e E P EE EE E E EE s air err BERE errr 1291 Illustration 55 Entering the number 1 291 To clear a seed and decrease the number of the corresponding amount you should click with the right button of the mouse on the corresponding seed gap by holding down the control key CTRL While holding down the CTRL key and clicking wi
87. resentation of number 100 2x1x40 5x4x40 80 20 100 Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 30 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 4 2 Addition TO DO 2 4 3 Multiplication TO DO Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 31 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 5 Chirinos Theory 2008 Positional 10 Vertical 5 rows x 4 columns from 1 to 11 0 1 1 10 100 1000 Andres Chirino Riverea proposed for the Poma de Ayala s abacus a numeral system with positional notation in base 10 and expects to have the yupana upright vertical longest side 5CHU 2 5 1 Rrepresentation of a number Every seed will have a value which depends on the row R on the column C and on the position in the box CxR it is occupying Every row R of the yupana corresponds to a power of 10 according to the formula f R 40 Re 1 5 1 4 1 with R increasing from bottom to top The horizontal progression of the values is 1 2 3 4 5 6 7 8 9 10 11 and it is based on the position of the circles present in the drawing by Poma de Ayala so for instance in row 2 the seeds will have value 1 column 1 values 2 and 3 column 2 values 4 5 6 column 3 e 7 8 9 10 11 column 4 For detail
88. rs The sub menu Yupana Radicati see Illustration 9 has four choices also selectable via the function keys F1 F2 F3 and F4 which correspond to the following functions 1 Representation allows to represent a number on the yupana 2 Addition Allows to perform the 3 Subtraction allows you to perform the subtraction operation between two numbers 4 Multiplication allows you to perform the multiplication 3 6 1 Representation uw JEU A a4 4 by Kunturweb EIEE AA CEEI EEE kakiki e d d VEEE r M 1 m r T r r T r m A A a RE 0 Illustration 10 Empty Yupana corresponding to number 0 zero Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Yupana di Radicati di Primeglio Rappresentazione of two numbers e Doc Rev EN 0 7 1 07 02 14 67 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL The program shows a table in an upright position and empty corresponding to the number zero Illustration 10 The size of the table is different depending on the resolution of the screen and is able to work even on a netbook with resolution 800x480 This window allows to enter and represent four numbers in base 10 positional system one for each column Entering a number In each of the four colums it is possible to represent a number Each row starting from the bottom represent a power of
89. rst row may contain a minimum value of zero yupana empty and a maximum equal to 5x1x129 23x6x12922x24x 122 1 x 72x 1222 5 18 A8 72 143 The number 144 corresponds to a yupana filled solely with a seed placed in a hole of the box to the left of the second row Each number from 144 to 287 144 143 is represented with a single seed in the first box of the second row value 144 and with a number of seeds present in the first row up to a maximum of 143 seeds first line full See an example of representation in Table 36 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q e O Q Q Q Q Q Q Q Q Q Q e Q e e Q Q Q Q Q Q Q e Q Q Table 36 Representation of 1 496 742 The representation of each number is univoque Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 50 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 7 2 The yupana as a computational tool for astronomy In his article Kak also proposes that the yupana could be used for astronomical calculations Kak consider the arrangement of the circles of the drawing by Poma de Ayala He supposes that each row corresponds to the representation of a number between 1 and 143 so as five rows with power 129 See Table 37 e Q Q e Q e Q 56 e Q O Q Q Q Q Q Q e 79 e Q e e e Q e e O Q 29 e Q Q e O Q e e Q
90. s please see Table 14 Chirinos divides vertically every row of the yupana into six columns called huachos that correspond to the disposition of the cirlcles of the drawing by Poma de Ayala He then brings together the scores for quadrants called suyos and groups the suyos two by two into two main sections called sayas Finally he points out that the sum of the seeds of the left saya is 60 while the sum from the right saya is 6 and concludes that the Inca while adopting a positional numeral system in base 10 they also had the option of using the yupana as a sexagesimal calculation table Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 22 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Table 14 Chirinos Yupana scheme Particular attention is paid to the so called central circles called unique or chullas which are highlighted by blue color in Table 14 The remaining circles are called coupled boxes or pitu and are indicated by the color black in Table 14 In this theory there are number with multiple representations Chirinos identifies three types 1 Coupled representation when the number is represented only by pitu type boxes see an example in Table 15 2 Representation decoupled when the number is represented both by pitu boxes that chullas boxes see an example in Table 16 3 Rapprresentazione concrete
91. s to the Yupana by the Chroniclers of the Indies In the next page of the drawing mentioned above Poma de Ayala describes only approximately the method to make calculations used by the accountant of the empire Poma de Ayala wrote Major accountant of the whole kingdom Condor Chaua son of gods they called him Tawantin suyo rune quipoc Yncap haziendan chasquicoc major treasurer It is said that he had a great ability To test it the Inca ordered him to number count and adjust the natives of the kingdom He associated the subjects with the wool of the Andean Deer produced and with a food called quinoa and could determine how much quinoa and wool was produced His skill was great better that if he could write Major accountant or hatun hucha quipoc and Accounting minor or Huchuy hucha quipoc they use tables counting from one hundred thousand to ten thousand and one hundred and ten up to the unit Everything that happens in this kingdom they shall annotate parties weeks months and years In every town there are these accountants and count starting from one two and three Suc 1 yscay 2 quinza 3 Taua 4 pichica 5 zocta 6 Canchis 7 puzac 8 yscon 9 chunga 10 yscay chunga 20 quinza chunga 30 Taua chonga 40 pisca chunga 50 zocta chunga 60 Canchis chunga 70 pozac chunga 80 yscon chunga 90 pachaca 100 uaranga 1000 5 Information collected e They used tables to count Probably had an
92. sta org Doc Rev EN 0 7 1 07 02 14 21 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 3 William Glynn Burns Theory 1981 Positional 10 Horizontal 5 rows x 4 columns vertical in the original version 1 1 1 M or 5 3 2 M 1 10 100 1 000 10 000 The theory developed by the textile engineer William Glynn Burns is based on a numeral system with positional notation in base 10 corresponding to the horizontal progression growing from right to Jef SMOV amp ISLES The vertical progression instead consists of the numbers 5 3 2 from bottom to top which having sum 10 are sufficient to represent all the numbers from one to one hundred thousand The last column that of the one is meant to be used as a memory There are many variations of this numeral system proposed by various authors some put the yupana horizontally long side horizontal other vertically but the theory behind it 1s obviously the same 2 3 1 Representing a number since the horizontal arrangement of the Yupana has been more successful and has been used in many educational projects in various countries around the world I decided to adopt it for the development of Tk yupana Referring then to a table of 5 columns and 4 rows longest side horizontal the circles of the Yupana by Poma de Ayala are mnemonic indicating the maximum number of items seeds that can fit in a box
93. sual control is achieved by breaking down the addends into small numbers in one case and with the help of memory in the other e All Spanish chroniclers from Poma de Ayala on describe the numeral system of the Incas in base 10 this would exclude the numeral system of De Pasquale This view is strongly expressed in an interesting article published in the Journal of Mathematics amp Culture by M Leonard and C Shakiban s 5 Here the numeral system of Glynn is reviewed in the light of linguistic and cultural considerations This study which expects to have the yupana in an upright position as shown by Poma de Ayala and that places in high regard the importance of the five fingers of the hand and then number five in the Andean culture highlights two factors not negligible First of all thanks to the vertical arrangement of the yupana there is a direct correspondence between it and the khipu Another tool used by the Incas to record events and quantity 8 59 Secondly based on the statements of the linguist Pilares Casass l about the way of counting of the Ayamara speaking populations there is a correlation with the sequence of numbers 5 3 2 1 of the yupana These populations in fact adopt a sign value numeral system or count in a manner similar to that of the Romans the number 5 qallqu which would represent the first completion fingers of the hand are derived the word 6 maqallqu formed by 5 and 1 ma 7 paqallqu 5 and 2
94. t least 1n the initial version is an example of sign value notation systems for it was defined a set of basic symbols I V X L C D M respectively corresponding to 1 5 10 50 100 500 1000 The value of the other numbers was obtainable from the sum of the values of the fundamental symbols l1 PENES ES 3 I I I III 4 I I I I III 5 V 6 V I VI 7 V 4 I4 I VIl S8 V I 4 I1 4 I VIII 9 V I4 I14 I14I1 VIII 10 xX Note that in a purely additive system the representation of a number do not depend on the position of the basic symbols i e the number 8 could also be written IVII or HIV as the sum of the values of the symbols is the same as the conventional writing In positional systems the value of the symbols used to represent a number depends on the position occupied by the symbol Among the positional numeral systems we distinguish different notations with respect to many types of base The base is the number of digits that a unique positional numeral system used to represent all numbers An example of a positional numeral system is the Arabic the most commonly used today Each 12 Later IV subtraction of number I from number V N A 13 Later IX subtraction of number I from number X N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 10 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL number is represented by sequences of 10 digits
95. t rev gt tar zip from the web page http kunturweb altervista org pag it tk yupana html and put it in a folder Unzip the file thyupana_ lt rev gt zip with your favourite unzip program A directory called tkyupana_ lt reyv gt will be created It contains the tcl script and all the files 3 Enter the directory fkyupana rev by double clicking on it Run the program tkyupana tcl by double clicking on it Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 57 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 3 The main menu Tk Yupan When you start the program an icon representing a stylized condor is displayed Figure 1 Clicking with the left mouse button on the condor or pressing the F10 key you receive a drop down menu Illustration 2 from which you can access the various functions of the program by topic which are described in Table 40 Figure 1 Menu Illustration 2 Drop down menu of tkYupana Note that while for the Yupana by Florio and Yupana by Glynn the algorithms for calculating the sum and the multiplication have been implemented for the other theories the work 1s still in progress and the yupanas run only to represent numbers or for the addition Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 58 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunt
96. ter any number from 0 row and empty table to number 39 full row To enter the number 40 must clear the entire first row and insert one seed in the first box to the right of the second column Every number does not have a unique representation as can be seen in Illustration 46 and Illustration 47 where are shown two representations of number 9 A us Tk Yupana base 40 De Pasquale vw x _ Tk Yupana d by Kunturweb E EXER Totale 9 E RE EXE Ex EA AE EA A AN ee EH MAA N E E AN Illustration 46 A representation of the number 9 given by 3x2 2x1 1 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 85 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL uw ma Pe by Kunturweb TE Totale 9 BE EXER ESI SUE pu HERR ERR SB HEC EDI EDT LER LI ER wa pen HE ERR Ei Ah fel sty se bn s Zee Zee eee eee el el ENE Illustration 47 Another representation of the number 9 given by 3x3 To clear a seed decrease the number of the corresponding amount you have to click with the right button of the mouse on the corresponding seed box by holding down the Control key CTRL p While holding down the CTRL key and clicking with the right mouse Ctri e e button on a seed it is removed from the Yupana You can enter any number from one to 102 399 990 The result 1s displayed
97. tervista org Doc Rev EN 0 7 1 07 02 14 19 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 2 3 Subtraction To subtract two numbers it is necessary to represent the minuend and the subtrahend in the first two columns of the yupana see previous paragraph Subsequently starting from the column of units the seeds of the first column must be moved in the third or fourth column subtracting from time to time the seeds of column 2 If the result is negative a seed belonging to the row power above must be transformed into 10 seeds of the current row power before subtracting the seeds of the 2 4 column O O O O OO OO 0 QQQ O Q Q O O O 90 0 90 90 2 9 O O O 90 0 0 OO 0 QQQ O QO Q O O O 90 0 90 QQQ O O O 90 OO OO O0 OO O O Q Q O O O 90 0 90 90 2 9 e O O O O O O O0 O0 OO 9 e O OQO O O O O O QO e O 9 6 O O O O O OO O0 OO 9 e O O O O O O O QO e O 9 6 O O O O O OQO O0 O0 QQQ e O O O O O O O QO e O 9 QO O O O O OQO O0 O0 OO O O 9 O O O OO 9 O 9 e O O O O OO O0 OO 9 e 6 O O O O OQO OO e e OQO e O e O O O O O OO 9 e 9 e O OQ O O QOQ e OQ e O O O OQO O0 O0 OO 9 e o O O O OO O O 9 e e O O OO OO 9 e eo e e O O QO QOQ e e O e o e oO O O QO OO 9 Q e O O Q Q Q O 9 e O O OO OO O 6 0009 e e O QO OQO Q e e e O e eo O OO OO 9 e 60600 e e D O O O ee e e O e 6 O O OO OO 9 e 60600 e e D O O O e e e Table 7 Process for the
98. th the right mouse button on a gap a seed is removed from the Yupana EN Author Kunturweb Web http kunturweb altervista org e mail kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 97 Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 11 2 Addition This window allows you to add two numbers The numbers are inserted through the panel on the upper left of the window which involves the insertion of the first and second addend as well as the possibility to select the speed with which the yupana is filled see Illustration 56 Illustration 56 Yupana by Florio addition Entering the 1 term You can enter the first term using the field s arrows or by entering the number directly using the keypad Once you leave the field the second term is enabled otherwise disabled and the first number is represented in the yupana with coloured seeds according to the rules described in the previous paragraph Yupana di Florio Addizione Illustration 57 Entering number 1 234 Entering the 2 term Yupana di Florio Addizione Once you have entered the first term and only then you can enter the second term using the arrows or by directly entering the field by typing on the keyboard When you leave the field the button Calculate 1s enabled Illustration 58 Entering 2 term 5 678 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 98 e mai
99. the condor upper left picture a drop down menu is displayed from which you can select the following functions 1 Clean the Yupana remove all the seeds from the table and clears the counter of the total F10 e F1 2 Help Displays a help file F10 e F12 3 Close Closes the window F10 e F8 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 9 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 10 Yupana di Kak 2014 Yupana by Kak Representation Tk Yupana p by Kunturweb Baku Bele 1 z Image 51 Empty Yupana corrispondente al numero 0 zero When you open the window Yupana by Kak the program presents an empty table in an upright position corresponding to the number zero Image 51 The size of the table is different depending on the resolution of the screen and is able to work even on a netbook with 800x480 resolution In this case have not yet been implemented algorithms but it is only possible to represent numbers in the non uniform positional numeral system 3 10 1 Representation The numbering of the columns grows from left to right and the rows grows from the bottom upwards The boxes of the table have a different weight depending on the row and column in which they are located according to the scheme shown in section 2 7 1 30 TkYupana stato testato su un Asus eee PC 701 4G con sistema operativ
100. the terms By clicking on the Calculate button the multiplication of two numbers entered previously begins the operation will be performed step by step in particular at the first press of the Calculate button is presented the SS SS SS ees first step that will be run the row and column and the operation are highlighted in yellow 2 Illustration 41 The words of the Calculate button is changed with the number of the step in question e g Step 1 3 To continue with the operation you need to press the button again Yupana di Glynn Moltiplicazione Illustration 41 First press of calculate button next step is shown in yellow Note that if the speed is set to 0 every step of the calculation is done instantly and the result displayed on the yupana otherwise the seeds will be placed in yupana with increasing speed depending on the number entered in the Speed field Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 81 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Yupana di Glynn Moltiplicazione The seeds placed in each step of the calculation have different colours this not E because they have different weights but for ai educational purposes only I By clicking on the Step 1 3 take the filling EE of yupana and the words change to Step
101. thor Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 70 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Entering the subtrahend as Once inserted the minuend and only then l you can enter the subtrahend using the arrows or by directly entering the field with the keyboard When you leave the field the number is represented in the second column by black seeds and the Calculate button is enabled which allows you to compute the sum Please see Illustration 25 Illustration 18 Inserimento del sottraendo 1234 Subtraction dicuends Ena T By clicking on the Calculate button the subtraction of the two numbers you entered previously is performed If the speed is set to 0 the calculation is done instantly and the result difference appears in the fourth column of the yupana If the speed is set to a value greater than zero the filling of the table is done gradually and with increasing speed according to the number entered in field speed Please see Illustration 23 Illustration 19 Sottrazione 5555 3 6 4 Multiplication dw Yupana di Radicati di Primeglio Moltiplicazione Q3 Q9 This window allows to multiplicate two numbers together The numbers are inserted through the panel on the left which involves the insertion of the multiplicand multiplier as well as the ability to select the speed with which the
102. tion of the mask that concerned the multiplier is activated to set the multiplier see section 2 6 3 Illustration 61 Entering number 111 To clear a seed from the table and decrease the number of the corresponding amount you have to click with the right button of the mouse on the corresponding seed box by holding down the Control key CTRL While holding down the CTRL key and clicking with the right mouse button on a cell the corresponding seed is removed from the Yupana Entering the multiplier Once inserted the multiplicand M it is possible to change the breakdown of the multiplier m as sum of two terms different from 3 and 2 acting on the numbers in the upper left corner relative to the columns 2 and 3 Illustration 62 Illustration 62 Fields for the decomposition of the multiplier Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 101 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Changing those numbers will change J accordingly the number of boxes displayed in columns 2 and 3 respectively and the same multiplicative factors factors that appear under the columns themselves In this regard see the Illustration 63 in which the multiplier 4 005 is divided in two 4 000 5 Illustration 63 Example of decomposition of the multiplier 4005 in 4000 5 The format 1s Column X A 10 S where
103. tration 45 Empty Yupana number zero When you open the window Yupana by De Pasquale the program presents an empty table in an upright position corresponding to the number zero Illustration 45 The size of the table is different depending on the resolution of the screen and is able to operate even on a net book with resolution 800x480 In this case have not yet been implemented algorithms but it is only possible to represent numbers in the positional numeral system in base 40 3 8 1 Representation The numbering of the columns increases from right to left and that of rows grows from the bottom upwards The boxes of the table have a different weight depending on the row and the column in which they are located according to the scheme shown in section 2 4 1 28 Tk yupana has been tested on an Asus Eee PC 701 4G with Linux Xandros operating system N A Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 84 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Clicking with the left mouse button on a box you put a seed in the Yupana To enter a number you have to click with the left button of the mouse on any of the boxes When the check box is selected you will see a seed in it the corresponding value is added to the total and it is then displayed next to the table Since the numbering is in base 40 in the first row at the bottom you can en
104. tri nella individuazione t 4te 1yck4 di una relazione matematica nella Yupana di Guaman Poma de Ayala e Incontri e disincontri tra Europa e America Salerno 14 15 maggio e 10 12 Dicembre 2008 O dipus Editore Salerno cinziaflorio G tiscali it SGIL GILDORF Thomas E Ethnomathematics of the Inkas Department 9 7 t9 3y 39fe of Mathematics University of North Dakota SHOL GONZ LEZ HOLGU N Diego 1608 Vocabulario dela Lengua 5 9 77 9 2 54EN General de todo Peru SLAU LAURENCICH Laura 2007 La yupana abaco e ordinatore dello spazio inca o esempio di sincretismo culturale Atti del XXVIII Convegno Internazionale di Americanistica Quaderni di Thule Argo Editrice SLES LEONARD Molly SHAKIBAN Cheri The Incan Abacus A http to ly jdOd Curious Counting Device Journal of Mathematics and Culture Nov 2010 5 2 University of St Thomas MN USA SMOV MORA Lyda VALERO Nydia La Yupana como Herramienta 579 785 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 107 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL Pedag gica en la Primaria Universidad Pedag gica Nacional SPAR PAREJA Diego 1986 Instrumentos Prehisp nicos de c lculo el 9 779 1 759L quipu y la yupana Revista Integracion Dep De Matematicas UIS Vol 4 No 1 enero junio 1986 SPIL PILARES CASAS Guido A 2005 Los sistemas num ri
105. u Camayuq decomposed in units tens hundreds etc Suppose we want to represent the number 3 204 It was broken down as 3 x 1 000 2 x 100 0x 10 4 x 1 Than the Khipu Camayuq needed 3 rows to represent the number He chose the seeds coloured according to the power of ten base 10 and distributed them in the first three boxes of column 1 grouping them by colour up to the complete number see Table 20 Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 37 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL SVs 3 red seeds 3 000 we 2 green seeds 200 S S e 4 black seeds 4 3 204 Table 20 Yupana by Florio representing number 3 204 Note that the insertion order from the thousands to the units adopted in the example is not decisive for the representation of the number in a sign value notation system the representation of numbers is not dependent on the position of the digits 2 6 2 Addition The addition according to the theory of Cinzia Florio occurred representing two or more numbers in as many columns and then summing them row by row The result was shown in the last column the first from left that of the result Suppose you want to add the number 3 204 to number 2 847 The procedure was as follows 1 Number 3 204 is represented in the yupana as we have already shown in the previous section see Table 20 or first c
106. uq and obviously depends on his experience in general it may be economical for the purposes of the calculation speed decompose the multiplier in a series of addends whose multiplication with the multiplicand is almost immediate as for example 16 10 6 or 165 100 30 30 5 b the terms in which the multiplier has been decomposed fix the values a and the number K of the columns subsequent to the first To the left of column 1 are created K columns one for each addend and the respective value corresponding to the number of seeds that can be inserted in each box and a power s corresponding to the power of ten of the addend under consideration are associated to each column c This would result in a checker board of N rows the number increasing from top to bottom multiplied by K 1 columns 1 column of the multiplicand K columns of the multiplier the number increasing from right to left Since here on each square of the board will be identified by the words nx k IxnxNelxkzK l and will have a value a and power s corresponding to the associated column Table 22 15 Note that entering a number in the method proposed by Cinzia Florio takes place from top to bottom in accordance with Jos Acosta they do not write by rows but from the top to down SACO Book VI Chapter IX N A 16 You can do a reverse discourse starting from the yupana represented by Poma de Ayala Given that it has only five rows you can enter the nu
107. urweb USER MANUAL 3 4 Conventions Some conventions have been adopted both in the development of tk yupana and writing this document 1 6 With the term of mask or window we mean a graphical to user interface GUI intended for a specific function which may be to allow the user to make a choice through a menu or to perform an operation with the abacus etc With the term table we mean a particular zone of the mask which reproduce schematically the Yupana drawn by Poma de Ayala Each table is equipped with a number of rows indicated with RX where X is an integer 1 2 3 and columns indicated with CY where Y is an integer 1 2 3 The numbers are variable depending on the theory chosen A box or house is an element of the table obtained by crossing a row and a column With the agreements of paragraph 3 a box may be written as RXx CY In each box there is a variable number of gaps that correspond to the empty spaces where it is possible to deposit the colored seeds The number of these gaps and their values varies depending on the theory that you are considering Unless otherwise stated to put a seed into a gap simply click the gap with the left button of the mouse 7 Unless otherwise stated to remove a seed from a gap you need to click the gap with the right button of the mouse while holding down the CTRL key Author Kunturweb Web http kunturweb altervista org Doc Rev E
108. wo groupings of successive columns Table 39 O e O O Q e Q e O O e Table 39 Moving three seeds from one box to another above and result below numeral system of Glynn Only the numeral system by Florio so far 1s compliant with the distribution of the balls on the table drawn by Poma de Ayala e According to Florio the piece of Fibonacci series is a pure coincidence and even contingent on the particular calculation done We hope that Tk yupana can allow users to experience and enjoy the use of different numeral systems applicable to the Incan Abacus and choose the one that best suits their needs Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 55 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 Tk yupana Tk yupana is a simple program that shows various theories applicable to the yupana drawn by Poma de Ayala on the El Primer Nueva Coronica y Buen Gobierno These theories are based on both positional numeral systems yupana base 10 by Henry Wassen and William Burns Glynn and yupana in base 40 by Nicolino De Pasquale and sign value numeral systems yupana by Cinzia Florio 3 1 Requirements Tk yupana is a tcl tk script so it is necessary to install tcl and tk programs for your OS before running the program LINUX According to your distribution you should install tkl 8 4 and tk 8 4 or g
109. wo numbers the first addend should be represented in the yupana see previous paragraph Subsequently starting from the column of units add as many seeds as there are units of the second addend If you complete the column ten seeds all the seeds placed are removed and replaced with a seed in the memory that is 10 with x equal to the number of the column starting from the right to the left this seed also applies a seed in a box in the column immediately to the left The three examples below are equivalent and are worth 100 10 tens 1 memory value 100 1 hundreds O O O O O OO OO OO OO OO OO 6 O9 O9 O9 O9 QQQ eee QQQ QQQ OOe QQQ 2 3 3 Multiplication The multiplication of two numbers according to Glynn use magic numbers 1 2 3 and 5 drawn by Poma de Ayala The first step is to find the four multiples of the multiplicand M1 M2 M3 M5 Secondly involves representing a multiplier in powers of 10 by the base 1 2 3 5 Then x is the exponent of the power of ten corresponding to the digit in question 0 for the unit one for the tens 2 for the hundreds and so on all digits from 0 99 999 are representable some in different 14 This is because the obliged choice of the numbers 2 3 5 does not meet Zeckendorf s theorem which states that every positive integer can be represented uniquely as the sum of one or more distinct non consecutive Fibonacci numbers As the numbers 2 and 3
110. wo numbers you have entered previously will begin The operation will be carried out step by step On the first pressure of the button the fields partial products are highlighted but the operation is not executed just to give the user the opportunity to understand the method At a subsequent press of the button the partial products are calculated see Illustration 23 and the central cells corresponding to rows 1 2 and 3 from the bottom and columns 2 3 and 4 from the left are Illustration 23 Riempimento dei prodotti parziali filled Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 72 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 2 1 step Summations xo bn T Next at a further pressure of the button rr ERR FF diagonal summations of the partial products are carried out The results are shown in column 5 Each press will change the message of the button itself to indicate the step under consideration Note that if the speed is set to 0 every step of the calculation is done instantly and the result suddenly displayed on the yupana otherwise the seeds will be placed in the yupana with increasing speed depending on the number entered in the speed field Illustration 24 Calcolo delle sommatorie 3 step Product Finally on further clicking on the Calculate button the operation ends The calculation of th
111. xt power Holding down the SHIFT key and clicking the left mouse button on a box full of seeds all the seeds are transformed into one or more iss seeds belonging to the cell of the first column of the next row Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 64 e mail kunturweb altervista org Tk yupana r0 7 Tk yupana by Kunturweb USER MANUAL 3 5 2 Addition The window allows you to add two numbers The numbers are inserted through the panel on the left which involves the insertion of the first and second addend as well as the ability to select the speed with which the yupana will be filled see Illustration 6 Entering the first term You can enter the first term using the arrows in the field or by entering the number directly using the keypad Once you leave the field the second term 1s enabled otherwise disabled and the first number is represented in yupana with white seeds according to the rules described in the previous paragraph Illustration 7 Entering 1 addend 1 234 Entering the 2 term Once you have entered the first term and only then you can enter the second term using the arrows or by directly entering the field with the keyboard When you leave the field the Calculate button is enabled Author Kunturweb Web http kunturweb altervista org Doc Rev EN 0 7 1 07 02 14 65 e mail kunturweb altervista org Tk

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