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Yupana Calculator Addition and Substractions during Inka Times

by Liliana Usvat and Sascha Pugar

A yupana is an abacus used to perform arithmetic operations dating back to the time of the Incas in 15 and 16 centuries.

The results of these calculation were recorded on Quipu. A quipu, or knot-record (also called khipu), was a method used by the Incas and other ancient Andean cultures to keep records and communicate information. This simple and highly portable device achieved a surprising degree of precision and flexibility.

It uses the Fibbonaci sequence 1, 2, 3, 5.

The Inka system used seeds ( we are using coins) on the table bellow to do mathematical calculations.

How the table work?

We have rows of one, rows of two, rows of three and rows of five.

We also have coloumns, They mean multiplication in base 10. For example one seed ( coin) in the row 2 and coloumn x1 means number 2. If we place a seed ( coin) in the row 3 and coloumn x 100 means 300. If we place a coin in the row 1 coloumn x 10,000 that number means 10,000.

A coin in row 2 coloumn x 1000 means 2000.

A coin in row 3 coloumn X 10 means 30.

A coin in row 5 coloumn X10 means 50.

The followig examples demonstrates the addtion and substraction of the followint numbers

418 + 327 = 745

We take the coins and place them on the table for 418 = 400+ 10 + 8

To write 400 : There is no 4 row on the table. we use 3 + 1 or 300 ( row 3 coloumn 100 and row 1 coloumn 100) to form 400.

To write 10 : For the tens we use one coin in the row 1 coloumn x 10

To write 8 : we place 2 coins in row 5 coloumn 1 and row 3 Coloumn 1

We take the coins and place them on the table for 327 = 300 + 20 + 7

To write 300 : We place a coin on the row 3 coloumn x 100

To write 20: We place a coin on the row 2 coloumn x 10

To write 7: We place a coin on the row 5 coloumn x 1 and another coin on the row 2 Coloumn x1

To add the two numbers we do the followings:

We take away the coins on the row 2 and 3 on coloumn x1 and replace it with one coin on the row 5 in coloumn x1

We have 15 wich are 3 coins worth of five in the coloumn x1 ( last coloumn) . We remove 2 coins worth of 5 and replace it with one of 10 (one coin in the row 1 coloumn x 10 ).

We have 2 coins in the row 1 coloumn x 10 . We remove them and replace in with one coin in the row 2 coloumn x 10. two tens is the same as a twenty.

One rule they used when they had 4 or 6 they used two different numbers that composed that number for example 4 = 3 + 1 instead of 4 =2 + 2 ot 6 = 5 + 1 instead of 6 = 3 + 3/

In the coloumn x 10 we reorganize 40 and write it as 30 + 10 so we have one coin in row 1 and another coin in row 3 for the coloumn x 10. It is easier to read that way.

in the x 100 coloumn we have 1 coin on the first row and 2 coins on the row 3 that means 100 + 3000 + 300= 700. We rearange that in 500 + 200 ( one coin in row 2 and one coin in row 5, coloumn x 100). which is easier to read.

The result is 745.

Next addition is

1128 + 2364 = 3492

To write that we place the coins on the following positions

1128= 1000 + 100+ 20+ (5 + 3)

2364 = 2000 + 300 + (50+10) + ( 3+1)

How to substract with yupana?

We need two counters for this.We use pennies ( for positive numbers) and nickels for negative numbers). In the past they used two type of seeds.

212 - 139 = 73

212 = 200+ 10+ 2

One pennie on row 2 coloumn x100 , one pennie on row 1 coloumn x10, one pennie one row 2 coloumn x1 .

We write the second number using nickels

139= 100 + 30 + ( 5 + 3 + 1)

One nickel on row 1 coloumn x 100 , one nickel on row 3 coloumn x 10 , and one nickel in row 5, 3 and 1 on coloumn x 1.

The nikels represent negative numbers and the pennies represent positive numbers.

We take the pennies and breake down into smaller numbers.

Two pennies break down in 2 of one and one pennie and one nickel on the coloumn x 1 cancel each other and are eliminated from the board.

one 10 is equivalent to two five . On the 5 row we have one nickel and one pennie that cancel each other and are eliminated from the board.

We need to eliminate all the nikels. we cannot leave any nickels on the board.

5 is broken donw in 2 and 3 and those ( coloumn x 1)

200 is broken down in 100 + 100 and one nikel and one pennie cancel each other and are eliminated from the board.

100= 50 + 50

50= 20+ 30

30 canlel out.

The answer is 73.

Another problem presented is :

6322 - 4816 = 1506 .

First number is the positive pennies, the second number is the negative nickels.

6322 = (5000+ 1000) +300 + 20 + 2

4816= (3000+ 1000) + (500+300) +10 +( 5 + 1)

Please see this attached video.

 

"Chance favors the prepared mind." - Louis Pasteur