ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual

Russian Peasant Multiplication - contributed by Hannu Hinkka

 

Russian Peasant Multiplication is a very old method. But this is not a primitive method of multiplication. It is a very interesting, powerful and inspiring example of how everyday problems have been resolved in past centuries.

However, here is a slightly more modern variation and technique which is adapted to the Abacus.

This method shows how we can make multiplication without multiplication tables by just doing addition.

Multiplication A * B , we use  temporary numbers (C) to help calculate. Three values (A,B,C)  are needed to work.

To start with place A on the left hand side of Abacus, B in the middle and C on right side. If B is an odd number then C=A. Otherwise C is zero. These are the initial values.

 
Multiplication rules for A*B 

     1.      Double A

2.     Halve B. Drop the decimal part of the quotient (if there has one). If the result is an odd number add A to C.

3.      Repeat steps 1 and 2 until B is 1

4.      Multiplication result  is the sum of A and C


A few words before we start.

The easiest way to double the number is to add it to itself.

In Examples 1 and 2, B is relatively small so that B / 2 is easy to calculate mentally. If B is a larger number finding half is not so easy. For this reason before Example 3 (see below), I've illustrated an easy method for finding half of a larger number. The method uses simple addition.

 
Now lets begin!

 
Example 1. 14*7=98

 A=14, B=7, C=0   

..A.....B....C.
001400007000000

 

We start with A=14, B=7, C=14, because 7 is an odd number we add A's value (14)  to C =0+14=14
 
..A.....B....C.

001400007000014

 

A=14+14=28,  B=7/2=3 (decimal part of the quotient dropped),  because 3 is an odd number we add A's value to C=28+14=42

..A.....B....C.
002800003000042

 

A=28+28=56, B=3/2=1 (decimal part of the quotient dropped). We stop because B=1.  

Finally we add A+C

..A.....B....C.
005600001000042
 

42  + 56 = 98

The result is 98.  

  

Example 2. 49*86=4214

 A=49, B=86, C=0   

....A....B....C.
0000490008600000

 

A=49+49=98, B=86/2=43, because 43 is an odd number we add A's value to C=98+0=98 

...A....B.....C.
0009800043000098

 

A=98+98=196, B=43/2=21 (decimal part of the quotient dropped), because 21 is an odd number we add A's value to C=196+98=294 

.A.....B....C..
019600021000294

 

A=196+196=392, B=21/2=10 (decimal part of the quotient dropped)

.A.....B....C..
039200010000294

 

A=392+392=784, B=10/2=5, because 5 is an odd number we add A's value to C=784+294=1078

.A......B....C...
07840000500001078

 

A=784+784=1568,B= 5/2=2 (decimal part of the quotient dropped)

A.......B....C...
15680000200001078

 

A=1568+1568=3136, B=2/2=1, we stop this because B=1

 

Finally add A+C 

A.......B....C...
31360000100001078

3136+1078=4214

The result is 4214


Here is a simple method to find half of a larger number using addition. This method is particularly well suited  for Russian Multiplication.


Rule for B/2

     1.      Add B to B

2.      Double

3.      Add B

4.      Divide by 10

 

Example 3. 173/2=86.5 (B=173) 

173+173=346    Add B to B

346+346=692    Double

692+173=865    Add B

865/10=86.5      Divided by ten

 

Example 4. 1952/2=976  (B=1952)

 1952+1952=3904    Add B to B

3904+3904=7808    Double

7808+1952=9760    Add B

9760/10=976            Divide by ten

  

 Example 5. 967*293=283331

 A=967, B=293, C=0   

 ....A......B......C..
000096700002930000
000

 

A=967, B=293, because 293 is an odd number we add A's value (967)  to C=967+0=967

 ....A......B......C..
000096700002930000967

A=967+967=1934

Lets calculate B/2=293/2  by the above rule.

293+293=586        Add B to B

586+586=1172      Double

1172+293=1465    Add B

1465/10=146.5      Divide by 10

B=146 (decimal part of the quotient dropped)

 

...A......B......C..
00019340001460000967

A=1934+1934=3868

Calculated B/2=146/2  by the above rule.

146+146=292

292+292=584

584+146=730

730/10=73

B=73, 73 is an odd, add A to C=3868+967=4835

 

...A.......B....C...
00038680000730004835

A=3868+3868=7736, B=72/2=36

 

 ...A.......B....C...
00077360000360004835

A=7736+7735=15472, B=36/2=18

 

..A........B....C..
00154720000180004835

A=15472+15472=30944, B=18/2=9, because 9 is an odd number we add A's value (30944) to C=30944+4835=35779

 

..A.........B....C....
0030944000009000035779

A=30944+30944=61888, B=9/2=4 (decimal part of the quotient dropped)

 

 .A.........B....C....
0061888000004000035779

A=61888+61888=123776, B=4/2=2

 

 .A..........B....C....
0123776000002000035779

A=123776+123776=247552, B=2/2=1, we stop this because B=1

  

Finally add A+C

 .A..........B....C....
0247552000001000035779

247552+35779 = 283331

The result is 283331

 

Abacus: Mystery of the Bead
Advanced Abacus Techniques
© May, 2011
Hannu Hinkka  Tampere  Finland
Email
hannuensio[at]gmail[dot]com