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..
000096700002930000000
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