Vedic sutra - all from 9 and last from 10
Problem
With this method you only ever need multiplication tables up to 5 times 5. It is one of many ancient Indian sutras and this one involves a cross subtraction method which, according to old historical traditions, is responsible for the acceptance of the ´ mark as the sign of multiplication. Here is a very simple example of the method. Can you give a good explanation of WHY it works?
Suppose we want to multiply 9 by 7. We subtract each number from 10 and, using these differences (or deficiencies), write:
| 9-1 |
| 7-3 |
| 6 3 |
The product has two parts, left and right.
To get the right part (or units digit) multiply the deficiencies
(1×3)
The left hand digit (tens digit) of the answer can be found in four
different ways. Why do they all give the same answer?
- Subtract 10 from the sum of the two given numbers (9+7=16, 16-10=6)
- Subtract the sum of the two deficiencies (1+3=4) from 10 and you get 6.
- Cross subtract (9-3=6)
- Cross subtract (7-1=6)
This gives the answer 63.
Here are some more examples. Try some of your own.
| 9-1 | 8-2 | 9-1 | 8-2 |
| 6-4 | 7-3 | 9-1 | 5-5 |
| 5 4 | 5 6 | 8 1 | 4 0 |
| Note: Here you have to express 5 times 2 as 1 ten and 0 units. |
Student Solutions
Well done Geoffrey, University College School, London for this excellent solution.
This problem is actually remarkably straightforward. A little bit of algebra is all that is needed.
Let x and y each denote 1 of the 2 numbers to be multiplied together.
The layout of the grids in the question then corresponds to:
| x | ( 10 - x ) |
| y | ( 10 - y ) |
The 4 ways of calculating the left hand digit are expressed algebraically as follows:
- x + y - 10
- 10 - [ ( 10 - x ) + ( 10 - y ) ] = 10 - ( 20 - x - y ) = x + y - 10
- x - ( 10 - y ) = x + y - 10
- y - ( 10 - x ) = x + y - 10
All 4 ways are algebraically equivalent, so this is why they give the same answer.
To explain why the Vedic Sutra works, we run through the whole method algebraically.
The right hand digit is ( 10 - x ) * ( 10 - y ).
The left hand digit is ( x + y - 10 ). We should remember that this
is the tens digit and multiply by 10 before summing both sides.
10 * ( x + y - 10 ) + ( 10 - x ) * ( 10 - y )
= 10x + 10y - 100 + 100 - 10x - 10y + xy
= xy
Using the Vedic Sutra, we get the result xy, which is indeed the product of x and y.