Difference between the number and the base is called the deviation. For example, if the number is 15, the deviation is (15 – 10) = +5 or simply 5. For the number 6 the deviation is (6 – 10) = -4.
Lets take a very simple multiplication ( 7 x 8 ) = 56.
Step 1: Write the numbers and their deviations from the base 10 which is the nearest base for both,
7 -3
8 -2
_____
Step 2: Now add any number with the deviation of the other.
In this case you can add 7 with -2 which will give 5 or,
add 8 with -3 to get 5. Note that in both case there should be the same result.
Write the result like this,
7 -3
8 -2
_____
5/
Step 3: Now multiply the two deviations which are -3 and -2 in this case.
The product is +6
Write the product as,
7 -3
8 -2
_____
5 /6
Hence the result is 5/6 or simply 56.
Now Lets work on a slightly more complicated problem.
( 94 x 108 )
Take the base 100.
94 -6
108 8
_______
Now the Left hand side will be ( 94 + 8 ) or (108 – 6) which is 102
So,
94 -6
108 8
_______
102/
Now the product of the deviation in this case is ( -6 x 8 ) = -48 which is negative.
Remember that we have chosen the base as 100.
94 -6
108 8
________
102/-48
Multiply the right hand side result with the base and subtract 48 to get the result.
102 X 100 = 10200
10200 – 48 =10152
Hence the result is 10152. (We have actually done the same thing in the first example which is, 5 x 10 + 6 = 56)
Now take another example (16 x 13) = 208. Here we take 10 as the base.
16 6
13 3
______
19/18
Here if you add 18 with the right hand side result, the multiplication will be a mess.
Remember we have taken 10 as the base and look at the deviations each of which has one digit. So you must put 8 in the right hand side and carry 1.
16 6
13 3
______
19/18
Now add the carry to the right hand side result to get (19 + 1) = 20.
16 6
13 3
______
20 /8
Hence the result is 20/8 or simply 208.
Exercise: Verify the results
1. 1003 x 997 (Take 1000 as base)
2. 97 x 89 (Take 100 as base)
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