Product of 2-digit numbers

The simplest case is when two numbers are not too far apart and their difference is even, for example, let one be 24 and the other 28. Find their average: (24 + 28)/2 = 26 and half the difference (28 - 24)/2 = 2. Subtract the squares:

28·24 = 26² - 2² = 676 - 4 = 672.

Another example:

19·31 = 25² - 6² = 625 - 36 = 589

since 25 = (19 + 31)/2 and (31 - 19)/2 = 6.

If the difference is odd add or subtract 1 from one of the numbers, compute the product as before and then subtract or add the other number. For example,

 37·34= 37·35 - 37
  = (36² - 1²) - 37
  = 1296 - 1 - 37
  = 1258.

The same product could be found differently:

 37·34= 37·33 + 37
  = (35² - 2²) + 37
  = 1225 - 4 + 37
  = 1258.

Why does this work?

First, recollect the formula for the difference of squares:

a² - b² = (a - b)(a + b).

Observe that, if A = a - b and B = a + b, then a = (A + B)/2 and B = (B - A)/2 which modifies the formula to

[(A + B)/2]² - [(B - A)/2]² = AB.

The ancient Babylonians used the same method but with a slight modification. The previous formula shows that

(A + B)² - (B - A)² = 4AB.

from which

AB = [(A + B)² - (B - A)²] / 4.

This requires to compute the square of the sum and the square of the difference of the two numbers and dividing the difference of the two by 4. Although the two approaches are equivalent, sometimes one and sometimes the other is more amenable to mental calculations.

Here is a case where the Babylonian method seems simpler:

 33·32= (65² - 1²)/4
  = (4225 - 1)/4
  = 4224/4
  = 1056.

(This is because the difference (1) is small and the sum (65) ends in 5, so that squaring it is easy.)


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