Mental arithmetic often calls for a healthy dose of opportunism. Take 51 × 49. An easy way to find the answer is to notice that 51 = 50 + 1 and 49 = 50 − 1, then use a standard identity: (50 + 1)(50 − 1) = 50 2 − 12, which is much easier to calculate. (It is 2,500 − 1 = 2,499.)
The trick readily extends to the product of any two integers a and b whose difference a − b is even (though it can be adapted for odd differences): write m for the mean of the two numbers, so that m = (a + b)/2, and d for half their difference, so that d = (a − b)/2. We then have a × b = (m + d)(m − d) = m2 − d2.
If, as in our previous example, m2 obligingly happens to be easy to calculate and d is not too large, we can begin by giving m2 as a first approximation, then refine it by subtracting d2. For example, for 64 × 56, we might say "Wait... it’s a little less than 3,600..." while mentally working out 3,600 − 16 = 3,584.
A head for squares
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As the trick above suggests, perfect squares are among the first things to learn for mental arithmetic, after the multiplication tables. Knowing the value of 632, for example, opens the door, via the standard identity, to quickly calculating products of the form (63 + b)(63 − b). But if our memory does not stretch that far, how can we find the square of a number such as 63?
One method well suited to two-digit numbers—that is, numbers of the form
n = 10a + b, where a and b are integers between 0 and 9—is based on the following easily verified algebraic identity:
(10a + b)2 = (10a + 2b) × a × 10 + b2. It is more useful to write this as n2 = (n + b) × a × 10 + b2.
In practice, to calculate 632, we work out 632 = (63 + 3) × 6 × 10 + 32. In terms of multiplication, this calculation requires only one multiplication by a single-digit number (6; multiplying by 10 is trivial, and the square of b is easy to find because b is a single digit).
The trick works in every case (and can be adapted to numbers with more digits), but when the units digit is greater than 5, it can be improved slightly by using the identity n2 = (n − b') × (a + 1) × 10 + b'2 instead, where b' = 10 − b.
The Trachtenberg method
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Jakow Trachtenberg was a Jewish engineer who was interned several times in concentration camps during the Second World War. While there, he devised a method for doing multiplication mentally, to pass the time. Each multiplication—by 3, by 4, and so on—has its own rule; here we consider multiplying a number n by 7. Start by placing a 0 to the left of the decimal representation of n, then apply the rule: "Double each digit, add 5 if it is odd, and add half the digit immediately to its right in the original number (rounding down if necessary)." Consider multiplying 294 by 7. First write it as 0 294. Double the 4, giving 8 as the units digit (there is no digit to its right). Next double the 9 and add 5 (since 9 is odd), then add 4/2 = 2, giving 25: the tens digit is therefore 5, with 2 carried. Next calculate 2 × 2 + [9/2] + 2 (the last 2 is the carry), which gives 10: the hundreds digit is therefore 0, with 1 carried. Finally, apply the same procedure to the 0, giving 0 × 2 + 2/2 + 1 = 2, so the thousands digit is 2. Thus 294 × 7 = 2,058.
