Mental math

Mental math is all about estimating. It's also about clever tricks to get close to the right answer. This is good not only for reasonableness checking, but also when close is good enough.

It's important to know how to do math problems in your head, because that way you can double-check the results you get from your computer. I was astonished when one of my kids did a problem on the calculator: find the square root of 80, got an answer of 28.28, and just wrote it down and went on to the next problem. "NO!" I hollered. "It must be less than NINE because the square root of 81 is NINE!" (I wonder why my kids grew up not loving math?)

Quick! Multiply 24 by 26! (The difference of two squares)
24 × 26 = 25² - 1²

25² is 625, so 24 × 26 = 624.

What if you didn't know 25² is 625? No problem:

25² = 20 × 30 + 5² = 600 + 25 = 625

These are all applications of one simple fact: if you have two numbers, the difference of their squares is equal to the product of their sum and their difference. In algebraic terms,

a²-b² = (a+b)(a-b)

What's the decimal expansion of 1/59?
1/59 is nearly 1/60. Set the division out thus: 0.0 1 ... 6 ) 1.0 ... Here we have the decimal for 1/59, obtained by dividing 1 by 60; as we obtain each digit we merely enter it in the dividend, one place later, and continue with the division. We "bring down" the digit we just appended to the dividend rather than bringing down a zero as we normally would.   0.0 1 6 ... 6 ) 1.0 1 ...

Continuing in this way,

0.0 1 6 9 4 9 1 5 2 5... 6 ) 1.0 1 6 9 4 9 1 5 2...

What we're really calculating when we do this is (1+1/59)/60, which is mathematically equivalent to 1/59. But this form has the distinct advantage that the outermost division operation is done with a simple one-digit divisor. The digits of the numerator (1+1/59) are supplied for us as we divide, so this new formulation of 1/59 really helps us divide quickly.

This trick helps us divide quickly any number ending in nines, such as 1/59, as you have seen, or 5/299, etc. For the latter example, you would shift digits by two places as you append them to the dividend.

0.0 1 6 7 2 2 4 0 8 0 2 6...    --   3 ) 5 0 1 6 7 2 2 4 0 8 0...

If you know the factors of numbers ending in nines, you can do even more problems. For example, to find 5/23 you need only find 15/69.

A slight variation on this trick is to subtract (rather than append) the digits from the dividend.

For example 5/23 is equal to 435/2001; and if we note that 435 is the same as 434.999999999..., we have another method, in which, as we obtain the digits, we subtract them from the dividend, so many places later. Thus in the present case 217 391 304 347 ...    - 2 ) 434 782 608 695 652 ...

For example, 217 from 999 gives 782, which we then divide by 2, obtaining 391; this, subtracted from 999, gives 608; and so on. These clever methods of division are credited to Alexander Craig Aitken, who lived from 1895 to 1967.