When I was a math teacher, one day I was helping my students check a column of numbers of the form n2 - 1 to see if any were prime. I impressed them by quickly factoring the whole list: I rattled off some big ones like, "Nope, 182 - 1 is 17 x 19, 192 - 1 is 18 x 20," and so on, and most of the students caught on to the pattern: n2 - 1 = (n - 1)(n+ 1). It's a specific case of the formula which should be known to anybody taking the SATs: (a + b)(a - b) = a2 - b2.
I knew the formula could be used for multiplying two numbers conveniently placed on either side of a nice round number, like 88 x 92 = (90 - 2)(90 + 2) = 902 - 22 = 8,100 - 4 = 8,096. What I didn't know until recently was it could be used for squaring big numbers:
Start with the formula
a2 - b2 = (a + b)(a - b)
Add b2 to both sides and you get
a2 = (a + b)(a - b) + b2
Now instead of squaring a number like 27 by multiplying it by itself, you can use a nice round number, in this case 3o, which is 27 + 3. Use this to solve 272.
272 = (27 + 3)(27 - 3) + 32 = 30 x 24 + 9 = 720 + 9 = 729.
The way I would have done this previously is to use the formula
(a - b)2 = a2 - 2ab + b2
272 = (30 - 3)2 = 302 - 2(30)(3) + 32 = 900 - 180 + 9 = 720 + 9 = 729.
What Arthur Benjamin uses in the video below to solve 57,6832 is the related formula
(a + b)2 = a2 + 2ab + b2
(57,000 + 683)2 = 57,0002 + 2(57,0000)(683) + 6832
Which one could argue is simplifying the job a bit, but each term of the above expression still makes me reach for my calculator.
Some folks get really good at using these binomial expansions to square big numbers. I was impressed by the job Art does in this video introducing his calculating tricks.
He made a couple of mistakes (the square of 722 is actually 521,284) but his performance is a great bit of publicity for his terrific book Secrets of Mental Math. Learning a few "tricks" can certainly serve to lessen math anxiety, so try it out!
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