How does Arthur Benjamin square 5 digit numbers in his head?

30Jan/0816

How does Arthur Benjamin square 5 digit numbers in his head?

Last month I blogged about mathemagician Arthur Benjamin and his amazing mental Math feats. Benjamin is a master of doing arithmetic in his head with lots of digits involved. In particular, he's able to square a 5-digit number without writing down partial results. How does he do it? I picked up a copy of Benjamin's Secrets of Mental Math to learn how. Here are the steps Benjamin provides for squaring 46,792.

1. First, Benjamin breaks the number into 46,000 + 792.

2. Then he does a little algebra. If a=46,000 and b=792, then (a+b)^2 = a^2 + 2ab + b^2 = (46,000)^2 + 2(46,000)(792) + 792^2.

3. This simplifies a bit to 1,000,000(46^2) + 2,000(46)(792) + 792^2.

4. Then Benjamin sets out to do the middle product: 2,000(46)(792).

5. He's not worried about the factor of 2,000 since he'll take the product of 46 and 792 and multiply it by 2 and add 3 zeros at the end.

6. At this point, Benjamin needs to decide how to multipy 46 x 792 in his head. Building up to this point in the book, Benjamin has shown a variety of methods to streamline multiplication of numbers of with different numbers of digits and he makes the point that you need to spend a few moments looking at the problem at hand and pick the method that will strain your brain the least.

7. For this example, Benjamin notices that 792 is close to 800 so he calculates 46(800-8) = 46x8x100 - 46x8.

8. He calculates that 46x8 = 368, adds two zeros to get 36,800 then subtracts 368 from that to get 36,432.

9. Now he has to multiply 36,432 by 2,000. He doubles 36,432 to get 72,864 then adds three zeros to get 72,864,000.

10. Now things get more interesting. Benjamin has just calculated the 2ab part of the a^2+2ab+b^2 equation. He's got two squares left to compute and he's got to add those two squares to the number he just computed. How is he going to remember this all?

11. Benjamin uses a phonetic code where he translates digits into syllables. The idea is that strings of syllables are easier to remember than strings of digits. I believe this is also how people memorize long strings of arbitrary digits quickly. He memorizes the first two digits, 72, and encodes 864 as "fischer". Then, Benjamin says, out loud, "72 fischer", a couple of times to anchor the digits and phonetic code into his memory. There are 3 zeros at the end. He'll use those later.

12. Now, onto computing 1,000(46^2). Benjamin doesn't say how he does this calculation but I imagine he notices that 46 is close to 50 and calculates 46(50-4) = 46x50 - 46x4. Multiplying by 50 is the same as multiplying by 100 and dividing by 2. So, 46x50 = 2,300. Subtract 46x4 = 184 from 2,300 gives us 2,116 which is 46^2.

13. Benjamin adds 6 zeros to 2,116 to get 2,116,000,000 which is 46,000^2. He adds 2,116,000,000 to the 72,864,000 that he memorized with that phonetic code "72 fischer" and gets 2,188,864,000.

14. Benjamin doesn't want to remember this new number, 2,188,864,000 so first he says "2 billion aloud" since that digit won't change in the subsequent calculation.

15. Next, Benjamin want to says "188 million" but before doing that he needs to see if adding the rest of the product, namely 792^2 will generate a carry. 800 is close to 800 so 792^2 is close to 800^2 which is 640,000. Adding 640,000 to 864,000 will definitely generate a carry so Benjamin says "189 million". Now Benjamin doesn't need to memorize the 2 or the 188 anymore and he's left with 864,000 to add 792^2 to.
15. Benjamin has already noticed that 792 is close to 800 in step 7 so he uses the fact that (a+b)(a-b) = a^2 - b^2, or (a+b)(a-b) + b^2 = a^2 to determine that 792^2 = (792+8)(792-8) + 8^2 = 800 x 784 + 64. Benjamin has, earlier, in the book, given techniques for multiplying single digit numbers by multiple digit numbers so he quickly determines that 800 x 784 + 64 = 627,264.

16. Now, all Benjamin has left to do is add 627,264 to 864,000, which he memorized as "fischer". He adds 864 to 627 to get 1,491. He has already used the 1 as a carry previously so he just says "491 thousand" aloud.

17. All that's left now is "264" which he says aloud as the rest of the answer.

There you have it. It takes quite a bit of mental Math effort to do this kind of calculation in your head but my intent, and Benjamin's, is to demystify mathematics to show you how you could do this kind of arithmetic in your head if you really wanted to.

Secrets of Mental Math goes into more detail than I provided and shows a number of clever Math magic tricks. This is a great book to have if you want to impress your friends with mental Math tricks.

Wild About Math! Making Math fun and accessible

How does Arthur Benjamin square 5 digit numbers in his head?

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