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MatheMUSEments - Arithmagic (7/4/2006)

By Ivars Peterson

When he was a kid, Arthur Benjamin liked to show off. Now, he's a math professor at Harvey Mudd College in Claremont, California—and a professional magician.

Unlike other magicians, Benjamin doesn't pull rabbits out of hats, make coins appear and disappear, or perform rope tricks. Instead, he amazes audiences by multiplying numbers in his head faster than someone using a calculator.Benjamin says that anyone—young or old—can learn to do the same feats of mental arithmetic. You, too, can look like a genius without really trying, he insists. All it takes is a memory for numbers and the trick of calculating from left to right, opposite to the way you were probably taught.

Suppose you want to multiply 27 by 8. Starting from the left, you would multiply 20 by 8 to get 160. Then, 7 times 8 equals 56, which is added to 160 to give 216.

How would you multiply 378 by 7? Multiply 300 by 7 to get 2,100. In the next step, 70 times 7 equals 490, which is added to 2,100 to give 2,590. Finally, 8 times 7 equals 56, which is added to 2,590 to give the answer 2,646.

One advantage of using the left-to-right method is that you can start saying your answer while you're still calculating it, Benjamin says. That's very handy for a mathemagician intent on impressing his audience.

Squaring a number means multiplying it by itself. Here's a quick way to square a two-digit number, such as 37. Pick a nice round number that's close to 37, such as 40. The number 40 is 3 more than 37. Calculate the number that is 3 less than 37, which is 34. Use left-to-right multiplication to calculate 34 times 40, which is 1,360. Then add the difference (3) multiplied by itself (9) to get the final answer: 1,369.

The trick is to choose the difference so that the multiplication is easy. For example, to square 59, choose a difference of 1. Go up to 60 and down to 58. Multiply 60 times 58 to get 3,480. Multiply 1 by itself to get 1, and add that to 3,480 to get the answer 3,481.

With practice, you can square two-digit numbers in your head faster than you can do it using a calculator. Benjamin uses similar tricks to square three-digit, four-digit, and even five-digit numbers. Imagine multiplying 79,635 by 79,635 in your head.

There are lots of tricks you can use for quick mental arithmetic. Can you find any other magical shortcuts?

Source: Science News for Kids