A Fast Method for Multiplying 2-Digit Numbers
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This course shows how to quickly multiply two-digit numbers without having to memorize thousands of possibilities.
Introduction
Most people can instantly answer "what is A times B?" when A and B are both in the range 1-9. This is because they have had the 1-9 multiplication table memorized since elementary school. People often encounter a situation where two-digit numbers need to be multiplied together. Essentially no one has the 11-99 multiplication table memorized (that's 89*88/2+89 = 4005 different combinations to memorize!), so we all rely on some heuristic, or our eyes glaze over, instead of doing the multiplication. Heuristics, like rounding to nice numbers before multiplying, are at times sufficient. But sometimes an exact answer is needed. This course provides a fast, exact method for multiplying any two 1- or 2-digit numbers. It is hardly slower than having the 11-99 multiplication table memorized.
The Theory
Our problem is to quickly find X = A*B where A and B are two 2-digit numbers and, without loss of generality, A >= B. With some creativity, we see that it is possible to rewrite A*B the difference of two perfect squares when their difference is even: A*B = (C + Y)*(C - Y) = C*C - Y*Y. Imagine if we had all 89 perfect squares of two-digit numbers greater than 10 memorized. Then finding A*B is as simple as doing two subtractions (finding Y = (A-B)/2 and finally C*C - Y*Y), which, with not much practice, can be whittled down to no more than a second or two of calculation.
Dealing with Odd Differences
Suppose A - B is odd. Then Y is not an integer in our equation X = A*B = (C + Y)*(C - Y) and we do not have Y*Y memorized. One way to deal with this is to use the "wishful thinking" problem solving strategy. We want A - B to be even, but it's not. Well, we can make the difference even anyways! Let's replace A or B, whichever feels easiest, with one more or less than A or B, respectively, to make the difference even. For example, let's add a B to both sides of our original equation, X = A*B. Now, we have X + B = (A+1)*B. We know how to quickly solve (A+1)*B, since A+1 - B must be even. All we have to do is subtract B from the result to get back to X! Can you think of another, better way to deal with odd differences?
Putting It All Together
We now know how to quickly find an exact solution to A*B = X where A and B are 1- or 2-digit numbers. To use this knowledge in practice, however, we must memorize the 89 perfect squares of two-digit numbers greater than 10.
Vocabulary Set: The Perfect Squares 10² through 99²
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