Learning to Multiply

Events during 1954.

During the Fourth Grade, when I was eight years old, I paid attention to the Teacher (Miss Trambarger) when she said interesting things, but mostly I spent most of the class time in creating a very large multiplication table. This was in 1954, far before computers, so I had hand-calculated each entry in a table which took many dozens of sheets of paper. I learned many interesting patterns, which later became very useful for me.

Within the giant multiplication table that I had created, I first noticed that the SQUARES were interesting. I noticed that memorizing just the first 25 of them made it very easy to know the square of nearly any number! So I memorized 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625.

Those are the squares of 1 through 25. But it turns out that the squares of 26 to 50 have the EXACT SAME last two digits (just with different hundreds numbers. So 26 squared is 676, 27 squared is 729, 28 squared is 784, and so on. Continuing to 47 squared being 2209, 48 squared being 2304, 49 squared being 2401, and 50 squared being 2500.

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This continues, forever! So, without having to have memorized it, or actually multiplying it out, we immediately know that 103 squared is 10609.

Determining the hundreds numbers is also a pattern, but I will let any readers to try to work that out for themselves.

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Further analysis of my multiplication table (as an eight-year-old), I noticed additional value in it. Say you want to know the product of 61 times 67. That is not that difficult a multiplication problem, but I am using it here as an example of how this works. There are two steps involved, and it only works if both numbers are either even or odd, and it is easiest if they are fairly similar numbers. For this example, we notice that there is a MIDDLE number which is 64. Using the method described above, we quickly know that 64 squared is 4096. Now we notice that each of our numbers are THREE away from that middle number. We square that difference, of 3, which is 9. To get the problem's answer, we just subtract 9 from 4096, which gives us the correct answer of 4087.

For an even simpler example, if we did not know what 7 times 9 was, we know from above that 8 squared is 64. We then subtract 1 squared (or 1) to get 63, the right answer. Similarly, for 9 times 5, we need 7 squared or 49 and subtract 2 squared or 4, and we see that the correct answer is 45.

It turns out that there are occasional multiplication problems where this is quickly helpful. Say you need to know 73 times 77. That is just 5625 minus 4 or 5621. Or 143 times 137. That is just 19600 minus 9 or 19591.

When you immediately give such correct answers, people think you are really smart, as though you actually KNOW all the possible answers! But you don't! You just know how to use some PATTERNS which exist in the way multiplication works.

However, this approach does NOT always work, and even when it does, the standard multiplying might be quicker and easier. If one number is odd and the other is even, it actually technically still works, but it is time-consuming to do! Say you want to know what 5 times 8 is. The middle number here is 6.5 and you then have to know that 6.5 squared is 42.25. Then the DIFFERENCE is 1.5, and that squared is 2.25. So the correct answer is 42.25 - 2,25 or 40. So even though it works for odd and even, is is often not worth the bother!

As to another common situation when it is not worth doing, if the two numbers are very far apart, it again is sometimes not worth doing this way. Consider 91 times 3. The middle number is 47, and 47 squared is 2209. The DIFFERENCE is 88/2 or 44, and that number squared is 1936. Then 2209 - 1936 gives 273, which is the correct answer, but it took several complicated steps to get an answer that you probably could have simply done by easy multiplication!


This presentation was first placed on the Internet in April 2012.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago