When I was but a small lad and not too good at arithmetic, I fear, my father used to ask me these questions:
"What is 2 × 2?" Again, I correctly said, "4."
"Now can you find more numbers that do that?"
After a brief moment of thought, I usually came up with the number "0". Of course, it works, as "0 + 0 = 0 × 0 = 0", but for some reason he usually insisted that I think some more. I don't recall if we ever came to some closure on this matter, but the idea has always stuck in my mind.
Using elementary algebra, it is easy to show that these are the only two whole numbers, or integers, whose sums equal their products. Just solve this quadratic equation:
But what would happen if we were to loosen the restriction that the same number must be used? That is to say, what if we were to allow two different values to be used in this fashion?
Somewhere along my mathematical travels I encountered just such a pair of numbers: 3 and 1.5. These numbers do indeed meet our requirment that their sum is equal to their product.
Then I promptly was shown that there is a wide variety of such number pairs, and an unlimited number of instances of this beautiful pattern. Here are some examples.
Mixed numbers | 1 1/3 and 4 |
Decimals | 1.25 and 5 |
Integers | -9 and 0.9 |
Radicals | sqrt(2) and 2 + sqrt(2) |
Complex Numbers | i and ½ - ½i |
The first two categories, and perhaps the third, should be studied by the upper elementary and middle school student carefully. It is easy to demonstrate that the mixed number and decimal types obey the following structure:
1 1 + ----- and n n-1as in 1 1/3 with 4, or 1 1/4 with 5, and so on. Some of those can be conveniently changed to decimal form and verified, if the fraction produces a terminating decimal.
Later, at a higher grade level, the algebra student should be encouraged to verify that the given expressions truly do have equal sums and products.
Now maybe that was what my father wanted me to learn, but I'll never know.
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