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Benjamin's Theorem and
the Power of Algebra

Arthur Benjamin in the book, Secrets of Mental Math, which he co-authored with Michael Shermer, tell an interesting little story.

When I was thirteen, my algebra teacher did a problem on the board for which the answer was 1082. Not content to stop there, I blurted out, "108 squared is simply 11,664!"

The teacher did the calculation on the board and arrived at the same answer. Looking a bit startled, she said, "Yes, that's right. How did you do it?" So I told her, "I went down 8 to 100 and up 8 ti 116. I then multiplied 116 × 100, which is 11,600, and just added the square of 8, to get 11,664."

She had never seen that method before. I was thrilled. Thoughts of "Benjamin's Theorem" popped into my head. I actually believed I had discovered something new. When I finally ran across this method a few years later in a book by Martin Gardner on recreational math, Mathematical Carnival (1965), it ruined my day! Still, the fact that I had discovered it for myself was very exciting to me.

What would Benjamin's Theorem have been?

Benjamin's Theorem: If an increment β is added and subtracted from a number α and those two numbers multiplied together and the square of the increment added the result is the square of the number α.

Stated verbally it is not obvious that the theorem is true. Stated algebraically it truth is obvious. One could say its proof is trivial or elegant.

Proof:

(α + β) ·(α − β) + β² = (α² − β²) + β² = α²

This illustrates the power of algelbra.

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