Elementary Functions

Abstract

Many high school students are puzzled by the following “proof”: Let a = b. Then

$$ a^2 = ab,{\mathbf{ }}a^2 - b^2 = ab - b^2 ,{\mathbf{ }}and{\mathbf{ }}(a + b)(a - b) = b(a - b). $$

Dividing by a − b, we have

$$ a + b = b,{\mathbf{ }}2b = b,{\mathbf{ }}and{\mathbf{ }}2 = 1. $$

The reader, of course, is not fooled by the invalid division by zero. So let us produce an absurdity without dividing by zero. Since 1/(−1) = (−1)/1, we take square roots to obtain

$$ \sqrt {(1/ - 1)} = \sqrt {( - 1/1)} ,{\mathbf{ }}\sqrt 1 /\sqrt { - 1} = \sqrt { - 1} /\sqrt 1 ,{\mathbf{ }}and{\mathbf{ }}1/i = i/1. $$

Cross multiplying, we have 1² = i² or 1 = −1.