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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.

Keywords

Line Segment Complex Number Elementary Function Trigonometric Function Mapping Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2006

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