False Proof – 1 = 2 | Math ∩ Programming

This is the first in a series of “false proofs.” Despite their falsity, they will be part of the Proof Gallery. The reason for putting them there is that often times a false proof gives insight into the nature of the problem domain. We will be careful to choose problems which do so.

Problem: Show 1 = 2.

“Solution”: Let $a=b \neq 0$ . Then $a^2 = ab$ , and $a^2 - b^2 = ab - b^2$ . Factoring gives us $(a+b)(a-b) = b(a-b)$ . Canceling both sides, we have $a+b = b$ , but remember that $a = b$ , so $2b = b$ . Since $b$ is nonzero, we may divide both sides to obtain $2=1$ , as desired.

Explanation: This statement, had we actually proved it, would imply that all numbers are equal, since subtracting 1 from both sides gives $0=1$ and hence $a=0$ for all real numbers $a$ . Obviously this is ridiculous.

Digging into the algebraic mess, we see that the division by $a-b$ is invalid, because $a=b$ and hence $a-b = 0$ .

Division by zero, although meaningless, is nevertheless interesting to think about. Much advanced mathematics deals with it on a very deep and fundamental level, either by extending the number system to include such values as $\frac{1}{0}$ (which still gives rise to other problems, such as $\frac{0}{0}$ and $0 \cdot \infty$ ), or by sidestepping the problem by inventing “pseudo” operations (linear algebra) and limiting calculations (calculus).

Math ∩ Programming

False Proof – 1 = 2

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