False Proof – 1 = 2

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).