>[!abstract] >The proof that $1 = 2$ is a classic case of a [[Falsidical paradox|falsidical paradox]] in mathematics. In the variant below, the hidden fallacy occurs on line 5 when dividing by $a − b$, which is zero since $a = b$. Since division by zero is undefined, the argument is invalid. >[!example] "Proof" >1. Let _a_ and _b_ be equal, nonzero quantities. >$ \begin{flalign} a = b && \end{flalign} $ >2. Multiply by _a_ >$ \begin{flalign} a^2 = ab && \end{flalign} $ >3. Subtract *b$^2$* >$ \begin{flalign} a^2 - b^2 = ab - b^2 && \end{flalign} $ >4. Factor both sides: the left factors as a difference of squares, the right by extracting b >$ \begin{flalign} (a - b)(a + b) = b(a - b) && \end{flalign} $ >5. Divide out *(a - b)* >$ \begin{flalign} a + b = b && \end{flalign} $ >6. Use the fact that _a_ = _b_ >$ \begin{flalign} b + b = b && \end{flalign} $ >7. Combine like terms on the left >$ \begin{flalign} 2b = b && \end{flalign} $ >8. Divide by *b* (allowed because *b* is non-zero) >$ \begin{flalign} 2 = 1 && \end{flalign} $ >[!related] >- **North** (upstream): [[Falsidical paradox]] >- **West** (similar): — >- **East** (different): [[Howler]] >- **South** (downstream): —