>[!abstract] >Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a howler. The following is an example of a howler involving anomalous cancellation: >$ \begin{flalign} \frac{16}{64} = \frac{1\cancel6}{\cancel64} = \frac{1}{4} && \end{flalign} $ >Here, although the conclusion 1/4 is correct, there is a fallacious, invalid cancellation in the middle step (Wikipedia, 2025). >[!abstract] Disguised division by zero example >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} $ > >The fallacy is in line 5: the progression from line 4 to line 5 involves division by *a − b*, which is zero since *a = b*. Since division by zero is undefined, the argument is invalid. >[!related] >- **North** (upstream): [[Gettier problem]] >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —