A mathematical fallacy is a series of steps which are seemingly correct, but contain a flawed argument to prove an obvious contradiction or absurdity, such as that one equals two. Mathematical fallacies differ from simple mistakes in that there is an element of concealment in the presentation of the proof, perhaps hidden by the algebraic notation being used; not only do they lead to an absurd result, they do so in a deliberately deceptive way.[1]
Example
The following appears to prove that one equals two:
\(\text{Let }a=b:\)
\(a\times{a}=b\times{a}\text{ (Multiply both sides by }a\text)\)
\(a^2=ab\)
\(a^2-b^2=ab-b^2\text{ (Subtract }b^2\text{ from both sides)}\)
\((a+b)(a-b)=b(a-b)\)
\(a+b=b\text{ (Divide both sides by }(a-b)\text{)}\)
\(a+a=a\text{ (Since }b=a\text{)}\)
\(2a=a\)
\(2=1\text{ (Divide both sides by }a\text{)}\)
The deception occurs in the fifth step of the “proof”, the division of both sides of the equation by \((a-b)\). As the initial premise is that \(a=b\), then that is a division by zero, which is an undefined operation; \(\frac{0}{0}\neq 1\) .
