Mathematical fallacy

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 1 = 2. 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 1 = 2.

Let $a=b$ :
$a\times{a}=b\times{a}$ (Multiply both sides by $a$ )
$a^2=ab$
$a^2-b^2=ab-b^2$ (Subtract $b^2$ from both sides)
$(a+b)(a-b)=b(a-b)$
$a+b=b$ (Divide both sides by $(a-b)$
$a+a=a$ (Since $b=a$ )
$2a=a$
$2=1$ (Divide both sides by $a$ )

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; $\frac{0}{0} \neq 1$ .

References

p. 9

Works cited

{4928910:AGQUTA58} modern-language-association creator asc 1 0 20838