Zero Equals One

Paradox

$\ds -20$

$=$

$\ds -20$

$\ds \leadsto \ \ $

$\ds 16 - 36$

$=$

$\ds 25 - 45$

$\ds \leadsto \ \ $

$\ds 4^2 - 9 \times 4$

$=$

$\ds 5^2 - 9 \times 5$

$\ds \leadsto \ \ $

$\ds 4^2 - 9 \times 4 + \dfrac {81} 4$

$=$

$\ds 5^2 - 9 \times 5 + \dfrac {81} 4$

$\ds \leadsto \ \ $

$\ds \paren {4 - \frac 9 2}^2$

$=$

$\ds \paren {5 - \frac 9 2}^2$

$\ds \leadsto \ \ $

$\ds 4 - \frac 9 2$

$=$

$\ds 5 - \frac 9 2$

$\ds \leadsto \ \ $

$\ds 0$

$=$

$\ds 1$

subtracting $4 - \dfrac 9 2$ from both sides

This is a falsidical paradox arising from incorrect reasoning about the nature of square roots.

While this line is correct:

$\paren {4 - \dfrac 9 2}^2 = \paren {5 - \dfrac 9 2}^2$

note that:

$4 - \dfrac 9 2 = -\dfrac 1 2$

but:

$5 - \dfrac 9 2 = \dfrac 1 2$

and the paradox is resolved.

$\blacksquare$