"Proof" that  1 = –1

We use the complex number  i which has the property  i² = –1

we know that  (–1)(–1) = 1

taking the square root of both sides preserves the equality, and we get:

1 = sqrt[1] = sqrt[(–1)(–1)] = sqrt[–1]sqrt[–1] = i·i = i²= –1

hence   1 = –1

Can this be correct? Obviously not. Hint: consider the quadratic polynomial (x²+1).

This apparent "paradox" can be easily explained: see solution. But hold on, before you dismiss it as completely trivial, it does lead to some further rather interesting mathematics, as is indicated at the end of the solution section.