Why is this -1=1 proof incorrect?

Question

We have

1=1
1²=1²

Since, (-1)²=1=1², we have
1²=(-1)²

So, taking square root on both sides gives
1=-1.

Obviously, this is wrong, but where is the mistake?

Is it because taking the square root on both sides of 1²=(-1)² should give
+or-1=+or-1, and not 1=-1?

If so, then why is it okay to write:: x²=4 <=> x=+or-2?

Shouldn't it then be:: x²=4 <=> +or-x=+or-2?

Thanks.

Answer 1

The trouble is that the squaring function cannot be inverted "cleanly", because if we have $x^2 = 4$, then there are two possibilities for what $x$ could be (namely $2$ and $-2$). Thus if we have $x^2 = a^2$, then we can remove the two squares, but we must account for the possibility of negative solutions by manually saying $x = \pm a$.

Looking at your particular case: $(1)^2 = (-1)^2$ we can apply the same logic to conclude $$ 1 = \pm(-1) = \pm 1 $$ which is true. $1$ does in fact equal $1$ or $-1$ (in this case we obviously know it's $+1$, but that doesn't mean that the plus-or-minus expression is wrong, just kind of silly).

So the short answer is, you cannot undo the squares without introducing (manually) the plus-or-minus into the expression. After you do that, you may be able to determine which of them it is, but that's your own job.

As for your other question, why isn't it that $x^2 = 4 \implies \pm x = \pm 2$? Well, actually, I guess it does, but it's unnecessary to put the $\pm$ on both sides. You only need to put it on one. I remember myself once wondering why does $\pm$ get placed on only one side? Don't you need to do it to both sides? But this is a misunderstanding of what is going on when we make the logical leap from $x^2 = 4$ to $x = \pm 2$. You might think we are taking the $\pm \sqrt{\ \ } $ of both sides, but that is not really true. $\pm \sqrt{\ \ }$ is not a function, so the usual rule of applying it to both sides doesn't really make sense. Really all we're doing is just remembering that there is always two solutions to a quadratic, and thus we use that principle (and no other) when we deduce that $x^2 = 4 \implies x = \pm 2$.

Answer 2

Claiming $(-1)^2 = (1)^2$ implies $-1 = 1$ is more than just bad arithmetic, it is bad logic.

For instance,

I am not seven feet tall. Angela Merkel is not seven feet tall. Therefore I am Angela Merkel. This is false (but you'll have to take my word on it).

Answer 3

$$x=2 \implies x^2=4 ~ ; ~ x=-2 \implies x^2=4$$

But ,

$$x^2=4 \not \implies x=2 ~~; x^2=4 \not \implies x=-2 $$

And also ;

$$x^2=4 \iff |x|=2 \implies x=\pm2~~ \text{(Both are possible)}$$