How can I prove that I need to take the same root of a number?

Question

For example : $$1=1$$ $$\sqrt1 = \sqrt1$$ $$-1 = 1$$ Since $-1$ and $1$ are both solutions to $\sqrt1$. I'm not sure where the mistake is. Any ideas?

Related: https://math.stackexchange.com/questions/2972378/why-for-real-number-we-have-just-one-square-root-whereas-for-complex-number-we-h/ — eyeballfrog, Mar 15 '22 at 17:09
Huh? Note that $\sqrt{x}$ is a well defined function it maps each positive number $x$, to the unique positive number $y$ which satisfies $y^2=x$. — Vivaan Daga, Mar 15 '22 at 17:11
Is that the definition of √a? Because isn't the whole point that there are negative and positive solutions? I didn't know that y in y^2 = x had to be positive. — SalvorHardin, Mar 15 '22 at 17:15
The point here is that indeed $x^2 =y$ has two solutions, $\sqrt y$ and $-\sqrt y$. — Cristian Gratie, Mar 15 '22 at 17:27
A site search for square root positive negative yields many duplicate questions. — Blue, Mar 15 '22 at 17:29
You could look at it this way: (Ignoring zero for this discussion ...) If the square root symbol were inherently ambiguous, so that it could be interpreted to mean either a positive or negative value, then we wouldn't need to write $\pm$ in front of it so often. That is, when we solve $p^2=3$ as $p=\pm\sqrt{3}$, we need the $\pm$ precisely because $\sqrt{3}$ alone means the positive value but the equation also admits the negative value as a solution. — Blue, Mar 15 '22 at 17:39
Does this answer your question? Square roots -- positive and negative — Hans Lundmark, Mar 15 '22 at 20:27

score 2 · Accepted Answer · answered Mar 15 '22 at 17:26

Quoted from a comment:

isn't the whole point that there are negative and positive solutions?

As comments have noted, if $x\ge0$ we define $\sqrt{x}$ not as an arbitrary $y$ with $y^2=x$, but as the unique such $y$ with $y\ge0$. This allows us to distinguish one such $y$, namely $\sqrt{x}$, from another, $-\sqrt{x}$ (well, technically they're the same if $x=0$; that's just the same root twice). So while both roots exist, we can unambiguously give them different labels. So while $1,\,-1$ both square to $1$, they're not both $\sqrt{1}$: one is $-\sqrt{1}$ instead.

How can I prove that I need to take the same root of a number?

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