Why does $\sqrt{x^2}=|x|$?

Question

By convention, we say that: $$\sqrt{x^2}=|x|$$ In fact, the above statement is how we define absolute value.

We would not write $\sqrt{4}=-2$. Although logically it is correct, by convention it is wrong. You have to say $\sqrt{4}=2$ unless the question specifically asks for negative numbers like this: $$-\sqrt{4}=-2$$ Why is this? I suspect it is because back then, square roots were used to calculate distances (e.g. with Pythagoras' theorem) and distances must be positive. Am I correct? Any other reasons why we only define square roots to be positive?

Edit: This entire topic is confusing for me because for example, when you are finding the roots of the function $f(x)=x^2-4$, you would set $f(x)=0$, so now the equation is $0=x^2-4$. This means that $x^2=4$, so $x=\pm\sqrt{4}=\pm 2$. Therefore the roots are $2, \ -2$. But normally we cannot say that $\sqrt{4}=\pm 2$. Hope this clarifies things a bit.

Answer 1

This convention makes the square root of non-negative numbers a well-defined single valued function.

This is the one and only reason behind this convention.

Answer 2

The equation $x^2 = n$ has solutions of $x$ and $-x$. But for convention, the reverse form $\sqrt{x}$ is taken only as the one nearest +1. This is because one needs to ensure that the same root of the first equation is being used throughout a structure. For example, if one wrote that a chord of an octagon is $1+\sqrt{2}$ then one typically does not want -0.414.. but +2.414..

There is a kind of conjucation that is used in geometry, that cycles through the roots of equations, and these rely on the solutions being a matched set. So for example, putting $a+b\sqrt 2$ by $a-b\sqrt 2$ will convert octagons into octagrams, and vice versa. This applies in the higher dimensions as well, as a kind of isomorph.

A similar process exists in the heptagon, where the chords of the heptagon, when cycled through the solutions, give the two stars {7/2} and {7/3}.