Definition:Square Root

Theorem
Let $$x\in \mathbb{R}: x \ge 0$$ be a positive real number.

Then from the definition of root, we have that $$\exists y \in \mathbb{R}: x = y^2$$, and we write $$y = \sqrt x$$.

From Even Powers are Positive, we have that $$y^2 = x \iff \left({-y}\right)^2 = x$$ and so we can also write $$y = \pm \sqrt x$$.

The number $$y = + \sqrt x$$ is called the positive square root of $$x$$, and $$y = - \sqrt x$$ is the negative square root of $$x$$.

Frequently, when written just as "$$\sqrt x$$", the positive one is being referred to by default.

Note also that square roots are so much more commonly used in mathematics than any other sort of root, $$\sqrt x$$ is frequently just called "root $$x$$"