Definition:Square Root/Positive Real

Definition
Let $x \in \R_{\ge 0}$ be a positive real number.

The square roots of $x$ are the real numbers defined as:
 * $x^{\left({1 / 2}\right)} := \left\{{y \in \R: y^2 = x}\right\}$

where $x^{\left({1 / 2}\right)}$ is the $2$nd root of $x$.

From Existence of Square Roots of Positive Real Number, we have that:
 * $y^2 = x \iff \left({-y}\right)^2 = x$

That is, for each (strictly) positive real number $x$ there exist exactly $2$ square roots of $x$.

Notation
For $y^2 = x$, the notation commonly used is:
 * $y = \pm \sqrt x$

Also see

 * Existence of Square Roots of Positive Real Number

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


 * Definition:Positive Square Root
 * Definition:Square Root of Negative Number


 * Definition:Complex Square Root


 * Even Powers are Positive