Definition:Square Root/Positive Real

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Definition

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

The square roots of $x$ are the real numbers defined as:

$x^{\paren {1 / 2} } := \set {y \in \R: y^2 = x}$

where $x^{\paren {1 / 2} }$ is the $2$nd root of $x$.


The notation:

$y = \pm \sqrt x$

is usually encountered.


From Existence of Square Roots of Positive Real Number, we have that:

$y^2 = x \iff \paren {-y}^2 = x$

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


Also see


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


Sources