# 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

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.9$: Roots - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates: Functions