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:

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

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

• Existence of Square Roots of Positive Real Number

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

• Definition:Complex Square Root

• Even Power is Non-Negative

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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): square root