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
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