Square root

Builds on Existence of square roots of nonnegative reals

Definition

Let $x$ be a real number with $x \ge 0$ . The square root of $x$ , written $\sqrt{x}$ , is the unique real number $y$ with $y \ge 0$ and $y^2 = x$ .

Remarks

The existence of square roots shows that exactly one such $y$ exists. The function $x \mapsto \sqrt{x}$ on $[0, \infty)$ is increasing, and $\sqrt{a b} = \sqrt{a} \sqrt{b}$ for $a, b \ge 0$ . For each real $x$ , $\sqrt{x^2} = |x|$ , the absolute value of $x$ .

Reading path

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Real numbers object
Set bounded above definition
Supremum definition
Completeness axiom statement
Existence of square roots of nonnegative reals statement
Square root definition

Where this leads

260 pages build on this one.

Tag	`0VB8BFRABH`
URI	`m2://0VB8BFRABH`
Kind	definition
Writing	W1 stated
MSC	26A09
Defines	Square root
Uses	Existence of square roots of nonnegative reals

Square root

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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