Nonnegative square root

For $a\ge0$ , its nonnegative square root is the unique real number $s\ge0$ satisfying $s^2=a$ , denoted $\sqrt a$ . The sign convention matters: if $a>0$ , the equation $x^2=a$ has two real solutions, $\sqrt a$ and $-\sqrt a$ .

Existence from completeness

For $a>0$ , let $S=\{x\ge0:x^2\le a\}$ . This set is nonempty and bounded above, so it has a supremum $s$ . If $s^2<a$ , a sufficiently small positive increment still has square below $a$ , contradicting the upper-bound property. If $s^2>a$ , a sufficiently small decrement is an upper bound of $S$ , contradicting minimality. Thus $s^2=a$ . Uniqueness follows because squaring is strictly increasing on the nonnegative real numbers.

Identities

For $a,b\ge0$ , $\sqrt{ab}=\sqrt a\sqrt b$ , and for real $x$ , $\sqrt{x^2}=|x|$ . The map is smooth for $a>0$ ; differentiability at zero is a separate issue.

References

John K. Hunter, Introduction to Analysis, Chapter 2: Numbers.