Definition:Square Root

Definition
Let $$x \in \R: x \ge 0$$ be a positive real number.

Then from the definition of root, we have that $$\exists y \in \R: x = y^2$$, and we write $$y = \sqrt x$$.

From Even Powers are Positive, we have that $$y^2 = x \iff \left({-y}\right)^2 = x$$ and so we can also write $$y = \pm \sqrt x$$.

The number $$y = + \sqrt x$$ is called the positive square root of $$x$$, and $$y = - \sqrt x$$ is the negative square root of $$x$$.

Frequently, when written just as "$$\sqrt x$$", the positive one is being referred to by default.

Note also that square roots are so much more commonly used in mathematics than any other sort of root, $$\sqrt x$$ is frequently just called "root $$x$$"

Square roots of primes
The square root of any prime is irrational, so can not be expressed precisely by a rational fraction.

The decimal expansions of the first few primes are:


 * $$\sqrt 2 \approx 1.41421 \ 35623 \ 73095 \ 0488 \ldots$$


 * $$\sqrt 3 \approx 1.73205 \ 08075 \ 68877 \ 2935 \ldots$$


 * $$\sqrt 5 \approx 2.23606 \ 79774 \ 99789 \ 6964 \ldots$$