Definition:Square Root

Definition

Positive Reals

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 or principal 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 $x$"

Negative Reals

Let $x \in \R$ be a real number.

Then the principal square root of $x$ is defined as:

$\sqrt x = \begin{cases} +\sqrt x & : x \ge 0 \\ i \left( {+\sqrt {\left( {-x} \right)}}\right) & : x < 0 \end{cases}$

where $i$ is the imaginary unit and $i^2 = -1$.

Hence we have $\sqrt{-1}=i$.

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:

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

This sequence is A002193 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

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

This sequence is A002194 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

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

This sequence is A002163 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

• One Equals Minus One

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $1.3, \ 1.4, \ 1.5$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.9$
• For a video presentation of the contents of this page, visit the Khan Academy.