Square root

The square root of a number x is a number that, multiplied by itself, results in x. For example, 5 × 5 = 25 so 5 is a square root of 25. Also, -5 × -5 = 25 so 5 is the other square root.

The symbol for the square root of x is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{x}} .

Two square roots

All numbers other than zero have two square roots that differ by a factor of -1:

$3\cdot 3=9$ and $-3\cdot -3=9$

When speaking of "the square root of x", people usually refer to the positive square root. For example:

${\sqrt {9}}=3$

When both square roots are to be considered, in mathematics the convention is to use a sign, pronounced "plus or minus" to signify them:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{9} = \pm 3}

This has important implications when solving equations, in that, if the square root of both sides is taken, one must be careful to allow for both the positive and negative square root. A well known example is completing the square to solve a quadratic equation.

Irrational numbers as square roots of whole numbers

Some square roots are relatively simple whole numbers, for example, 3 the square root of 9. Others are less amenable to expression, such as the square root of 2 (=1.414...), which has been proven to be an irrational number.

Square roots of negative numbers

Negative numbers have square roots that lie outside the real numbers: Multiplying a real number by itself always results in a positive number. The square roots of negative numbers involve what are called imaginary numbers:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1} = i} which for example leads to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{-9} = 3i}

The statements in "Two square roots for each number" also apply to negative numbers.

Alternate expressions and notations

Many computer languages and spreadsheet programs use "sqr(x)" or "sqrt(x)" to describe (or calculate) the square root of x. This is also the common notation used when mathematical symbols are not conveniently available.

The square root of a number can also be denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{\frac{1}{2}}} . This can readily be seen by the rule of adding powers when multiplying:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^{\frac{1}{2}+\frac{1}{2}} = x^{1} = x}

If superscript is not available, the common way of writing this is x^(1/2).