Square root

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The square root of a number x is a number that, multiplied by itself, results in x. In geometry, the positive square root is the length of one side of a square. Multiply the length by itself, and the product is the area of the square.

For example, 5 × 5 = 25 so 5 is a square root of 25. Also, -5 × -5 = 25 so -5 is the other square root. [1]

The usual symbol for the square root of x is \sqrt{x}. If you need both roots, use \pm \sqrt{x}.


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:

\pm \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.

Contents

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:

\sqrt{-1} = i which for example leads to \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 x^{\frac{1}{2}}. This can readily be seen by the rule of adding powers when multiplying:

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).

Notes

  1. "Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3, since (-3)2 = (+3)2 = 9." Wolfram Research - Eric Weisstein
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