Square root

The square root of a number X is the number that, multiplied by itself, results in X. The symbol for the square root of X is <math>\sqrt{X}</math>. Many computer languages and spreadsheet programs use "sqr(X)" or "sqrt(X)"to express this, as do people when mathematical symbols are not conveniently available.

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.

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, constructed via use the square root of -1, which is labelled i. Thus, the square root of -9 is 3i.

Technically, all numbers have two square roots, one positive and the other negative, since squaring the negative number results in a positive number. Thus, sqr(9)=plus or minus 3, written +/-3; likewise <math>\sqrt{-9} = \pm3i</math>.

The square root of a number can also be denoted as <math>X^{\frac{1}{2}}</math>, or X^(1/2) if superscript is not available. This can readily be seen by the rule of adding powers when multiplying: <math>X^{\frac{1}{2}} * X^{\frac{1}{2}} = X^{\frac{1}{2}+\frac{1}{2}} = X^{1} = X</math>