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 .
Two square roots for each number
All numbers have two square roots that differ by a factor of -1:
These two square roots don't have to be distinct. For example, the square root of zero has two values: +0 and -0. (While these may seem to be identical, in the mathematics of limits, they can have differing characteristics.)
When speaking of "the square root of x", people usually refer to the positive square root. For example:
Since in many situations only the positive square root has any meaning, to be clear that both square roots are to be considered, in mathematics the convention is to use a sign, pronounced "plus or minus" to signify the distinct solutions:
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:
- which for example leads to
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 . This can readily be seen by the rule of adding powers when multiplying:
If superscript is not available, the common way of writing this is x^(1/2).