Square Root - Properties

Properties

The principal square root function (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.

The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers).

For all real numbers x

\sqrt{x^2} = \left|x\right| = \begin{cases} x, & \mbox{if }x \ge 0 \\ -x, & \mbox{if }x < 0. \end{cases} (see absolute value)

For all non-negative real numbers x and y,

and

The square root function is continuous for all non-negative x and differentiable for all positive x. If f denotes the square-root function, its derivative is given by:

The Taylor series of √1 + x about x = 0 converges for |x| ≤ 1 and is given by

which is a special case of a binomial series.

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