Square root
A square root of a number is a number
such that
. Thus
is a square root of
if and only if
is the square of
.
The square root (or the principle square root) of a number is denoted
. For instance,
. When we consider only positive reals, the square root function is the inverse of the squaring function. However, this does not hold more generally because every positive real has two square roots, one positive and one negative. The notation
is used for the positive square root.
Square roots can also be written in exponent notation, so that is equal to the square root of
. Note that this agrees with all the laws of exponentiation, properly interpreted. For example,
, which is exactly what we would have expected. This notion can also be extended to more general rational, real or complex powers, but some caution is warranted because these do not give functions. In particular, if we require that
always gives the positive square root of a positive real number, then the equation
does not hold. For example, replacing
with
gives
on the left but gives
on the right.