# the nominal square root

(suggested new terminology)
definition: the nominal square root of x^2 is x.
definition: the actual square root is, in light of the existence of the nominal square root, a retronym for the (traditional) square root, that is, the actual square root of a nonnegative real real number x is that (unique) nonnegative number y such that x = y^2.
observation: nominal square roots obey the same manipulation-laws as actual square roots. for example, the nominal square root of a quotient is the quotient of the nominal square roots.
observation: for a binomial, the nominal square root and the actual square root are equal.
observation: if a binomial is a perfect square, then it contains at least one numeric literal.
note: during the course of the derivation of the quadratic formula, there is a certain step at which the nominal square root of a certain quotient is taken. the numerator is a binomial, and so we just put it under the square-root sign, but the denominator is a perfect square, and so its nominal square root is not equal to its actual square root. taking the nominal square root instead of the actual square root allows the derivation to go through without having to consider two separate cases of the algebraic sign of the leading coefficient.
keywords: mathematics, numeric literals