What is "monomial"

Monomial \Mo*no"mi*al\, n. [See Monome, Binomial.] (Alg.) A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.

Monomial \Mo*no"mi*al\, a. (Alg.) Consisting of but a single term or expression.

a. Relative to a polynomial consisting of one term. n. (context mathematics English) A single term consisting of a product of numbers and variables with positive integer exponents.

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two different definitions of a monomial may be encountered:

• For the first definition, a monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. The constant 1 is a monomial, being equal to the empty product and for any variable . If only a single variable is considered, this means that a monomial is either 1 or a power of , with a positive integer. If several variables are considered, say, x, y, z,  then each can be given an exponent, so that any monomial is of the form xyz with a, b, c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1).
• For the second definition, a monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is also a monomial in the second sense, because the multiplication by 1 is allowed. For example, in this interpretation  − 7x and (3 − 4i)xyz are monomials (in the second example, the variables are x, y, z,  and the coefficient is a complex number).

In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers.

Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope of "mononomial".