This article is about the mathematical term; Multiplicity is also the title of a 1996 film.

In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. For example, the term is used to refer to the value of the totient valence function , or the number of times a given polynomial equation has a root at a given point.

Multiplicity of a prime factor

In the prime factorization

60 = 2 × 2 × 3 × 5

the multiplicity of the prime factor 2 is 2; the multiplicity of the prime factor 3 is 1; and the multiplicity of the prime factor 5 is 1.

Multiplicity of a root of a polynomial

A real or complex number a is called a root of multiplicity k of a polynomial p if there exists a polynomial s with:

and

p(x) = (x − a)^ks(x).

If k = 1, then a is a simple root.

Example

The following polynomial p:

p(x) = x³ + 2x² − 7x + 4

has 1 and −4 as roots, and can be written as:

p(x) = (x + 4)(x − 1)²

This means that x = 1 is a root of multiplicity 2, and x = −4 is a 'simple' root (multiplicity 1).

In complex analysis

Let z₀ be a root of a holomorphic function f, and let n be the least positive integer m such that, the m-th derivative of f evaluated in z = z₀ differs from zero:

Then the power series of f about z₀ begins with the nth term, and f is said to have a root of multiplicity (or "order") n. If n = 1, the root is called a simple root (Krantz 1999, p. 70).

See also

• Collection
• Fundamental theorem of algebra
• Multiset

External link

"Multiplicity" on MathWorld