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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

The common reason to consider notions of multiplicity is to count correctly, without specifying exceptions (for example, double roots counted twice). Hence the expression counted with (sometimes implicit) multiplicity.

When mathematicians wish to ignore multiplicity they will refer to the number of distinct elements of a set.

Multiplicity of a prime factor

In the prime factorization In number theory, the integer factorization problem is the problem of finding a non-trivial divisor of a composite number; for example, given a number like 91, the challenge is to find a number such as 7 which divides it. ...

60 = 2 × 2 × 3 × 5

the multiplicity of the prime factor 2 is 2, while the multiplicity of the prime factors 3 and 5 is 1. Thus, 60 has 4 prime factors, but only 3 distinct prime factors.

Multiplicity of a root of a polynomial

Let F be a field and p(x) be a polynomial in one variable and coefficients in F. An element a ∈ F is called a root of multiplicity k of p(x) if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x). If k = 1, then a is a called a simple root. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

For instance, the polynomial 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 1 is a root of multiplicity 2, and −4 is a 'simple' root (of multiplicity 1).

The discriminant of a polynomial is zero if and only if the polynomial has a multiple root. In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...

Multiplicity of a zero of a function

Let I be an interval of R, let f be a function from I into R or C be a real (resp. complex) function, and let c ∈ I be a zero of f, i.e. a point such that f(c) = 0. The point c is said a zero of multiplicity k of f if there exist a real number l ≠ 0 such that

In a more general setting, let f be a function from an open subset A of a normed vector space E into a normed vector space F, and let c ∈ A be a zero of f, i.e. a point such that f(c) = 0. The point c is said a zero of multiplicity k of f if there exist a real number l ≠ 0 such that In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...

The point c is said a zero of multiplicity ∞ of f if for each k, it holds that

Example 1. Since

0 is a zero of multiplicity 1 for the function sine function.

Example 2. Since

0 is a zero of multiplicity 2 for the function 1 − cos.

Example 3. Consider the function f from R into R such that f(0) = 0 and that f(x) = exp(1 / x²) when x ≠ 0. Then, since

for each k ∈ N

0 is a zero of multiplicity ∞ for the function f.

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 at 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). Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

See also

• Zero (complex analysis)
• Set
• Fundamental theorem of algebra
• Fundamental theorem of arithmetic
• Multiset
• Algebraic multiplicity and geometric multiplicity of an eigenvalue

In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

References

• Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.