# Dictionary Definition

multiplicity

### Noun

1 the property of being multiple

2 a large number [syn: numerousness, numerosity]

# User Contributed Dictionary

## English

### Noun

#### Translations

- Italian: molteplicità (alla)

# Extensive Definition

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 number of times a given polynomial
equation has a root at a given point.

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- 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 called a simple root.

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.

### Geometric behavior

Let f(x) be a polynomial function. Then, if
f is graphed on a
Cartesian coordinate system, its graph will cross the x-axis at
real zeros of odd multiplicity and will not cross the x-axis at
real zeros of even multiplicity. In addition, if f(x) has a zero
with a multiplicity greater than 1, the graph will be tangent to
the x-axis and will have slope 0.

## 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 \ell\neq 0 such that

- \lim_\frac=\ell.

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 \in 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 \ell \neq
0 such that

- \lim_\frac=\ell.

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

- \lim_\frac=0.

Example 1. Since

- \lim_\frac=1,

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

Example 2. Since

- \lim_\frac=\frac 12,

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/x^2) when x \neq 0. Then,
since

- \lim_\frac=0 \mboxk \in \mathbb

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

## In complex analysis

Let z_0 be a root of a holomorphic function f, and let n be the least positive integer such that, the nth derivative of f evaluated at z_0 differs from zero. Then the power series of f about z_0 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).We can also define the multiplicity of the zeroes
and poles of a meromorphic
function thus: If we have a meromorphic function f = \dfrac,
take the Taylor
expansions of g and h about a point z0, and find the first
non-zero term in each (denote the term numbers m and n
respectively). if m = n, then the point has non-zero value. If m
> n, then the point is a zero of multiplicity m - n. If m <
n, then the point has a pole of multiplicity n - m.

## See also

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

## References

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

multiplicity in German: Vielfachheit

multiplicity in Esperanto: Obleco

multiplicity in Spanish: Multiplicidad

multiplicity in Dutch: Meervoudig nulpunt van
een polynoom

multiplicity in Japanese: 重根 (多項式)

multiplicity in Swedish:
Multiplicitet

# Synonyms, Antonyms and Related Words

Proteus, abundance, allotropism, allotropy, barrel, countlessness, diversification,
diversity, great deal,
her infinite variety, heterogeneity, heteromorphism, infinitude, innumerability, lashings, lot, manifoldness, manyness, mass, mess, much, multifariousness,
multifoldness,
multiformity,
multitudinousness,
nonuniformity,
numerousness,
omnifariousness,
omniformity,
peck, plenty, polymorphism, power, profuseness, profusion, rifeness, shapeshifter, sight, swarmingness, teemingness, variation, variegation, variety, variousness