In
mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
and
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
it holds that
:
where is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and ...
(). The formula is named after
Abraham de Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He move ...
, although he never stated it in his works. The expression is sometimes abbreviated to .
The formula is important because it connects
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
and
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that is real, it is possible to derive useful expressions for and in terms of and .
As written, the formula is not valid for non-integer powers . However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the th
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
, that is,
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
such that .
Example
For
and
, de Moivre's formula asserts that
or equivalently that
In this example, it is easy to check the validity of the equation by multiplying out the left side.
Relation to Euler's formula
De Moivre's formula is a precursor to
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
which establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
One can derive de Moivre's formula using Euler's formula and the
exponential law
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
for integer powers
:
since Euler's formula implies that the left side is equal to
while the right side is equal to
:
Proof by induction
The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer , call the following statement :
:
For , we proceed by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
. is clearly true. For our hypothesis, we assume is true for some natural . That is, we assume
:
Now, considering :
:
See
angle sum and difference identities.
We deduce that implies . By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, is clearly true since . Finally, for the negative integer cases, we consider an exponent of for natural .
:
The equation (*) is a result of the identity
:
for . Hence, holds for all integers .
Formulae for cosine and sine individually
For an equality of
complex numbers, one necessarily has equality both of the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s and of the
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of both members of the equation. If , and therefore also and , are
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s, then the identity of these parts can be written using
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...
s. This formula was given by 16th century French mathematician
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
:
:
In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact valid even for complex values of , because both sides are
entire
Entire may refer to:
* Entire function, a function that is holomorphic on the whole complex plane
* Entire (animal)
Neutering, from the Latin ''neuter'' ('of neither sex'), is the removal of an animal's reproductive organ, either all of it or a c ...
(that is,
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
on the whole
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
) functions of , and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for and :
:
The right-hand side of the formula for is in fact the value of the
Chebyshev polynomial
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshe ...
at .
Failure for non-integer powers, and generalization
De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power . If a complex number is raised to a non-integer power, the result is
multiple-valued
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
(see
failure of power and logarithm identities). For example, when , de Moivre's formula gives the following results:
:for the formula gives , and
:for the formula gives .
This assigns two different values for the same expression , so the formula is not consistent in this case.
On the other hand, the values 1 and −1 are both square roots of 1. More generally, if and are complex numbers, then
:
is multi-valued while
:
is not. However, it is always the case that
:
is one of the values of
:
Roots of complex numbers
A modest extension of the version of de Moivre's formula given in this article can be used to find
the th roots of a complex number (equivalently, the power of ).
If is a complex number, written in
polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
as
:
then the th roots of are given by
:
where varies over the integer values from 0 to .
This formula is also sometimes known as de Moivre's formula.
Analogues in other settings
Hyperbolic trigonometry
Since , an analog to de Moivre's formula also applies to the
hyperbolic trigonometry. For all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s ,
If is a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
(but not necessarily an integer), then will be one of the values of .
Extension to complex numbers
The formula holds for any complex number
:
where
:
Quaternions
To find the roots of a
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quate ...
there is an analogous form of de Moivre's formula. A quaternion in the form
:
can be represented in the form
:
In this representation,
:
and the trigonometric functions are defined as
:
In the case that ,
:
that is, the unit vector. This leads to the variation of De Moivre's formula:
:
Example
To find the
cube root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
s of
:
write the quaternion in the form
:
Then the cube roots are given by:
:
matrices
Consider the following matrix
. Then
. This fact (although it can be proven in the very same way as for complex numbers) is a direct consequence of the fact that the space of matrices of type
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
.
References
*.
External links
De Moivre's Theorem for Trig Identitiesby Michael Croucher,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
{{Spoken Wikipedia, date=2021-06-05, En-De Moivres Formula-article.ogg
Theorems in complex analysis
Articles containing proofs