Complex numbers a + bi can also be represented by 2 × 2 matrices that have the form

`{\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.}`

gives

Here the entries a and b are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of 2 × 2 matrices.

A simple computation shows that the map

`{\displaystyle a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}`

gives

is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.

The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector (x, y) corresponds to the multiplication of x + iy by a + ib. In particular, if the determinant is 1, there is a real number t such that the matrix has the form

`{\displaystyle {\begin{pmatrix}\cos t&-\sin t\\\sin t&\;\;\cos t\end{pmatrix}}.}`

gives

In this case, the action of the matrix on vectors and the multiplication by the complex number

`{\displaystyle \cos t+i\sin t}`

gives

are both the rotation of the angle t.]]>

The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients)

`a_0, ..., a_n`

gives

, the equation`{\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0}`

gives

has at least one complex solution z, provided that at least one of the higher coefficients

`a_1, ..., a_n`

gives

is nonzero. This property does not hold for the field of rational numbers

`{\displaystyle \mathbb {Q} }`

gives

(the polynomial

`x^2 - 2`

gives

does not have a rational root, because`\sqrt{2}`

gives

is not a rational number) nor the real numbers`{\displaystyle \mathbb {R} }`

gives

(the polynomial`x^2 + 4`

gives

does not have a real root, because the square of x is positive for any real number x).Because of this fact,

`{\displaystyle \mathbb {C} }`

gives

is called an algebraically closed field. It is a cornerstone of various applications of complex numbers, as is detailed further below. There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.]]>For any complex number z, with absolute value

`{\displaystyle r=|z|}`

gives

and argument

`{\displaystyle \varphi }`

gives

, the equation`{\displaystyle z=r(\cos \varphi +i\sin \varphi )}`

gives

holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as

`{\textstyle z=r\operatorname {\mathrm {cis} } \varphi }.`

gives

In electronics, one represents a phasor with amplitude r and phase

`\varphi`

gives

in angle notation:`{\displaystyle z=r\angle \varphi .}`

gives

If two complex numbers are given in polar form, the product and division can be computed as

`{\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).}`

gives

if

`{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.}`

gives

(These are a consequence of the trigonometric identities for the sine and cosine function.) In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of

`{\displaystyle (2+i)(3+i)=5+5i.}`

gives

Because the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees,

or

`\frac{\pi}{4}}`

gives

.Thus, the formula

`{\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)}`

gives

holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of

`\pi`

gives

]]>In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:

`{\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .}`

gives

Euler's formula relates the complex exponential function of an imaginary argument, which can be thought of as describing uniform circular motion in the complex plane, to the cosine and sine functions, geometrically its projections onto the real and imaginary axes, respectively.

In 1748, Euler went further and obtained Euler's formula of complex analysis:

`{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }`

gives

by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's *A Treatise of Algebra.*

The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.

Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

]]>The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:

`{\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).}`

gives

For example, the first few powers of the imaginary unit i are

`{\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots }`

gives

The n nth roots of a complex number z are given by

`{\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}`

gives

for

`0 \leq k ≤ n - 1`

gives

]]>De Moivre's formula is a precursor to Euler's formula

`{\displaystyle e^{ix}=\cos x+i\sin x,}`

gives

with x expressed in radians rather than degrees, 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 for integer powers

`{\displaystyle \left(e^{ix}\right)^{n}=e^{inx},}`

gives

since Euler's formula implies that the left side is equal to

`{\displaystyle \left(\cos x+i\sin x\right)^{n}}`

gives

while the right side is equal to

`{\displaystyle \cos nx+i\sin nx.}`

gives

**Example**

For

`{\displaystyle x=30^{\circ }}`

gives

and

`{\displaystyle n=2},`

gives

,de Moivre's formula asserts that

`{\displaystyle \left(\cos(30^{\circ })+i\sin(30^{\circ })\right)^{2}=\cos(2\cdot 30^{\circ })+i\sin(2\cdot 30^{\circ }),}`

gives

or equivalently that

`{\displaystyle \left({\frac {\sqrt {3}}{2}}+{\frac {i}{2}}\right)^{2}={\frac {1}{2}}+{\frac {i{\sqrt {3}}}{2}}.}`

gives

In this example, it is easy to check the validity of the equation by multiplying out the left side.

]]>In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it holds that

`{\displaystyle {\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,}`

gives

where

i is the imaginary unit

`(i^2 = -1)`

gives

.The formula is named after Abraham de Moivre, although he never stated it in his works. The expression cos x + i sin x is sometimes abbreviated to cis x.

The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x.

As written, the formula is not valid for non-integer powers n. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers z such that

`z^n = 1.`

gives

Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when x is an arbitrary complex number.

]]>More generally, the division of an arbitrary complex number

`{\displaystyle w=u+vi}`

gives

by a non-zero complex number

`{\displaystyle z=x+yi}`

gives

equals

`{\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.}`

gives

Using the conjugate, the reciprocal of a nonzero complex number

`{\displaystyle z=x+yi}`

gives

can be computed to be

`{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}`

gives

Using the conjugate, the reciprocal of a nonzero complex number

`{\displaystyle z=x+yi}`

gives

can be computed to be

`{\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}`

gives

For any complex number z = x + yi , the product

`{\displaystyle z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}}`

gives

is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root

`{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}`

gives

By Pythagoras' theorem,

`{\displaystyle |z|}`

gives

is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that

`{\displaystyle |z|=1}.`

gives

If

`{\displaystyle z=x=x+0i}`

gives

is a real number, then

`{\displaystyle |z|=|x|}`

gives

: its absolute value as a complex number and as a real number are equal.]]>

The product of two complex numbers is computed as follows:

`{\displaystyle (a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.}`

gives

For example,

`{\displaystyle (3+2i)(4-i)=3\cdot 4-(2\cdot (-1))+(3\cdot (-1)+2\cdot 4)i=14+5i.}`

gives

In particular, this includes as a special case the fundamental formula

`{\displaystyle i^{2}=i\cdot i=-1.}`

gives

This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number x always satisfies

`{\displaystyle x^{2}\geq 0}`

gives

.This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number x always satisfies

`{\displaystyle x^{2}\geq 0}`

gives

.With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.

]]>The multiplication of complex numbers is slightly different from the multiplication of natural numbers. Here we need to use the formula of

`i^2 = -1`

gives

.For the two complex numbers

`z_1 = a + ib, z_2 = c + id`

gives

the product is

`z_1 \cdot z_2 = (ca - bd) + i(ad + bc)`

gives

.The multiplication of complex numbers is polar form is slightly different from the above mentioned form of multiplication. Here the absolute values of the two complex numbers are multiplied and their arguments are added to obtain the product of the complex numbers.

]]>**Commutative Law: For two complex numbers**

`z_1,z_2`

gives

is`z_1 + z_2 = z_2 + z_1`

gives

.Associative Law: For the given three complex numbers

`z_1, z_2, z_3`

gives

we have

`z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3`

gives

Additive Identity: For a complex number

`z = a + ib`

gives

, there exists`0 = 0 + i0`

gives

, such that`z + 0 = 0 + z = 0`

gives

.Additive Inverse: For the complex number

`z = a + ib`

gives

, there exists a complex number`-z = -a -ib`

gives

such that`z + (-z) = (-z) + z = 0`

gives

Here -z is the additive inverse.

**Subtraction of Complex Numbers**

The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part. For the complex numbers

`z_1 = a + ib, z_2 = c + id`

gives

we have

`z_1 - z_2 = (a - c) + i(b - d)`

gives

.]]>The various operations of addition, subtraction, multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows.

**Addition of Complex Numbers**

Th addition of complex numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added to the real part and the imaginary part is added to the imaginary part. For two complex numbers of the form

`z_1 = a + id`

gives

and

`z_2 = c + id`

gives

,the sum of complex numbers

`z_1 + z_2 = (a + c) + i(b + d)`

gives

.The complex numbers follow all the following properties of addition.

Closure Law: The sum of two complex numbers is also a complex number. For two complex numbers

`z_1 \text {and} z_2`

gives

,the sum of

`z_1 + z_2`

gives

is also a complex number.]]>