Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation \(i^{2}= -1\). Since no real number satisfies the above equation, i was called an imaginary number by René Descartes. Every complex number can be expressed in the form \(a + bi\), where a and b are real numbers, a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols \(\mathbb C\) or C. Despite the historical nomenclature, "imaginary" numbers are not fictitious: they are no less real mathematically than real numbers, and they are essential to the scientific description of the physical world.

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation \((x+1)^2 = -9\) has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions \(x=-1+3i\) and \(x=-1-3i\).

Addition, subtraction and multiplication of complex numbers are defined, taking advantage of the rule \(i^{2}=-1\), along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse, allowing division by complex numbers other than zero. This makes the complex numbers a field with the real numbers as a subfield. Because of these properties, ⁠\(a + bi = a + ib\)⁠, and which form is written depends upon convention and style considerations.

The complex numbers also form a real vector space of dimension two, with \(\{1,i\}\) as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of \(i\) are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.

The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

A complex number is an expression of the form a + bi, where a and b are real numbers, and i is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, 2 + 3i is a complex number.

For a complex number a + bi, the real number a is called its real part, and the real number b (not the complex number bi) is its imaginary part. The real part of a complex number z is denoted Re(z), \(\mathcal{Re}(z)\), or \(\mathfrak{R}(z)\); the imaginary part is Im(z), \(\mathcal{Im}(z)\), or \(\mathfrak{I}(z)\): for example, \(\operatorname{Re}(2 + 3i) = 2\), \(\operatorname{Im}(2 + 3i) = 3\).

A complex number z can be identified with the ordered pair of real numbers \((\Re (z),\Im (z))\), which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram. The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. It is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i.

The set of all complex numbers is denoted by \(\Complex\) (blackboard bold) or C (upright bold).

In some disciplines such as electromagnetism and electrical engineering, j is used instead of i, as i frequently represents electric current, and complex numbers are written as a + bj or a + jb.

Two complex numbers \(a =x+yi\) and \(b =u+vi\) are added by separately adding their real and imaginary parts. That is to say:

\[a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.\] Similarly, subtraction can be performed as \[a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.\]

The addition can be geometrically visualized as follows: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent.

The product of two complex numbers is computed as follows:

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

For example, \((2-i)(3+4i) = 2 \cdot 3 - ((-1) \cdot 4) + (2 \cdot 4 + (-1) \cdot 3)i = 10 +5i.\) In particular, this includes as a special case the fundamental formula

\(i^2 = i \cdot i = -1.\)

This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number.

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 complex conjugate of the complex number z = x + yi is defined as \(\overline z = x-yi.\) It is also denoted by some authors by \(z^*\). Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: \(\overline{\overline{z}}=z.\) A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division.

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

\(z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2\)

is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root \[|z|=\sqrt{x^2+y^2}.\] By Pythagoras' theorem, \(|z|\) 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 \(|z| = 1\), known as the unit complex numbers. If \(z = x = x + 0i\) is a real number, then \(|z|= |x|\): its absolute value as a complex number and as a real number are equal.

Using the conjugate, the reciprocal of a nonzero complex number \(z = x + yi\) can be computed to be

\[\frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2} = \frac{x - yi}{x^2 + y^2} = \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.\] More generally, the division of an arbitrary complex number \(w = u + vi\) by a non-zero complex number \(z = x + yi\) equals \[\frac{w}{z} = \frac{w\bar{z}}{|z|^2} = \frac{(u + vi)(x - iy)}{x^2 + y^2} = \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.\] This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The argument of z (sometimes called the "phase" φ) is the angle of the radius Oz with the positive real axis, and is written as arg z, expressed in radians in this article. The angle is defined only up to adding integer multiples of \(2\pi\), since a rotation by \(2\pi\) (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval \((-\pi,\pi]\), which is referred to as the principal value. The argument can be computed from the rectangular form x + yi by means of the arctan (inverse tangent) function.

For any complex number z, with absolute value \(r = |z|\) and argument \(\varphi\), the equation

\(z=r(\cos\varphi +i\sin\varphi)\)

holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as \(z = r \operatorname\mathrm{cis} \varphi\). In electronics, one represents a phasor with amplitude r and phase φ in angle notation:\[z = r \angle \varphi .\]

If two complex numbers are given in polar form, i.e., z₁ = r₁(cos φ₁ + i sin φ₁) and z₂ = r₂(cos φ₂ + i sin φ₂), the product and division can be computed as \[z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).\] \[\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right), \text{if }z_2 \ne 0.\] (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 \[(2+i)(3+i)=5+5i.\] Because the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula \[\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right)\] 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 π: \[\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)\]

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: \[z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).\] For example, the first few powers of the imaginary unit i are \(i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots\).

The n nth roots of a complex number z are given by \[z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)\] for 0 ≤ k ≤ n − 1. (Here \(\sqrt[n]r\) is the usual (positive) nth root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values. For any \(z \ne 0\), there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely

\(z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.\)

In general there is no natural way of distinguishing one particular complex nth root of a complex number. (This is in contrast to the roots of a positive real number x, which has a unique positive real n-th root, which is therefore commonly referred to as the n-th root of x.) One refers to this situation by saying that the nth root is an n-valued function of z.

The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients) a₀, ..., a_n, the equation \[a_n z^n + \dotsb + a_1 z + a_0 = 0\] has at least one complex solution z, provided that at least one of the higher coefficients a₁, ..., a_n is nonzero. This property does not hold for the field of rational numbers \(\Q\) (the polynomial x − 2 does not have a rational root, because √2 is not a rational number) nor the real numbers \(\R\) (the polynomial x + 4 does not have a real root, because the square of x is positive for any real number x).

Because of this fact, \(\Complex\) 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.

The field of complex numbers is defined as the (unique) algebraic extension field of the real numbers later in #Abstract algebraic definitions.

The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis ('irreducible case'). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna, though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture". This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term \(\sqrt{81 - 144}\) in his calculations, which today would simplify to \(\sqrt{-63} = 3i\sqrt{7}\). Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced the negative value by its positive \(\sqrt{144 - 81} = 3\sqrt{7}.\)

The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:

Condensed: the full section is in Wikipedia.

While the above concrete definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.

One definition of the complex numbers is that they form a field denoted ⁠\(\C\)⁠ that contains the real field ⁠\(\R\)⁠, and is generated over ⁠\(\R\)⁠ by a distinguished element denoted ⁠\(i\)⁠ such that ⁠\(i^2=-1\)⁠. Equivalently, ⁠\(\C\)⁠ is the splitting field of the polynomial ⁠\(x^2+1\)⁠ over the real field ⁠\(\R\)⁠.

Existence of the field can be established by means of various models. One model is to regard the complex field as the set ⁠\(\R^2\)⁠ equipped with addition and multiplication laws on ordered pairs ⁠\((a,b)\)⁠ and ⁠\((a',b')\)⁠: \[(a,b) + (a',b') = (a+a',b+b'),\quad (a,b)(a',b') = (aa' - bb', ab' + a'b).\] In this model the element ⁠\(i\)⁠ corresponds to the element ⁠\((0,1)\)⁠, and the real numbers are embedded as the set of pairs ⁠\((a,0)\)⁠. Another is the standard construction of a splitting field using a quotient of the polynomial ring ⁠\(\R[X]\)⁠, given below.

Because the field ⁠\(\C\)⁠ has different models, they are not all literally the same mathematical object, but they are all isomorphic, up to a ring isomorphism preserving the real numbers. More precisely, given two models ⁠\(\C_1=\R(i_1)\)⁠ and ⁠\(\C_2=\R(i_2)\)⁠, there is a unique isomorphism ⁠\(\phi:\C_1\to\C_2\)⁠ which is linear over the real numbers and such that ⁠\(\phi(i_1)=i_2\)⁠. More generally, any non-trivial finite extension field of the reals is isomorphic to the complex field. Moreover, the isomorphism is unique, provided it preserves the real subfield, up to the action of the Galois group (complex conjugation).

One approach to \(\C\) is via polynomials, i.e., expressions of the form \[p(X) = a_nX^n+\dotsb+a_1X+a_0,\] where the coefficients a₀, ..., a_n are real numbers. The set of all such polynomials is denoted by \(\R[X]\). Since sums and products of polynomials are again polynomials, this set \(\R[X]\) forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number \(p(i) = a_n i^n + \dotsb + a_1 i + a_0\), i.e., the value obtained by setting \(X = i\). This defines a function

\(\R[X] \to \C\)

This function is surjective since every complex number can be obtained in such a way: the evaluation of a linear polynomial \(a+bX\) at \(X = i\) is \(a+bi\). However, the evaluation of polynomial \(X^2 + 1\) at i is 0, since \(i^2 + 1 = 0.\) This polynomial is irreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts of abstract algebra then imply that the kernel of the above map is an ideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is an isomorphism

\(\R[X] / (X^2 + 1) \stackrel \cong \to \C\)

between the quotient ring and \(\C\). Some authors take this as the definition of \(\C\). This definition expresses \(\C\) as a quadratic algebra.

The field \(\Complex\) is algebraically closed by the fundamental theorem of algebra, and is therefore the algebraic closure of \(\R.\)

Complex numbers a + bi can also be represented by 2 × 2 matrices that have the form \[\begin{pmatrix} a & -b \\ b & \;\; a \end{pmatrix}.\] 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 \[a+ib\mapsto \begin{pmatrix} a & -b \\ b & \;\; a \end{pmatrix}\] 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 polar form representation of complex numbers explicitly gives these matrices as scaled rotation matrices. \[r (\cos \theta + i \sin \theta)\mapsto \begin{pmatrix} r \cos \theta & -r \sin \theta \\ r \sin \theta & \;\; r \cos \theta \end{pmatrix}\] In particular, the case of r = 1, which is \(|a + ib| = \sqrt{a^2+b^2} = 1\), gives (unscaled) rotation matrices.

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).

Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, \(\mathbb{C}\), endowed with the metric \[\operatorname{d}(z_1, z_2) = |z_1 - z_2|\] is a complete metric space, which notably includes the triangle inequality \[|z_1 + z_2| \le |z_1| + |z_2|\] for any two complex numbers z₁ and z₂.