In mathematics, a complex number is a number of the form , where and are real numbers, and is the imaginary unit, a number that satisfies the equation ; the number is called the of , and is called the .
The complex numbers form a number system which extends the real numbers and supports the same basic arithmetic operations of addition, subtraction, multiplication, and division, satisfying the same associative, commutative, and distributive laws. This makes the complex numbers a field. The set of complex numbers is denoted by either of the symbols or .
Because any nonzero real number, whether positive or negative, squares to a positive number, taking the square root of a negative number was historically considered impossible. When, in the 16th century, such square roots were found helpful for determining the real solutions of cubic equations, they were originally considered a mere symbolic contrivance, and in the 17th century were given the name imaginary numbers. Despite the historical nomenclature,For an extensive account of the history of "imaginary" numbers, from initial skepticism to ultimate acceptance, see "imaginary" numbers are not fictitious: they are no less valid mathematically than real numbers, and they are essential to the scientific description of the physical world."Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.", .
Complex numbers allow solutions to all non-constant polynomial equations with real or complex coefficients, even those that have no solutions in real numbers, a result known as the fundamental theorem of algebra. The complex numbers also form a real vector space of dimension two, with as a standard basis. Thus each complex number can be uniquely associated to a point with coordinates in a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations; in particular, addition by a complex number corresponds to translation of the plane and multiplication by a complex number corresponds to scaling and rotation. Conversely some geometric objects and operations can be expressed in terms of complex numbers, enabling the solution of geometric problems using algebraic methods.
Definition and basic operations
A complex number is an expression of the form , where and are real numbers, and is an abstract symbol, the imaginary unit, whose meaning will be explained further below. For example, is a complex number.
For a complex number , the real number is its real part, and the real number (not the complex number ) is its imaginary part. The real part of a complex number is denoted , , or ; the imaginary part is , , or : for example, , .
A complex number can be identified with the ordered pair of real numbers , which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is 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 can be regarded as a complex number , whose imaginary part is 0. A purely imaginary number is a complex number , whose real part is zero. It is common to write , , and ; for example, .
The set of all complex numbers is denoted by (blackboard bold) or (upright bold).
In some disciplines such as electromagnetism and electrical engineering, is used instead of , as frequently represents electric current, and complex numbers are written as or .
Addition, additive inverse, and subtraction
Two complex numbers and are added by separately adding their real and imaginary parts. That is to say:
In particular, the additive inverse of must be because gives zero by the above addition formula. Similarly, subtraction can be performed as
The addition can be geometrically visualized as follows: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent.
Multiplication
The product of two complex numbers can expanded using the distributive property and the identity :
For example,
Using the sum rule from the previous section and this product rule, the familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. Namely, addition and multiplication always satisfy the associative and commutative properties, and multiplication is distributive over addition.
Complex conjugate, absolute value, argument and division
The complex conjugate of the complex number is defined as . It is also denoted by some authors by . Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number: 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 , the product
is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root
By Pythagoras' theorem, 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 , known as the unit complex numbers. If is a real number, then : its absolute value as a complex number and as a real number are equal.
Using the conjugate, the reciprocal of a nonzero complex number can be computed to be
Every complex number has an additive inverse and every nonzero complex number has a multiplicative inverse ; the complex numbers form a field, much as the rational or real numbers do.
More generally, the division of an arbitrary complex number by a non-zero complex number equals
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. Extract of page 570 Extract of page 37
The argument of , sometimes called the "phase", is the angle of the radius measured anticlockwise from the positive real axis. It is written as and is expressed in radians in this article. The angle is defined only up to adding integer multiples of , since a rotation by (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument (also known as choosing a principal value) is to require it to be within the interval , though there are other possibilities.Other authors, including , chose the argument to be in the interval . The argument plays a role with complex logarithms; see below. It can be computed from the rectangular form by means of the arctan (inverse tangent) function.
Polar form
For any complex number z, with absolute value and argument , the equation
holds. This identity is referred to as the polar form of z. It is sometimes written as or . In electronics, one represents a phasor with amplitude and phase in angle notation:
If two complex numbers are given in polar form, i.e., and , the product and division can be computed as
(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
Because the real and imaginary part of are equal, the argument of that number is 45 degrees, or (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
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 : Extract of page 112
Powers and roots
When is an integer, the -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:
For example, the first few powers of the imaginary unit are .
When is a positive integer, the -th roots of a complex number are given by
for integer satisfying and where is the usual (positive) -th root of the positive real number . Because sine and cosine are periodic, other integer values of do not give additional values. For any , these complex -th roots are distinct. For example, there are 4 fourth roots of 1, namely
For , the -th root is a multivalued function (or specifically, an -valued function) of . One of these values can be designated as the principal value, typically via the technique of branch cuts of the complex plane.
Fundamental theorem of algebra The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any coefficients that are complex numbers , the equation
has at least one complex solution z, provided that at least one of the higher coefficients is nonzero. This property does not hold for the field of rational numbers (the polynomial does not have a rational root, because is not a rational number) nor the real numbers (the polynomial does not have a real root, because the square of is positive for any real number ).
Because of this fact, is 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.
History
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 (). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers 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".O'Connor and Robertson (2016), "Girolamo Cardano." 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.Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998.
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.