The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of , . When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written . The field of complex numbers includes the field of real numbers as a subfield.
The set of complex numbers is implemented in the Wolfram Language as Complexes. A number can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.
Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible" (Wells 1986, p. 22).
Through the Euler formula, a complex number
may be written in "phasor" form
Here, is known as the complex modulus (or sometimes the complex norm) and is known as the complex argument or phase. The plot above shows what is known as an Argand diagram of the point , where the dashed circle represents the complex modulus of and the angle represents its complex argument. Historically, the geometric representation of a complex number as simply a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, "imaginary" numbers became accepted partly through their visualization.
Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. This property is not so surprising however when they are viewed as being elements in the complex plane, since points in a plane also lack a natural ordering.
The absolute square of is defined by , with the complex conjugate, and the argument may be computed from
The real and imaginary parts are given by
de Moivre's identity relates powers of complex numbers for real by
A power of complex number to a positive integer exponent can be written in closed form as
The first few are explicitly
(Abramowitz and Stegun 1972).
and complex division
can also be defined for complex numbers. Complex numbers may also be taken to complex powers. For example, complex exponentiation obeys
where is the complex argument.