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...

corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number z = x + i y , {\displaystyle...

In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle...

In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and...

Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended...

are strongly related: A complex logarithm of a nonzero complex number z {\displaystyle z} , defined to be any complex number w {\displaystyle w} for which...

consist of various extensions or modifications of the complex number system. In modern mathematics, number systems are considered important special examples...

for which the polynomial evaluates to zero. As another example, the complex number 1 + i {\displaystyle 1+i} is algebraic because it is a root of x4 +...

multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane or Gauss plane. In complex analysis,...

{\displaystyle \mathbb {T} } is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized...

(in the early 19th century). An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers...

{\displaystyle \Gamma (z)} is due to Legendre. If the real part of the complex number z is strictly positive ( ℜ ( z ) > 0 {\displaystyle \Re (z)>0} ), then...

base b {\displaystyle b} and any real number exponent x {\displaystyle x} . More involved definitions allow complex base and exponent, as well as certain...

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite...

resistance, which has only magnitude. Impedance can be represented as a complex number, with the same units as resistance, for which the SI unit is the ohm...

investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics...

respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and is also...

complex, a coordination complex with more than one bond Complex number, an extension of real numbers obtained by adjoining imaginary numbers Complex,...

{\displaystyle q} as a complex number. In the other direction, any formula for an algebraically constructible complex number can be transformed into...

In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals...

In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer...

described in § Other meanings below. Numbers from various number systems, like integers, rationals, complex numbers, quaternions, octonions, ... may have multiple...

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition...

integer or of a rational number is generally not a rational number, neither a constructible number. Every nonzero real or complex number has exactly three cube...

periodic; in the case of Dirichlet function, any nonzero rational number is a period. Using complex variables we have the common period function: e i k x = cos...

physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude A and initial phase...

in the complex numbers. That is, for any non-constant polynomial P {\displaystyle P} with rational coefficients there will be a complex number α {\displaystyle...

in late 1938 with Stibitz at the helm. He led the development of the Complex Number Calculator (CNC), completed in November 1939 and put into operation...

called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by...

}}y<0\\{\text{undefined}}&{\mbox{if }}r=0.\end{cases}}} Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying...