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...

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...

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...

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...

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...

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 +...

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

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...

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 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...

(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...

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

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

{\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...

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...

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

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...

programming languages provide a complex data type for complex number storage and arithmetic as a built-in (primitive) data type. A complex variable or value is usually...

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

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...

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...

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...

}}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...

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...

mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater...

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...

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

number theory, by contrast, relies on complex numbers and techniques from analysis and calculus. Algebraic number theory employs algebraic structures such...