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...

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

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...

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,...

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...

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 +...

{\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...

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...

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...

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...

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 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 \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...

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

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

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 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...

}}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...

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...

numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i. Imaginary numbers...

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...

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...

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

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...

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...

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

unit. In general, any non-zero complex number has n distinct complex-valued nth roots, equally distributed around a complex circle of constant absolute value...