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...

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...

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...

{\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...

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,...

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...

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

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...

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 mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree...

(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...

uncountable complex numbers. In that sense, almost all complex numbers are transcendental. All rational numbers are algebraic. Any rational number, expressed...

{\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...

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...

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...

Complex modulus may refer to: Modulus of complex number, in mathematics, the norm or absolute value, of a complex number: | x + i y | = x 2 + y 2 {\displaystyle...

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...

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

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

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along...

1938 with Stibitz at the helm. Their Complex Number Computer, completed 8 January 1940, was able to calculate complex numbers. In a demonstration to the...

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

discussion of powers of the complex number i, see § nth roots of a complex number. The limit of a sequence of powers of a number greater than one diverges;...

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

a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate...