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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 unit and satisfying the equation =; every complex number can be expressed in the form +, where a and b are real numbers.
Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler , Gauss , Riemann , Cauchy , Gösta Mittag-Leffler , Weierstrass , and many more in the 20th century.
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem.
A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex 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 sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Figure 1. This Argand diagram represents the complex number lying on a plane.For each point on the plane, arg is the function which returns the angle . 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 the line joining the origin and z, represented as a point in the complex plane, shown as in ...
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called the real axis, is formed by the real numbers, and the vertical y-axis, called the imaginary axis, is formed by the imaginary numbers.
Lagrange reversion theorem (mathematical analysis, combinatorics) Lambek–Moser theorem (combinatorics) Lamé’s theorem (computational complexity theory) Lami's theorem ; Landau prime ideal theorem (number theory) Lasker–Noether theorem (commutative algebra) Lattice theorem (abstract algebra) Laurent expansion theorem (complex analysis)