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Concept
Complex plane
Summary
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers.
The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
The complex plane is sometimes known as the Argand plane or Gauss plane.
In complex analysis, the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts:
for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.
In the Cartesian plane the point (x, y) can also be represented in polar coordinates as
In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2 (in radians), and some care must be taken to define the more complete arctangent function for points (x, y) when x ≤ 0. In the complex plane these polar coordinates take the form
where
Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Without the constraint on the range of θ, the argument of z is multi-valued, because the complex exponential function is periodic, with period 2π i. Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any non-zero integer.
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