Complex number

In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying i^2 = -1. Every complex number can be written in the form x+iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. Pairs of complex numbers can be added, subtracted, multiplied, and divided in a manner similar to that of real numbers. Formally, one says that the set of all complex numbers forms a field.

Definition

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:

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• $(\; a\; ,\; b\; )\; +\; (\; c\; ,\; d\; )\; =\; (\; a\; +\; c\; ,\; b\; +\; d\; )\; ,$
• $(\; a\; ,\; b\; )\; cdot\; (\; c\; ,\; d\; )\; =\; (\; ac\; -\; bd\; ,\; bc\; +\; ad\; ).\; ,$

So defined, the complex numbers form a field, the complex number field, denoted by C.

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We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1).

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In C, we have:

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• additive identity ("zero"): (0, 0)
• multiplicative identity ("one"): (1, 0)
• additive inverse of (a,b): (−a, −b)
• multiplicative inverse (reciprocal) of non-zero (a, b): $left(\{aover\; a^2+b^2\},\{-bover\; a^2+b^2\}\; ight).$

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R.

Related Topics:
Topological closure - Algebraic number - Algebraic closure

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Geometry

Choose a zero point 0 and a unit point 1 in the plane.

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Consider similar triangles.

Related Topics:
Similar - Triangle

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If riangle(0,A,B) sim riangle (X,B,A), then X=A+B . Thus points are added.

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If riangle (0,1,A) sim riangle (0,B,X) , then X=AB . Thus points are multiplied.

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If riangle (0,1,A) is mirror to riangle (0,1,X) , then

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X= ar{A} is the complex conjugate.

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The line (0, 1) is called the real axis. The points on the real axis are called real numbers. They satisfy the equation X-ar{X}=0 .

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The unit circle has centre at 0 and goes through 1. The points on the unit circle are called directions. They satisfy the equation Xar{X}-1=0 because riangle (0,1,X) sim riangle (0,ar{X},1) .

Related Topics:
Unit circle - Direction

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The two points of intersection between the real axis and the unit circle satisfy X^2-1=0 . They are X=+1 and X=-1 . These are the two square roots of unity. The factorization of the polynomial is X^2-1=(X-1)(X+1) .

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The two solutions to X^2+1=0 are called the imaginary units: imath and ar{imath}=-imath . The factorization of the polynomial is X^2+1=(X-imath )(X+imath ) . Geometrically riangle (0,1,imath) sim riangle(0,imath,-1) so that imath^2=-1 .

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The line (0, imath ) is called the imaginary axis. The points on the imaginary axis are called imaginary numbers. They satisfy the equation X+ar{X}=0 .

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Any point X is the sum of a real part rac{X+ar{X}}{2} and an imaginary part rac{X-ar{X}}{2} . (Usually the imaginary part is defined as rac{X-ar{X}}{2imath} which, however, is neither imaginary nor a part).

Related Topics:
Real part - Imaginary part

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The function Y=A+BX maps the real axis onto the line (A,A+B), and the unit circle onto the circle with centre in A and radius B.

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So the geometrical construction of points is translated into the solution of algebraic equations (see Constructible number).

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Coordinates

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A complex number is viewed as a point or a position vector on the two dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand).

Related Topics:
Position vector - Cartesian coordinate system - Jean-Robert Argand

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The Cartesian coordinates are the real part x and the imaginary part y, while the circular coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have

Related Topics:
Circular coordinates - Absolute value or modulus - Mod-arg form - Euler's formula

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: z = x + iy = r (cos phi + isin phi ) = r e^{i phi}. ,

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Additionally the notation r cis φ is sometimes used.

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Note that the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent.

Related Topics:
Modulo - Integer

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By simple trigonometric identities,

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we see that

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:r_1 e^{iphi_1} cdot r_2 e^{iphi_2}

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