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8 Complex (Imaginary) Numbers

Imaginary numbers got a foot hold intellectually because they showed up in solutions to quadratic equations. Consider the quadratic equation x2+1=0 => x2=-1, x = square root of (-1). There is no number that we can square to get (-1). This is why it is imaginary.

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  Definitions

Equality

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di if and only if a = c and b = d.

Notation and operations

The set of all complex numbers is usually denoted by C, or in blackboard bold by \mathbb{C} (Unicode ℂ). The real numbers, R, may be regarded as "lying in" C by considering every real number as a complex: a = a + 0i.

Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i2 = −1:

\,(a + bi) + (c + di) = (a + c) + (b + d)i
\,(a + bi) - (c + di) = (a - c) + (b - d)i
\,(a + bi)(c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i

Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.

In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

[edit] The complex number field

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

(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 (a field is an algebraic structure in which addition, subtraction, multiplication, and division are defined and satisfy certain algebraic laws. For example, the real numbers form a field).

Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane.

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).

In C, we have:

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

[edit] The complex plane

Geometric representation of z and its conjugate in the complex plane.
Geometric representation of z and its conjugate in the complex plane.

A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand see figure at right). The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.

[edit] Polar form

Alternatively, the complex number z can be specified by polar coordinates. The polar coordinates are r =  |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument of z. For r = 0 any value of φ describes the same number. To get a unique representation, a conventional choice is to set arg(0) = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is called the polar form of the complex number.

[edit] Conversion from the polar form to the Cartesian form

x = r \cos \varphi
y = r \sin \varphi

[edit] Conversion from the Cartesian form to the polar form

r = \sqrt{x^2+y^2}
\varphi =  \begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0\\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\ +\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\ -\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\ undefined & \mbox{if } x = 0 \mbox{ and } y = 0. \end{cases}

The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function which is often named atan2 and processes these internally. A formula that uses the arccos function requires fewer case differentiations:

\varphi =  \begin{cases} +\arccos\frac{x}{r} & \mbox{if } y \geq 0 \mbox{ and } r \ne 0\\ -\arccos\frac{x}{r} & \mbox{if } y < 0\\ undefined & \mbox{if } r = 0. \end{cases}

[edit] Notation of the polar form

The notation of the polar form as

z = r\,(\cos \varphi + i\sin \varphi )\,

is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as

z = r\,\mathrm{e}^{i \varphi}\,,

which is called exponential form.

[edit] Multiplication, division, exponentiation, and root extraction in the polar form

Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.

By simple trigonometric identities we see that

r_1\,e^{i\varphi_1} \cdot r_2\,e^{i\varphi_2}  = r_1\,r_2\,e^{i(\varphi_1 + \varphi_2)} \,

and that

\frac{r_1\,e^{i\varphi_1}}{r_2\,e^{i\varphi_2}}  = \frac{r_1}{r_2}\,e^{i (\varphi_1 - \varphi_2)}. \,

Exponentiation with integer exponents is just as simple; according to De Moivre's formula,

\big(r\,e^{i\varphi}\big)^n = r^n\,e^{in\varphi}. \,

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

The addition of two complex numbers is just the vector addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication by i corresponds to a counter-clockwise rotation by turn (90 degrees ) or (π/2 radians). The geometric content of the equation i 2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1)  (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

All the roots of any number, real or complex, may be found with a simple algorithm. The nth roots are given by

\sqrt[n]{r e^{i\varphi}}=\sqrt[n]{r}\ e^{i\left(\frac{\varphi+2k\pi}{n}\right)}

for k = 0, 1, 2, , n − 1, where \sqrt[n]{r} represents the principal nth root of r.