Definition:Complex Number

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[edit] Informal Definition

A complex number is a number in the form $a + b \imath$ or $a + \imath b$ where:

$a$ and $b$ are real numbers;
$\imath$ is the square root of $- 1$ , i.e. $\sqrt {-1}$ .

The set of all complex numbers is denoted $\mathbb{C}$ .

[edit] Formal Definition

A complex number is an ordered pair $\left({x, y}\right)$ where $x, y \in \mathbb{R}$ are real numbers, on which the operations of addition and multiplication are defined as follows:

[edit] Complex Addition

Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.

Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:

$\left({x_1, y_1}\right) + \left({x_2, y_2}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left({x_1 + x_2, y_1 + y_2}\right)$

[edit] Complex Multiplication

Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.

Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:

$\left({x_1, y_1}\right) \left({x_2, y_2}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$

[edit] Equivalence of Definitions

The two definitions as given above are equivalent.

The $a + \imath b$ notation proves more convenient; the ordered pair version is generally used only for the formal definition as given above.

[edit] Real Part

The real part of a complex number $a + \imath b$ is the coefficient $a$ .

The real part of a complex number $z$ is often denoted $\Re \left({z}\right)$ or $\operatorname{Re} \left({z}\right)$ or $\mathrm {re} \left({z}\right)$ .

[edit] Imaginary Part

The imaginary part of a complex number $a + \imath b$ is the coefficient $b$ (note: not $\imath b$ ).

The imaginary part of a complex number $z$ is often denoted $\Im \left({z}\right)$ or $\operatorname{Im} \left({z}\right)$ or $\mathrm {im} \left({z}\right)$ .

[edit] Wholly Real

The complex number $z = a + \imath b$ is called wholly real or completely real, or entirely real, etc. iff $b = 0$ .

[edit] Wholly Imaginary

The complex number $z = a + \imath b$ is called wholly imaginary or completely imaginary, or entirely imaginary, etc. iff $a = 0$ .

[edit] Notation

When $a$ and $b$ are symbols representing variables or constants, the form $a + \imath b$ is usually seen.

When $a$ and $b$ are actual numbers, for example 3 and 4, it usually gets written $3 + 4 \imath$ .

The symbol $\imath$ can also be seen as $i$ .

When mathematics is applied to engineering, in particular electrical and electronic engineering, the symbol $\jmath$ or $j$ is usually used, as $i$ is the standard symbol used to denote the flow of electric current, and to use it also for $\sqrt {-1}$ would cause untold confusion.

[edit] Complex Plane

Because a complex number can be expressed as an ordered pair, we can plot the number $x + \imath y$ on the Real Number Plane $\mathbb{R}^2$ :

This representation is also known as an Argand Diagram or a Gauss Plane, but as it is difficult to establish exactly who had precedence over the concept of plotting complex numbers on a plane, the more neutral term complex plane is usually preferred nowadays.

[edit] Real Axis

Complex numbers of the form $\left({x, 0}\right)$ , being wholly real, appear as points on the $x$ -axis.

[edit] Imaginary Axis

Complex numbers of the form $\left({0, y}\right)$ , being wholly imaginary, appear as points on the $y$ -axis.

[edit] Polar Form

The polar form of a complex number $x + \imath y$ is written $\left \langle {r, \theta} \right \rangle$ , where:

$x = r cosθ$ ;
$y = r sinθ$ ;

and $θ$ is measured in radians.

Thus $x + \imath y$ can be expressed $r \left({\cos \theta + \imath \sin \theta}\right)$ .

The value $r$ is the modulus of $x + \imath y$ :

$\left|{x + \imath y}\right| = \sqrt {x^2 + y^2} = \sqrt {r^2 \left({\cos^2 \theta + \sin^2 \theta}\right)} = r$ .

Definition:Complex Number

From ProofWiki

Contents

[edit] Informal Definition

[edit] Formal Definition

[edit] Complex Addition

[edit] Complex Multiplication

[edit] Equivalence of Definitions

[edit] Real Part

[edit] Imaginary Part

[edit] Wholly Real

[edit] Wholly Imaginary

[edit] Notation

[edit] Complex Plane

[edit] Real Axis

[edit] Imaginary Axis

[edit] Polar Form

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