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:

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 = rcosθ;
  • y = rsinθ;

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.