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Use the template, Luke! Also add modulus/phase material.
{{Math-m}}
A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.
A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.
For a given field, the field containing it (strictly speaking, the smallest such field) that is algebraically closed is called its algebraic closure. The field of real numbers is not algebraically closed; its closure is the field of complex numbers.
For a given field, the field containing it (strictly speaking, the smallest such field) that is algebraically closed is called its algebraic closure. The field of real numbers is not algebraically closed; its closure is the field of complex numbers.
==Polar coordinates, modulus, and phase==
Complex numbers are often depicted in 2-dimensional Cartesian analytic geometry; this is called the ''complex plane''. The real part is the x-coordinate, and the imaginary part is the y-coordinate. When these points are analyzed in [[polar coordinates]], some very interesting properties become apparent. The representation of complex numbers in this way is called an ''Argand diagram''.
If a complex number is represented as <math>x + yi\,</math>, its polar coordinates are given by:
:<math>r = \sqrt{x^2 + y^2}\,</math>
:<math>\theta = \tan^{-1}\frac{y}{x}\,</math>
Transforming the other way, we have:
:<math>x = r \cos\theta\,</math>
:<math>y = r \sin\theta\,</math>
The radial distance from the origin, <math>r\,</math>, is called the ''modulus''. It is the complex equivalent of the absolute value for real numbers. It is zero if and only if the complex number is zero.
The angle, <math>\theta\,</math>, is called the ''phase''. (Some older books refer to it as the ''argument''.) It is zero for positive real numbers, and <math>\pi\,</math> radians (180 degrees) for negative ones.
The multiplication of complex numbers takes a particularly interesting and useful form when represented this way. If two complex numbers <math>z_1\,</math> and <math>z_2\,</math> are represented in modulus/phase form, we have:
:<math>z_1 = r_1 (\cos\theta_1 + i\sin\theta_1)\,</math>
:<math>z_2 = r_2 (\cos\theta_2 + i\sin\theta_2)\,</math>
:<math>z_1 * z_2 = r_1 * r_2 * ((\cos\theta_1 \cos\theta_2 - \sin\theta_1 \sin\theta_2) + i (\sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2))\,</math>
But that is just the addition rule for sines and cosines!
:<math>z_1 * z_2 = r_1 * r_2 * (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))\,</math>
So the rule for multiplying complex numbers on the Argand diagram is just:
:Multiply the moduli.
:Add the phases.
It is a common belief that complex numbers have a weaker connection to physical reality than real numbers. Observables in [[Physics]] for example weight, energy, pressure etc. are usually represented as [[real]] [[numbers]], and the SI system of units relies on real numbers. However, the transformation between a SI base unit, e.g. an inductance/capacitance/resistance value and a complex impedance is arbritrary and set by convention, and the "natural" representation depends on the measurement method. As a matter of fact, a number of measurement devices (network analysers, lock in amplifiers) directly output real and imaginary component (where the imaginary component is obviously a real voltage/current value). Also, the [[index of refraction]] is often expressed as a complex number whose imaginary component indicates [[absorption]] loss as light propagates through the medium.
It is a common belief that complex numbers have a weaker connection to physical reality than real numbers. Observables in [[Physics]] for example weight, energy, pressure etc. are usually represented as [[real]] [[numbers]], and the SI system of units relies on real numbers. However, the transformation between a SI base unit, e.g. an inductance/capacitance/resistance value and a complex impedance is arbritrary and set by convention, and the "natural" representation depends on the measurement method. As a matter of fact, a number of measurement devices (network analysers, lock in amplifiers) directly output real and imaginary component (where the imaginary component is obviously a real voltage/current value). Also, the [[index of refraction]] is often expressed as a complex number whose imaginary component indicates [[absorption]] loss as light propagates through the medium.