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Define the operations -- they're too important to be left in a different article.
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>.
__TOC__
Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). This representation is known as the Argand diagram.
Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate''). This representation is known as the Argand diagram.
Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]]. This is the subject of [[complex analysis]].
Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]]. This is the subject of [[complex analysis]].
The complex numbers are defined as a 2-dimensional vector space over the [[real number]]s. That is, a complex number is an ordered pair of numbers: (a, b). The familiar real numbers constitute the complex numbers with second component zero. That is, x corresponds to (x, 0).
The second component is called the ''imaginary part''. Its unit basis vector is called <math>i</math>. The first component is called the ''real part''. Its unit basis vector is just 1. Thus, the complex number <math>(a, b)\,</math> can also be written <math>a + bi\,</math>.
The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.
==Arithmetic operations==
The field operations are defined as follows:
===Addition===
Addition is just the standard addition on the 2-dimensional vector space. That is, add the real parts (first components), and add the imaginary parts (second components).
Letting the complex numbers w and z be defined by their respective components:
:<math>w = (a, b) = a + bi\,</math>
:<math>z = (c, d) = c + di\,</math>
we have:
:<math>w + z = (a+c, b+d) = a+c + (b+d)i\,</math>
===Multiplication===
Multiplication has a special definition. It is this definition that gives the complex numbers their important properties.
:<math>w * z = (a*c - b*d) + (a*d + b*c)i\,</math>
:::When we multiply <math>(0+1i)*(0+1i)\,</math>, this definition gives <math>i^2 = -1\,</math>
Writing out the product <math>w * z\,</math> in the obvious way, we get the same answer:
:<math>(a + bi) * (c + di) = a*c + b*d*i^2 + (a*d + b*c)i = (a*c - b*d) + (a*d + b*c)i\,</math>
This means that, using <math>i^2 = -1\,</math>, one can perform arithmetic operations in a completely natural way.
===Division===
Division requires a special trick. We have:
:<math>\frac{w}{z} = \frac{a + bi}{c + di}\,</math>
To get the individual components, the denominator needs to be real. This can be accomplished by multiplying both numerator and denominator by <math>(c - di)\,</math>. We get:
:<math>\frac{w}{z} = \frac{(a + bi)*(c - di)}{(c + di)*(c - di)} = \frac{(a*c + b*d) + (b*c - a*d)i}{c^2 + d^2}\,</math>
This division will fail if and only if <math>c^2 + d^2 = 0\,</math>, that is, <math>c\,</math> and <math>d\,</math> are both zero, that is, the complex denominator <math>z\,</math> is exactly zero (both components zero). This is exactly analogous to the rule that real division fails if the denominator is exactly zero.
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.
==Polar notation==
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where
: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]]
: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]]
If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand plane') from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the complex plane.
If a line is drawn on the [[complex plane]] (also known as an 'Argand diagram' or the 'Argand plane') from the origin to a given complex number, the length of that line will be <math>\rho</math> and the angle it makes to the real (horizontal) axis will be <math>\theta</math>. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by <math>e^{i\theta}</math> is equivalent to an anticlockwise rotation through an angle <math>\theta</math> in the complex plane.
==Complex Numbers as Matrices==
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
:[a -b]
:[a -b]
<math>f(x+y)=f(x)+f(y)</math>,
<math>f(x+y)=f(x)+f(y)</math>,
and <math>f(x*y)=f(x)*f(y)</math>.
and <math>f(x*y)=f(x)*f(y)</math>.
==See Also==
==See Also==