Difference between revisions of "Complex number"
(Define the operations -- they're too important to be left in a different article.) |
(Fundamental theorem of algebra) |
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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. | 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. | ||
| + | |||
| + | ==Fundamental theorem of algebra== | ||
| + | The complex numbers form an [[algebraic closure|algebraically closed]] field. This means that any <math>n^{th}\,</math> degree polynomial can be factored into n first degree (linear) polynomials. Equivalently, such a polynomial has n roots (though one has to count all multiple occurrences of repeated roots. This statement is the Fundamental Theorem of Algebra, first proved by [[Gauss|Carl Friedrich Gauss]] around 1800. The theorem is ''not true'' if the roots are required to be real. But when the roots are allowed to be complex, the theorem applies even to polynomials with complex coefficents. | ||
| + | |||
| + | The simplest polynomial with no real roots is <math>x^2 + 1\,</math>, since -1 has no real square root. But if we look for roots of the form <math>a + bi\,</math>, we have: | ||
| + | :<math>(a + bi)^2 + 1 = a^2 - b^2 + 1 + 2abi = 0\,</math> | ||
| + | For the imaginary part to be zero, one of a or b must be zero. For the real part to be zero, a must be zero and <math>b^2\,</math> must be 1. This means that <math>b = \pm 1\,</math>, so the roots are <math>(0, 1)\,</math> and <math>(0, -1)\,</math>, or <math>\pm{}i\,</math>. | ||
| + | |||
| + | These two numbers, <math>i\,</math> and <math>-i\,</math>, are the square roots of -1. | ||
| + | |||
| + | Similar analysis shows that, for example, 1 has three cube roots: | ||
| + | *<math>1\,</math> | ||
| + | *<math>-\frac{1}{2} + \frac{\sqrt{3}}{2}i\,</math> | ||
| + | *<math>-\frac{1}{2} - \frac{\sqrt{3}}{2}i\,</math> | ||
| + | One can verify that | ||
| + | <math>(x-1) (x + \frac{1}{2} - \frac{\sqrt{3}}{2}i) (x + \frac{1}{2} + \frac{\sqrt{3}}{2}i) = x^3 - 1\,</math> | ||
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. | ||
Revision as of 01:03, October 2, 2009
A complex number is a number composed of two parts - a real component and an imaginary component, of the form 
, where a and b are real numbers and 
.
Contents
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.
The complex numbers form an algebraically closed field but do not permit a non-trivial ordering that is preserved under operations. They are the algebraic closure of the real numbers. One notable consequence, and a very natural way of seeing the necessity of complex numbers is the fact that all matrices of full rank over a vector space over real numbers repesent transformations, which, after a base transformation, are equivalent to a diagonal matrix of the same size with complex entries on the diagonal. Thus, any linear linear equation of motion of arbitrary order and dimension of real numbers can be represented in this way and be decomposed into eigenvectors (or modes). The evolution of the system is fully described by the complex amplitudes.
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 numbers. 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 
. The first component is called the real part. Its unit basis vector is just 1. Thus, the complex number 
can also be written 
.
The complex numbers form a 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:
we have:
Multiplication
Multiplication has a special definition. It is this definition that gives the complex numbers their important properties.
- When we multiply
, this definition gives 
- When we multiply
Writing out the product 
in the obvious way, we get the same answer:
This means that, using , one can perform arithmetic operations in a completely natural way.
Division
Division requires a special trick. We have:
To get the individual components, the denominator needs to be real. This can be accomplished by multiplying both numerator and denominator by 
. We get:
This division will fail if and only if 
, that is, 
and 
are both zero, that is, the complex denominator 
is exactly zero (both components zero). This is exactly analogous to the rule that real division fails if the denominator is exactly zero.
Fundamental theorem of algebra
The complex numbers form an algebraically closed field. This means that any 
degree polynomial can be factored into n first degree (linear) polynomials. Equivalently, such a polynomial has n roots (though one has to count all multiple occurrences of repeated roots. This statement is the Fundamental Theorem of Algebra, first proved by Carl Friedrich Gauss around 1800. The theorem is not true if the roots are required to be real. But when the roots are allowed to be complex, the theorem applies even to polynomials with complex coefficents.
The simplest polynomial with no real roots is 
, since -1 has no real square root. But if we look for roots of the form , we have:
For the imaginary part to be zero, one of a or b must be zero. For the real part to be zero, a must be zero and 
must be 1. This means that 
, so the roots are 
and 
, or 
.
These two numbers, 
and 
, are the square roots of -1.
Similar analysis shows that, for example, 1 has three cube roots:
One can verify that
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 can also be written in the form 
, where
-
is the square of the number's magnitude -
,where 
is the phase
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 
and the angle it makes to the real (horizontal) axis will be . This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by 
is equivalent to an anticlockwise rotation through an angle 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
- [a -b]
- [b a],
with mapping as a function f to the above matrix.
We can see that F and F' are isomorphic because:
The function f is clearly 1-to-1 and onto,
,
and 
.
