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{{Math-m}}

{{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>.

The '''complex numbers''' are a set of numbers which have important applications in the analysis of periodic, oscillatory, or wavelike phenomena.  Mathematicians denote the set of complex numbers with an ornate capital letter: <math>\mathbb{C}</math>.  They are the 5th item in this hierarchy of types of [[number]]s:

 

*The "[[natural number]]s", 1, 2, 3, ...  (There is controversy about whether zero should be included.  It doesn't matter.)

*The "[[integer]]s"&mdash;positive, negative, and [[zero]]

*The "[[real number]]s", including irrational numbers

*The "complex numbers", which give solutions to polynomial equations

 

A complex number is composed of two parts, or components&mdash;a ''real'' component and an ''imaginary'' component.  Each of these components is an ordinary (that is, real) number.  The complex numbers form an "extension" of the real numbers:  If the imaginary component of a complex number is zero, that number is essentially identical to the real number that is its real component.

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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.

<!-- This paragraph taken out.  The closure stuff is treated below.  The ordering stuff, while true, doesn't seem relevant to what we are saying.  The matrix material seems unmotivated and unclear.  (In any case, that's not the "necessity of complex numbers" -- the necessity arose from finding roots of polynomials.)  If someone can put the material about equations of motion / eigenvectors / evolution / amplitudes into a focused and clear explanation, please do so, but it probably wouldn't belong on this page.

The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|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.

The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|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 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>.  Numbers with real part of zero are sometimes called "pure imaginary", with the term "complex" reserved for numbers with both components nonzero.

 

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>.

While the "invisible" nature of the imaginary component may be disconcerting at first (and the word "imaginary" may be an unfortunate term for it), the complex numbers are just as genuine as the Dedekind cuts and Cauchy sequences that are used in the definition of the "real" numbers.

The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.

The complex numbers form a [[Field (mathematics)|field]], with the mathematical operations defined as shown below.

==Arithmetic operations==

==Arithmetic operations==

==Fundamental theorem of algebra==

==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 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. (This failure is what led to the development of complex numbers in the first place.) 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:

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>-\frac{1}{2} - \frac{\sqrt{3}}{2}i\,</math>

*<math>-\frac{1}{2} - \frac{\sqrt{3}}{2}i\,</math>

One can verify that

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>

<math>(x-1) \left(x + \frac{1}{2} - \frac{\sqrt{3}}{2}i\right) \left(x + \frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = x^3 - 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.

:Add the phases.

: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.

One can use this property to find all of the n^th roots of a number geometrically.  For example, by using these trigonometric formulas:

The three cube roots of 1, listed above, can all be seen to have moduli of 1 and phases of 0 degrees, 120 degrees, and 240 degrees respectively.  When raised to the third power, the phases are tripled, obtaining 0, 360, and 720 degrees.  But they are all the same angle&mdash;zero.  So the cubes of these numbers are all just one.  Similarly, <math>i\,</math> and <math>-i\,</math> have phases of 90 degrees and 270 degrees. When those numbers are squared, the phases are 180 and 540, both of which are the same angle&mdash;180 degrees. So their squares are both -1.

 

 

[[File:Argandcbrt1.jpg|thumb|left|400px|The three cube roots of +1.]]

[[File:Argandsqrtm1.jpg|thumb|right|400px|The two square roots of -1.]]

 

 

A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous ''Euler formula'':

 

The ''complex conjugate'', or just ''conjugate'', of a complex number is the result of negating its imaginary part.  The conjugate is written with a bar over the quantity: <math>\overline{z}\,</math>.  All real numbers are their own conjugates.

 

All arithmetic operations work naturally with conjugates&mdash;the sum of the conjugates is the conjugate of the sum, and so on.

It follows that, if P is a polynomial with real coefficients (so that its coefficients are their own conjugates)

:<math>\overline{P(z)} = P(\overline{z})\,</math>

If <math>z\,</math> is a root of a real polynomial, then, since zero is its own conjugate, <math>\overline{z}\,</math> is also a root.  This is often expressed as "Non-real roots of real polynomials come in conjugate pairs."  We saw that above for the cube roots of 1&mdash;two of the roots are complex and are conjugates of each other.  The third root is its own conjugate.

 

The higher mathematical functions (often called "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series ([[Taylor series]]).  They can be extended to handle complex arguments in the completely natural way, so these functions are defined over the complex plane.  They are in fact "complex [[analytic function]]s".  Just about any normal function one can think of can be extended to the complex numbers, and is complex analytic.  Since the power series coefficients of the common functions are real, they work naturally with conjugates. For example:

==Polar notation==

==Complex functions==

The complex number <math>a+bi</math> can also be written in the form <math>\rho e^{i\theta}</math>, where

The general study of functions that take a complex argument and return a complex result is an extremely rich and useful area of mathematics, known as [[complex analysis]].  When such a function is differentiable, it is called a ''[[analytic function|complex analytic function]]'', or just ''analytic function''.

: <math>\rho^2=a^2+b^2</math> is the square of the number's [[Absolute value|magnitude]]

: <math>\tan\theta=\frac{b}{a}</math>,where <math>\theta</math> 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 <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==

==Applications==

The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form

Complex numbers are extremely important in many areas of pure and applied mathematics and physics.  Basically, any description of oscillatory phenomena can benefit from a formulation in terms of complex numbers.

We can see that F and F' are isomorphic because:

The function f is clearly 1-to-1 and onto,

and <math>f(x*y)=f(x)*f(y)</math>.

*[[Arithmetic with complex number]]

*Theoretical mathematics:

::'''number theory'''&mdash;This seems improbable, but it is true.  The Riemann zeta function, and the Riemann hypothesis, are very important in number theory.  For a long time, the only proofs of the prime number theorem used complex numbers.  The Wiles/Taylor proof of [[Fermat's Last Theorem]] uses modular forms, which use complex numbers.

::'''trigonometry'''&mdash;finding arcsine and arccosine functions of numbers with [[absolute values]] more than 1 or arcsecant and arccosecant functions with absolute values between 0 and 1.

[[Category:Mathematics]]

[[Category:Mathematics]]

[[category:complex analysis]]

[[Category:Complex Analysis]]