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Line 105: + + + + + + + + + + + + + + + + − 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.+ − − ==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 <math>a+bi</math> 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, − <math>f(x+y)=f(x)+f(y)</math>, − and <math>f(x*y)=f(x)*f(y)</math>. − ==See Also==+ − + + + + + + + + + + + + + + + + + − +
Add conjugates; rewrite applications; take out the (irrelevant IMO) 2x2 matrix stuff.
==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:
of which there is the famous case (for θ = π):
of which there is the famous case (for θ = π):
:<math>e^{i\pi} = -1\,</math>
:<math>e^{i\pi} = -1\,</math>
==Conjugates==
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—the sum of the conjugates is the conjugate of the sum, and so on.
:<math>\overline{w+z} = \overline{w} + \overline{z}\,</math>
:<math>\overline{w*z} = \overline{w} * \overline{z}\,</math>
:<math>\overline{w/z} = \overline{w} / \overline{z}\,</math>
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—two of the roots are complex and are conjugates of each other. The third root is its own conjugate.
==Other functions==
The higher mathematical functions (often call "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power 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:
:<math>\sin(\overline{z}) = \overline{\sin(z)}\,</math>
:<math>\log(\overline{z}) = \overline{\log(z)}\,</math>
==Applications==
==Applications==
While it may seem improbable that an artificial construction like this, which does not arise in ordinary measurements, has extensive applications, complex numbers in fact are extremely important in many areas of pure and applied mathematics and physics. Basically, any description of oscillatory phenomena can be formulated in terms of complex numbers.
The applications include these:
*[[Arithmetic with complex number]]
*Theoretical mathematics:
::algebra (including finding roots of polynomials)
::linear algebra—vector spaces, inner products, Hermitian and unitary operators, Hilbert spaces, etc.
*Applied mathematics
::eigenvalue/eigenvector problems
::Fourier and Laplace transforms
::linear differential equations
::stability theory
*Electrical engineering
::alternating current circuit analysis
::filter design
::antenna design
*Theoretical physics
::quantum mechanics
::fundamental particle physics
::electromagnetic radiation
::optics
[[Category:Mathematics]]
[[Category:Mathematics]]
[[category:complex analysis]]
[[Category:Complex analysis]]