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Line 91: + + + + + − − ==Polar notation== − 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>\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.
Euler's formula, remove now-redundant polar stuff.
:Multiply the moduli.
:Multiply the moduli.
:Add the phases.
:Add the phases.
A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous ''Euler formula'':
:<math>z = r e^{i\theta}\,</math>
of which there is the famous case (for θ = π):
:<math>e^{i\pi} = -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.
==Complex Numbers as Matrices==
==Complex Numbers as Matrices==