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DeMoivre's theorem
:Multiply the moduli.
:Multiply the moduli.
:Add the phases.
:Add the phases.
One can use this property to find all of the n<sup>th</sup> roots of a number geometrically. For example, by using these trigonometric formulas:
::<math>\sin(30) = \cos(60) = \frac{1}{2}\,</math>
::<math>\sin(60) = \cos(30) = \frac{\sqrt{3}}{2}\,</math>
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—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—180 degrees. So their squares are both -1.
One can apply this property to the n<sup>th</sup> roots of any complex numbers. They lie equally spaced on a circle. This is DeMoivre's theorem.
===Euler's formula===
A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous ''Euler formula'':
A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous ''Euler formula'':
:<math>e^{i\pi} = -1\,</math>
:<math>e^{i\pi} = -1\,</math>
==Applications==
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.