# Waves: Part 4 - Complex Numbers

In this part we look at the most elegant way of finding and defining a sine wave.

Fig.1 a) Real numbers live in one dimension, b) Complex numbers are two dimensional and the real axis lies across the center. The imaginary axis is vertical and has units of i. A complex number can by specified by a and b, c) A complex number can also be specified by the Modulus, which is the radius and the Argument which is an angle.

Figure 2 - Various powers of i shown in the complex plane.

Fig.3 The graph shown has the characteristic that the slope is equal to y.

Fig.4 Euler’s equation links the exponential form of a complex rotation with the sine and cosine representation.

Fig.5 a) a pair of contra rotations. The vertical motion is coherent and adds. The horizontal motion cancels, b) The contra rotation requires a sine wave to have equal amounts of positive and negative frequency, hence the pair of sidebands obtained in modulation.

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# Waves: Part 6 - Wave Fronts

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