Complex Numbers and Harmonic Trigonometric Functions

Figure: A complex number maps perfectly into the two-dimensional $xy$ coordinate system in both Cartesian and Plane Polar coordinates. The latter are especially useful, as they lead to the Euler representation of complex numbers and complex exponentials.

We already reviewed very briefly the definition of the unit imaginary number $i = +\sqrt{-1}$ . This definition, plus the usual rules for algebra, is enough for us to define both the imaginary numbers and a new kind of number called a complex number $z$ that is the sum of real and imaginary parts, $z = x + iy$ .

If we plot the real part of $z$ ($x$ ) on the one axis and the imaginary part ($y$ ) on another, we note that the complex numbers map into a plane that looks just like the $x$ -$y$ plane in ordinary plane geometry. Every complex number can be represented as an ordered pair of real numbers, one real and one the magnitude of the imaginary. A picture of this is drawn above.

From this picture and our knowledge of the definitions of the trigonometric functions we can quickly and easily deduce some extremely useful and important True Facts about: