Look at trigonometric form of a complex number. The best way is to picture a complex plane. The abscissa is the real axis and the ordinate is the imaginary axis. Every point in the plane corresponds to a single complex number. The Cartesian coordinates of that point (x, y) are the real and imaginary part of the complex number z. When we write:
z = x + I*y (1)
we say the complex number is in its algebraic form.
Now, instead of using Cartesian coordinates, one can use polar coordinates. The distance of the point from the center is called a modulus:
ρ = |z| = Sqrt[x2 + y2] (2)
The angle with the positive real axis is called the argument of a complex number (it ranges from 0 to 2*Pi, remember angles are measured in radians in math):
φ = arg(z) = arctan(y/x) (3)
If one knows the polar coordinates, one can go back to Cartesian coordinates by:
x = ρ*cos(φ)
y = ρ*sin(φ) (4)
If you substitute (4) into (1), you get
z = ρ*[cos(φ) + I*sin(φ)] (5)
This is the trigonometric form of a complex number. Why is this useful. It turns out that when you multiply two complex numbers in this form, you get:
z1*z2 = ρ1*[cos(φ1) + I*sin(φ1)]*ρ2*[cos(φ2) + I*sin(φ2)]
z1*z2 = ρ1*ρ2*{[cos(φ1)*cos(φ2) - sin(φ1)*sin(φ2)] + I*[sin(φ1)*cos(φ2) + cos(φ1)*sin(φ2)]}
If you refresh your addition formulas from trigonometry, you will notice that the real and imaginary parts in the parentheses are exactly:
cos(φ1 + φ2) and sin(φ1 + φ2)
So, we can write:
z1*z2 = ρ1*ρ2*[cos(φ1 + φ2) + I*sin(φ1 + φ2)] (6)
Thus, the product of two complex numbers written in trigonometric form is again in trigonometric form. Furthermore, we can write:
|z1*z2| = |z1|*|z2| (7)
arg(z1*z2) = arg(z1) + arg(z2). (8)
Equation (7) says that the modulus of a product of complex numbers is a product of their moduli and equation (8) that the argument of a product is a sum of the arguments.
The last feature is very important. Let us look at complex numbers with |z| = 1. Geometrically, they lie on a circle with a unit radius and center in the origin in the complex plane. According to (5), they are given by the complex function of a real argument:
h(t) = cos(t) + I*sin(t) (9)
This function, according to (6) obeys the fundamental functional equation for exponential functions:
h(u + v) = h(u)*h(v) (10)
It is only left to find the basis of this exponential. In order to do this, let us find the derivative h'(t) from (9):
h'(t) = -sin(t) + I*cos(t),
where we used [cos(t)]' = -sin(t) and [sin(t)]' = cos(t). Now, because -1 = I2, we can take one I in front of both terms:
h'(t) = I*[cos(t) + I*sin(t)]
But, the term in the angle brackets is just h(t):
h'(t) = I*h(t). (10)
Every exponential function can be expressed as y(x) = exp(k*x). The derivative is then, according to the chain rule, y'(x) = k*exp(k*x) = k*y(x). Comparing this with (10), we see that k = I. So we can (formally) write:
h(t) = exp(I*t). (11)
Combining (5), (9) and (11), we can write the exponential form of a complex number as:
z = ρ*exp(I*φ). (12)
Now consider z = -1. For it x = -1, y = 0. If you plot it in the complex plane and use (2) and (3), you will find that ρ = 1 and φ = π. From here, we immediately arrive at:
-1 = exp(I*π). (13) Q.E.D.