We know that representing a real number is as simple as plotting a single point on a horizontal number line. But a complex number in the form of z = a + bi is represented by the point (a,b) on a rectangular coordinate plane with the horizontal axis representing the real numbers and the vertical axis representing the imaginary numbers. This coordinate system is known as the complex plane. Every point in the complex plane represents a number, just as every number represents a point. But how do we find the polar form of a complex number? As a tutor for nearly 14 years, I have used the examples in the next several paragraphs to help explain the often confusing topic of changing complex numbers to polar form and vice versa.
First, we must understand how to plot complex numbers. For example, suppose z = 4 – 3i. The complex number is represented by the point (4, -3) in the complex plane. Recall that the horizontal axis is the real number and the vertical axis is the complex number. So (4, -3) is plotted on the rectangular coordinate system. Now suppose that z = 5. There is no complex component to this number, so the point to plot is (5,0). Now suppose that z = -2i. There is no real number component to this number, so the point to plot is (0, -2).
How do we find the absolute value of a complex number? First, recall that absolute value is the distance from zero, in this case, the distance from the origin. Suppose the complex number is z = 3 + 4i. When plotting the point at (3,4) and constructing a right triangle, you will notice that the legs of the triangle are lengths 3 and 4, with the hypotenuse unknown. The hypotenuse is the distance from the origin, hence the absolute value of z, denoted as |z|. Using the Pythagorean Theorem, we get sqrt(a^2 + b^2) = sqrt(25) = 5. Therefore, the distance is 5.
What about the polar form of a complex number? A complex number in the form z = a + bi is said to be in rectangular form. Suppose the absolute value of z is denoted as r. Now we let B be the angle in standard position whose terminal side passes through (a,b). What we have is a right triangle with adjacent side, a and opposite side, b with hypotenuse r. Now using basic trig functions we know cosB = a/r, sinB = b/r and tanB = b/a. From this we know a = rcosB and b = rsinB. Using simple substitution into z = a + bi, we get z = r(cosB + isinB). This is called the polar form of a complex number.
Now, how do we write the polar form of a complex number? Suppose z = 2 + 2i. Therefore a = 2 and b = 2. From the definition, we know the polar form of z is r(cosB + isinB). First we get the value of r to be sqrt(2^2 + 2^2) = sqrt(8) = 2sqrt(2). To get angle B we use the fact that tanB = b/a, therefore tanB = 1 and B is 45 degrees, or Pi/4. Simply substitute 2sqrt(2) for r and Pi/4 for B to get z = 2sqrt(2)(cos Pi/4 + isin Pi/4).
We can also write complex numbers in rectangular form. The method is very simple. In the previous example, we changed z = 2 + 2i to the complex number, z = 2sqrt(2)(cos Pi/4 + isin Pi/4). We use exact value for cos Pi/4 and sin Pi/4 to get z = 2sqrt(2)(sqrt(2)/2 + isqrt(2)/2). Therefore z = 2 + 2i. Notice how we get the original complex number that we started with.
One can extend this topic to products and quotients of complex numbers in polar form, powers of complex numbers in polar form and roots of complex numbers in polar form. Such topics are more suited for another, more advanced discussion on the topic.