5. The Real and Imaginary Axes; Rectangular Form
 

Each point on the straight line through the origin O and the unity point U is expressible as  The set of all these points is closed under addition and multiplication. (See Problem 16 above.) In fact, these points behave just like the real numbers under addition, subtraction, multiplication, and division. For this reason, the line through O and U is called the real axis and we identify each real number with a point on the real axis in the following manner.

The real number zero is identified with the origin O. A positive (real) number r is identified with the point  The material in the previous section makes it natural for its negative -r to represent the point  In particular, we have U = e⁰ⁱ = 1.

The line perpendicular to the real axis at the origin is called the imaginary axis. The imaginary unit point  is designated as i. Then the points  may be written as ri and points as -ri. An important fact is that  that is, i² = -1.

Now every point on the real axis is represented by a real number a and every point on the imaginary axis by a pure imaginary number, i.e., a number bi with b real. Note that a and b may be positive, zero, or negative. See Figure 9.
 

Figure 9

Figure 10

Let P be any point in the Argand Plane. Then the foot of the perpendicular from P to the real axis has a representation as some real number a. Similarly, the foot of the perpendicular from P to the imaginary axis is a pure imaginary number bi. The rule for adding points shows that P = a + bi. [The parallelogram with vertices 0, a, P, bi turns out to be a rectangle in this case. See Figure 10.]

The representation a + bi, with a and b real, is called the rectangular form of a complex number. The real number a is called the real part and the real number b is called the imaginary part of the complex number.

It can be shown that in rectangular form the complex numbers have the following rules:

ADDITION 

SUBTRACTION 

MULTIPLICATION 
 

EXAMPLE 1. Convert the polar form to rectangular form a + bi.    Solution: Let H and V be the feet of the perpendiculars from P to the real axis and the imaginary axis, respectively. We see that  is a 45o, 45o, 90o triangle. Hence the lengths of its sides are in the ratio  Since the hypotenuse has length  the two equal sides must have length 5. Then  and  Hence P = H + V = -5 + 5i. See Figure 11.

Figure 11

EXAMPLE 2. Convert the rectangular form to polar form.  Solution: Let C = -7 and  [See Figure 12.] Clearly  is a right triangle. Since the ratio of the length of side DO to the length of the side QD is  it is a 30o, 60o, 90o triangle and the hypotenuse OQ has twice the length of the shortest side QD, that is, the hypotenuse has length 14. Also, the counterclockwise angle from ray OU to ray OQ is 240o i.e.,  Hence

Figure 12

EXAMPLE 3. Find the rectangular form of  Solution: First we note that  radians equals 75o, or 30o plus 45o. Using a 30o, 60o, 90o triangle and a 45o, 45o, 90o triangle [see Figure 13], one finds that  Multiplying these two complex numbers, one has

Figure 13