Complex Numbers, Roots of Unity (C)

Here we recount the elementary facts about arithmetic with complex numbers. We introduce the symbol i as a solution to z² + 1 = 0. We define the set of complex numbers, $C \equiv \{a + bi: a, b\in(-\infty,\infty)\}$. a is called the real part of a+bi and b is called the imaginary part of a+bi. Addition and subtraction of complex numbers is performed by adding real parts and adding imaginary parts separately:

$$(a+bi) \pm (c+di) \equiv (a\pm c) + (b\pm d)i$$

Multiplication is a little trickier:

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

You will not go wrong with addition, subtraction, and multiplication if you just treat the complex numbers as polynomials in i, with the additional rule that i² = -1.

Division is the hardest. You have to remember the trick of rationalizing the denomimator:

$$\frac{c+di}{a+bi} = \frac{c+di}{a+bi}\frac{a-bi}{a-bi} = \frac{(ac+bd) + (ad-bc)i}{a^2+b^2}$$

so

$$\frac{c+di}{a+bi} = \frac{ac+bd}{a^2+b^2} + \frac{ad-bc}{a^2+b^2}i$$

The quantity a-bi is called the complex conjugate of a+bi, and is denoted by $\overline{a+bi}$. It is easy to check that if z and w are complex numbers then

• $\overline{zw} = \overline{z}\cdot\overline{w}$
• $\overline{z/w} = \overline{z}/\overline{w}$
• $\overline{z\pm w} = \overline{z}\pm \overline{w}$

It is often useful to picture (a+bi) as the ordered pair (a,b). Complex numbers are often graphed, an the horizontal axis is called the the real axis and the vertical axis is called the imaginary axis. The distance from a + bi to 0 is called the modulus or absolute value of a+bi, and is usually denoted by |a+bi|. Of course,

• $\vert a+bi\vert = \sqrt{a^2+b^2}$;
• $\vert z\vert^2 = z\cdot\overline{z}$.

The counter-clockwise angle made by the positive real axis and the line joining an non-zero complex number to 0 is calle the argument of the complex number. For more on this, see the section on CiS $\ref{CiS}$ in the trigonometry section. In this section we describe in detail how to extract roots of complex numbers. One can easily verify that

$$\pm \frac{1}{\sqrt{2}}\pm \frac{1}{\sqrt{2}}i$$

are the four fourth roots of -1.

Exercises Here are some exercises for you to complete:

1.

Simplify the following into the form a+ bi:

$$(3+4i)(7-2i)\;\;\;(3-4i)+(2+6i)\;\;\;(6-9i)-(4+2i)\;\;\;\frac{9-8i}{2-4i}$$

2.

Find |12-5i|.

3.

What is the argument of 3-3i?

4.

Show that $\overline{zw} = \overline{z}\cdot\overline{w}$.

5.

Show that $\overline{z+w} = \overline{z}+\overline{w}$.

6.

Suppose that p(z) = z² + 6z - 4. Find p(2-i).