Chapter 2 — Complex Numbers
Topics in this chapter:
2.1 The Square Root of −1
The equation $x^2 + 1 = 0$ has solutions $x = \pm\sqrt{-1}$. We define:
\[i = \sqrt{-1}, \qquad i^2 = -1\]- Pure imaginary number: a real number times $i$, like $2i$, $\pi i$, $-7i$.
- Complex number: a number of the form $a + bi$ where $a, b \in \mathbb{R}$.
- Every real number is also a complex number (with $b = 0$).
Powers of $i$
The powers of $i$ cycle with period 4:
| $i^1$ | $i^2$ | $i^3$ | $i^4$ | $i^5$ | $i^6$ | $i^7$ | $i^8$ |
|---|---|---|---|---|---|---|---|
| $i$ | $-1$ | $-i$ | $1$ | $i$ | $-1$ | $-i$ | $1$ |
To evaluate $i^n$: divide $n$ by 4 and use the remainder.
Example: $i^{17}$: $17 = 4 \times 4 + 1$, so $i^{17} = i^1 = i$.
$i^{69}$: $69 = 4 \times 17 + 1$, so $i^{69} = i$.
$i^{1972}$: $1972 = 4 \times 493$, so $i^{1972} = i^0 = 1$.
Warning: You cannot combine radicals of negative numbers: $\sqrt{-1} \cdot \sqrt{-1} \neq \sqrt{(-1)^2} = 1$. The correct answer is $i \cdot i = i^2 = -1$. The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ only works when $a, b \ge 0$.
2.2 Complex Number Operations
A complex number $z = a + bi$ has:
- Real part: $\text{Re}(z) = a$
- Imaginary part: $\text{Im}(z) = b$ (note: $b$, not $bi$)
- Conjugate: $\overline{z} = a - bi$
Addition & Subtraction
Add real and imaginary parts separately:
\[(a + bi) + (c + di) = (a+c) + (b+d)i\]Example 2-2: $(3 + 4i) + (-3 + 8i) = (3-3) + (4+8)i = 12i$.
Multiplication
Use the distributive law and $i^2 = -1$:
\[(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\]Example 2-3: $(3 + 4i)(-3 + 8i) = -9 + 24i - 12i + 32i^2 = -9 + 12i - 32 = -41 + 12i$.
Division
Multiply numerator and denominator by the conjugate of the denominator:
\[\frac{a + bi}{c + di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{(a+bi)(c-di)}{c^2 + d^2}\]Example 2-4: $\dfrac{3 + 4i}{-3 + 8i} = \dfrac{(3+4i)(-3-8i)}{(-3)^2 + 8^2} = \dfrac{-9 - 24i - 12i - 32i^2}{73} = \dfrac{23 - 36i}{73} = \dfrac{23}{73} - \dfrac{36}{73}i$.
Key Properties of the Conjugate
| Property | Formula |
|---|---|
| Double conjugate | $\overline{\overline{z}} = z$ |
| Sum | $\overline{z + w} = \overline{z} + \overline{w}$ |
| Product | $\overline{z \cdot w} = \overline{z} \cdot \overline{w}$ |
| Quotient | $\overline{z/w} = \overline{z}/\overline{w}$ |
| Product with conjugate | $z \cdot \overline{z} = a^2 + b^2$ (always real, always $\ge 0$) |
| The quantity $ | z | = \sqrt{a^2 + b^2} = \sqrt{z \cdot \overline{z}}$ is called the modulus (or absolute value) of $z$. |
Exercises:
- Exercise 2-1: What is $i^{17}$? $i^{69}$? $i^{1972}$?
- Exercise 2-2: $(-\frac{1}{2} + i) + (2 - \frac{1}{2}i) = ?$
- Exercise 2-3: Find the general formula for $(z_1 + z_2 i) + (w_1 + w_2 i)$.
- Exercise 2-4: $(-\frac{1}{2} + i)(2 - \frac{1}{2}i) = ?$
- Exercise 2-5: Find the general product $(z_1 + z_2 i)(w_1 + w_2 i)$.
- Exercise 2-6: Simplify $(z_1 + z_2 i)(z_1 - z_2 i)$.
- Exercise 2-7: $\dfrac{-\frac{1}{2} + i}{2 - \frac{1}{2}i} = ?$
- Exercise 2-8: Find the general quotient formula.
- Exercise 2-9: Prove $\overline{\overline{z}} = z$.
- Exercise 2-10: What is the conjugate of a real number $a$? Of a pure imaginary $bi$?
- Exercise 2-11: Show $\overline{z+w} = \overline{z} + \overline{w}$.
- Exercise 2-12: Show $\overline{zw} = \overline{z}\,\overline{w}$.
- Exercise 2-13: Show $\overline{z/w} = \overline{z}/\overline{w}$.
- Exercise 2-14: What is $\text{Re}(z) + i\,\text{Im}(z)$?
Exercises & Problems
Problems to Solve for Chapter 2:
- Find $\dfrac{1-i}{1+i}$. (MAG 1987)
- Which are true? (MAG 1987)
- (i) $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
- (ii) $\overline{z_1 z_2} = \overline{z_1}\,\overline{z_2}$
- (iii) $\overline{z_1/z_2} = \overline{z_1}/\overline{z_2}$
- Evaluate $\sqrt{-1} \cdot (\sqrt{-1})^2 \cdot \sqrt{(-1)^2}$. (MAG 1991)
- Find $i^{-18} + i^{-9} + i^0 + i^9 + i^{18}$. (MAG 1991)
- Find $\text{Re}[(a+bi)(c+di)]$ in terms of $a, b, c, d$. (MAG 1991)
- Evaluate $(2+i)^3$. (MAG 1991)
- Find $(1+i)^4(2-2i)^3$. (MAG 1987)
- Simplify $\dfrac{\sqrt{-6}\sqrt{2}}{\sqrt{3}}$. (MAG 1990)
- If $F(x) = 3x^3 - 2x^2 + x - 3$, find $F(1+i)$. (MAG 1990)
- Which of the following are true? (MAG 1987)
- (i) $\overline{z + 3i} = \overline{z} - 3i$
- (ii) $\overline{iz} = -i\overline{z}$
- (iii) $(2+i)^2 = 3 + 4i$