Training Complex Numbers Division & Conjugates
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Division & Conjugates

24 min Complex Numbers

Division & Conjugates

Complex Conjugate

The conjugate of $z = a+bi$ is $\bar{z} = a - bi$.

$$z \cdot \bar{z} = a^2 + b^2 \quad (\text{always a real number})$$

Modulus

$$|z| = \sqrt{a^2 + b^2}$$

Note: $z \cdot \bar{z} = |z|^2$.

Division

To divide, multiply numerator and denominator by the conjugate of the denominator:

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

Example 1

Find the conjugate and modulus of $z = 3 - 4i$.

$\bar{z} = 3 + 4i$. $|z| = \sqrt{9+16} = 5$.

Example 2

Compute $\dfrac{2+3i}{1-i}$.

$$\frac{(2+3i)(1+i)}{(1-i)(1+i)} = \frac{2+2i+3i+3i^2}{1+1} = \frac{-1+5i}{2} = -\frac{1}{2} + \frac{5}{2}i$$

Example 3

Compute $\dfrac{5}{2+i}$.

$$\frac{5(2-i)}{(2+i)(2-i)} = \frac{10-5i}{5} = 2 - i$$

Practice Problems

1. Find $\bar{z}$ for $z = 5+2i$.
2. Find $|z|$ for $z = 3+4i$.
3. Compute $\frac{1}{i}$.
4. Compute $\frac{4+2i}{1+i}$.
5. Compute $\frac{6}{3-2i}$.
6. Find $z \cdot \bar{z}$ for $z = 2+5i$.
7. Compute $\frac{1+i}{1-i}$.
8. Find $|z|$ for $z = -1+i$.
9. Compute $\frac{3-i}{2+i}$.
10. Find $\bar{z}$ for $z = -4i$.
11. Show that $z + \bar{z} = 2\text{Re}(z)$.
12. Compute $\frac{i}{2+3i}$.
Show Answer Key

1. $5-2i$

2. $5$

3. $\frac{-i}{1} = -i$

4. $\frac{(4+2i)(1-i)}{2} = \frac{6-2i}{2} = 3-i$

5. $\frac{6(3+2i)}{13} = \frac{18+12i}{13}$

6. $4+25 = 29$

7. $\frac{(1+i)^2}{2} = \frac{2i}{2} = i$

8. $\sqrt{1+1} = \sqrt{2}$

9. $\frac{(3-i)(2-i)}{5} = \frac{5-5i}{5} = 1-i$

10. $4i$

11. $(a+bi)+(a-bi) = 2a = 2\text{Re}(z)$ ✓

12. $\frac{i(2-3i)}{13} = \frac{3+2i}{13}$

Complex Number Basics Polar Form & De Moivre's Theorem