Complex Number Arithmetic
Master addition, subtraction, multiplication, conjugates, and modulus of complex numbers.
Addition and Subtraction
To add or subtract complex numbers, combine the real parts and the imaginary parts separately:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) − (c + di) = (a − c) + (b − d)i
On the Argand diagram, addition corresponds to vector addition — use the parallelogram rule.
Multiplication
Multiply complex numbers using the distributive law (like expanding brackets), remembering that i2 = −1:
(a + bi)(c + di) = ac + adi + bci + bdi2
= (ac − bd) + (ad + bc)i
A useful shortcut: z × z̄ = |z|2, where z̄ is the conjugate of z.
Conjugate and Modulus
The complex conjugate of z = a + bi is:
z̄ = a − bi
The conjugate reflects z across the real axis on the Argand diagram.
Key properties of the conjugate:
z + z̄ = 2a (real)
z − z̄ = 2bi (imaginary)
z × z̄ = a2 + b2 = |z|2
|z| = √(a2 + b2)
Key Vocabulary
Complex Conjugate
For z = a + bi, the conjugate is z̄ = a − bi. The imaginary part flips sign.
Modulus
|z| = √(a2 + b2), the distance from z to the origin on the Argand diagram.
Distributive Law
The rule used to expand brackets when multiplying complex numbers.
Real and Imaginary Parts
For z = a + bi: Re(z) = a and Im(z) = b. Both are real numbers.
Worked Examples
Calculate (3 + 2i) + (1 − 5i).
Step 1: Add real parts: 3 + 1 = 4.
Step 2: Add imaginary parts: 2 + (−5) = −3.
Answer: (3 + 2i) + (1 − 5i) = 4 − 3i.
Calculate (2 + 3i)(4 − i).
Step 1: Expand: 2(4) + 2(−i) + 3i(4) + 3i(−i) = 8 − 2i + 12i − 3i2.
Step 2: Since i2 = −1: = 8 − 2i + 12i + 3 = 11 + 10i.
Answer: (2 + 3i)(4 − i) = 11 + 10i.
Find z × z̄ for z = 3 + 4i.
Step 1: The conjugate is z̄ = 3 − 4i.
Step 2: z × z̄ = (3 + 4i)(3 − 4i) = 9 − 12i + 12i − 16i2 = 9 + 16 = 25.
Answer: z × z̄ = 25 = |z|2.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Calculate (4 + 3i) − (2 + 7i).
Question 2
What is the conjugate of 6 − 2i?
Question 3
Calculate (1 + i)(1 − i).
Question 4
What is |3 − 4i|?
Question 5
If z = 2 + i, what is z × z̄?
Key Concepts Summary
- ● Add/subtract complex numbers by combining real and imaginary parts separately.
- ● Multiply by expanding brackets and replacing i2 with −1.
- ● The conjugate of a + bi is a − bi (flip the sign of the imaginary part).
- ● z × z̄ = |z|2 = a2 + b2 (always a real number).
- ● To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.