Add complex numbers by combining real parts and imaginary parts separately, exactly as with like terms.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Multiplying
Expand as usual, then replace i² with −1. That replacement is the only new step.
The conjugate
The conjugate of a + bi is a − bi. Multiplying a number by its conjugate always gives a real result.
Why it goes real
(a + bi)(a − bi) = a² − b²i² = a² + b². The imaginary terms cancel and the i² turns the sign.
Dividing
To divide, multiply top and bottom by the conjugate of the denominator. The denominator becomes real.
Finish in standard form
An answer should end as a + bi, with the real part first and a single i term.
Treat i like a variable, then substitute
Add, subtract and multiply exactly as with polynomials, then replace every i² with −1. That single substitution is the only rule distinguishing complex from ordinary algebra.
Answers go in a + bi form
Collect the real parts and the i terms separately. Leaving an answer with an i² in it is unfinished, in the same way as leaving an unsimplified fraction.
The conjugate clears i from a denominator
Multiplying a + bi by a − bi gives a² + b², a real number. It is the exact analogue of using a conjugate to rationalise a surd, and it is how complex division is performed.
They live on a plane
Plotting a + bi at (a, b) makes multiplication by i a 90° rotation. Complex arithmetic becomes geometric, which is why complex numbers are the natural language for anything rotational.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 2)
- Vector b(1, 3)
- a + b(4, 5)
The dashed arrow is b again, moved to the tip of a. The sum closes the triangle, and its components are just the x parts added and the y parts added.
Step 3: Watch an Example
One step at a time.
Watch Tomas Multiply Two Complex Numbers
Tomas computes (3 + 2i)(1 + 4i).
- Step 1
Expanding gives 3 + 12i + 2i + 8i².
Step 4: Your Turn
Practice makes it stick.
The Sum
Problem 1 of 2
(3 + 2i) + (1 + 5i). What is the imaginary coefficient?
The Conjugate
Problem 2 of 2
(3 + 4i)(3 − 4i). What is the result?
Combine Them
1 of 8
(2 + 3i) + (4 + i). Real part?
2 of 8
Same sum. Imaginary coefficient?
3 of 8
(5 + 2i) − (3 + 6i). Real part?
4 of 8
Same difference. Imaginary coefficient?
5 of 8
(1 + i)(1 − i). What is the result?
6 of 8
(2 + 5i)(2 − 5i). What is the result?
7 of 8
Put the division process in order.
- 1Multiply top and bottom by it.
- 2Replace every i squared with negative one.
- 3Write the answer as a + bi.
- 4Write the conjugate of the denominator.
8 of 8
(3 + i)(3 − i). What is the result?
Step 5: Quick Check
Show what you know.
Question 1 of 2
(4 + 3i)(4 − 3i). What is the result?
Question 2 of 2
Why does multiplying by the conjugate give a real number?
What You Learned
- Add and subtract by combining real and imaginary parts separately.
- Multiply by expanding, then replace i² with −1.
- Divide by multiplying top and bottom by the conjugate.