Adding complex numbers collects the real parts and the imaginary parts separately.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Multiplying
Expand as usual, then replace every i² with −1. That last step is the only new rule.
The conjugate
The conjugate of a + bi is a − bi. Multiplying them gives a real number, a² + b².
Dividing
Multiply top and bottom by the conjugate of the bottom. The denominator becomes real.
Treat i like a letter, then substitute
Add, subtract and multiply as if i were a variable, then replace every i² with −1. That single substitution is the only rule that distinguishes complex arithmetic from ordinary polynomial arithmetic.
Powers of i cycle
i, −1, −i, 1, then repeat. Any power of i can be reduced by dividing the exponent by 4 and reading the remainder. The cycle of four is a consequence of the 90° rotation repeating after four quarter turns.
The conjugate clears i from a denominator
Multiplying a + bi by a − bi gives a² + b², a real number. That is how you rationalise a complex denominator, and it is the exact analogue of using a conjugate to clear a surd.
Complex roots come in pairs
A polynomial with real coefficients has complex roots in conjugate pairs. So a cubic with real coefficients must have at least one real root — there is no way to pair up three complex roots. This is a genuinely useful structural fact.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 2)
- Vector b(1, 4)
- a + b(4, 6)
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 Naomi Multiply Two Complex Numbers
Naomi expands (2 + 3i)(1 + 4i).
- Step 1
She expands as usual: 2 + 8i + 3i + 12i².
Step 4: Your Turn
Practice makes it stick.
Adding
Problem 1 of 2
(3 + 2i) + (4 + 5i). What is the real part?
The Square
Problem 2 of 2
What is i²?
Real and Imaginary
1 of 8
(1 + i) + (5 + 3i). Imaginary part?
2 of 8
(6 + 2i) − (2 + 5i). Real part?
3 of 8
What is i⁴?
4 of 8
(2 + i)(2 − i). What is the result? It is real.
5 of 8
Conjugate of 3 + 7i. What is its imaginary part?
6 of 8
(1 + 2i)(1 − 2i). Result?
7 of 8
Order the steps for dividing by a complex number.
- 1Multiply top and bottom by it
- 2Replace every i squared with −1
- 3Split into real and imaginary parts
- 4Find the conjugate of the denominator
8 of 8
Is 5 a complex number? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
(4 + 3i)(4 − 3i). Result?
What You Learned
- Add and subtract by collecting real and imaginary parts separately.
- Multiply as usual, then replace i² with −1.
- The conjugate turns a complex denominator into a real one.