Complex Numbers
“i is a '90° turn', and turning twice lands you on −1.”
The formula
i² = −1, z = a + biHow to read it: introduce a number i whose square is −1, and build z from a real part a and an imaginary part b
- i
- — the imaginary unit — a number whose square is −1
- a
- — the real part — the 'real' portion of z
- bi
- — the imaginary part — a real b times i
The hook
Faced with the wall 'no number squares to a negative', mathematicians simply invented one that does — the imaginary unit i.
In plain words
Introduce a new number i whose square is −1, and every number can be written as a+bi, a real part a plus an imaginary part bi. That's a complex number.
The intuition
Real numbers live on a 1D number line. Complex numbers add a second axis (the imaginary axis) and spread into a 2D plane. Multiplying by i rotates a point 90° in that plane — so multiplying by i twice (a 180° turn) flips direction and lands on −1.
How it's built
A complex number z=a+bi has two pieces — real part a and imaginary part b. Add by combining real and imaginary parts separately; multiply with the distributive law, and whenever i² appears, replace it with −1. That single rule i²=−1 drives every calculation.
Example
(2+i)(2−i) = 2² − (i)² = 4 − i² = 4 − (−1) = 4 + 1 = 5. The imaginary part vanishes and a clean real number remains.
Common mistake
i is not a 'fake number that doesn't exist'. The name 'imaginary' misleads, but complex numbers are essential tools for handling real physical quantities in electronics and quantum mechanics.
Where it's used
Alternating current and signal processing, quantum mechanics, fluid and electromagnetic simulation, fractals — complex numbers shine wherever rotation and waves intertwine.
Where it came from
In the 1500s Cardano met square roots of negatives while solving cubics; long dismissed as 'imaginary', they became respectable numbers through Euler and Gauss.
Prerequisites
Quick check
What is i²?
- 1
- −1✓
- i
- −i
Practice
Find the real part of (3 + 2i) + (1 + 4i).
Answer: 4
- Add the real parts: 3 + 1 = 4
- (imaginary part is 2+4=6i)
Takeaway: Add complex numbers by real and imaginary parts separately.
Find the value of i⁴.
Answer: 1
- i⁴ = (i²)² = (−1)²
- = 1
Takeaway: Powers of i cycle back every 4 steps.
Find the value of (2 + i)(2 − i).
Answer: 5
- Difference of squares: 2² − i²
- = 4 − (−1) = 5
Takeaway: A complex number times its conjugate is real.
Find i³ and show your steps.
Answer: undefined
- i³ = i²·i
- = (−1)·i = −i
Takeaway: The single rule i²=−1 resolves every power of i.