What is a Complex Number?
A complex number extends the real numbers by including the imaginary unit i, where i² = −1. It has the form z = a + bi and can be plotted on a 2D plane called the Argand diagram.
A complex number z = a + bi has a real part a and an imaginary part b, with i² = −1. Its modulus (distance from the origin) is |z| = √(a² + b²).
- •Points on a single number line
- •Only magnitude, no direction
- •Cannot represent √−1
- •Always ordered (can compare with < or >)
- •Points on a 2D plane (Argand diagram)
- •Have magnitude and direction (angle)
- •z = a + bi, where i² = −1
- •Not ordered — no meaningful < or >
Try it: interactive calculator
Step-by-step worked examples
Find the modulus of z = 3 + 4i.
a=3, b=4 |z| = √(3² + 4²) = √(9+16) |z| = √25 = 5
Add (2 + 3i) and (5 − i).
Add real parts: 2 + 5 = 7 Add imaginary parts: 3 + (−1) = 2 Result: 7 + 2i
Multiply (1 + 2i)(3 − i).
Expand: 1×3 + 1×(−i) + 2i×3 + 2i×(−i) = 3 − i + 6i − 2i² Since i² = −1: 3 + 5i + 2 = 5 + 5i
Flashcards
Quick quiz
Q1.What is i²?
Q2.What is the modulus of z = 6 + 8i?
Q3.What is the conjugate of 4 − 5i?
Q4.Where is a complex number a + bi plotted?
The full card deck, worked steps and AI-tutor support for “What is a Complex Number?” are in Notek — study by hand before your exam.
Common mistakes
Treating i like a normal variable and writing i² = i². — Correct: i² always simplifies to −1 — replace it whenever it appears.
Adding complex numbers by multiplying real and imaginary parts together. — Correct: Add real parts together and imaginary parts together separately.
Thinking complex numbers can be ordered like real numbers (z1 < z2). — Correct: There's no meaningful < or > for complex numbers — only their moduli can be compared.
Forgetting to replace i² = −1 when multiplying two complex numbers. — Correct: Expand fully with FOIL, then replace every i² with −1 before simplifying.
FAQ
What is a complex number?
A number of the form a + bi, combining a real part a and an imaginary part b, where i² = −1.
What is the formula for the modulus of a complex number?
|z| = √(a² + b²), the distance of z = a + bi from the origin.
What are examples of complex numbers?
3 + 4i, −2 + i, and 5 (a real number, which is also complex with b = 0) are all complex numbers.
How do you calculate the modulus of a complex number?
Square the real and imaginary parts, add them, and take the square root: |z| = √(a² + b²).