Complex numbers
Complex numbers
- A complex number is $z = x + iy$, where $i^2 = -1$ — $x$ is the real part, $y$ the imaginary part.
- The conjugate $z^* = x - iy$. Real-coefficient polynomials have non-real roots in conjugate pairs.
- The modulus $|z| = \sqrt{x^2 + y^2}$; the argument is the angle from the positive real axis.
Practice
What is the modulus of the complex number 3 + 4i?
|z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
Practice
The conjugate of z = 5 + 2i is:
The conjugate flips the sign of the imaginary part: z* = 5 − 2i.
Practice
What is the value of i²?
By definition of the imaginary unit, i² = −1.
Argand diagram & division
- Plot $z$ as a point on an Argand diagram.
- Polar form: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ — multiplying multiplies moduli and adds arguments.
- To divide, multiply top and bottom by the conjugate of the bottom: $\dfrac{3+i}{1-i} = 1 + 2i$.
You've got it
Key idea
- $z = x + iy$, $i^2 = -1$; conjugate $z^* = x - iy$; modulus $|z| = \sqrt{x^2+y^2}$
- plot on the Argand diagram; polar form $z = re^{i\theta}$
- divide by multiplying top and bottom by the bottom's conjugate