Complex numbers

The number that doesn't exist

• What is $\sqrt{-1}$? For centuries, mathematicians said it doesn't exist. Then they discovered that pretending it does — calling it $i$ — unlocks solutions to equations that otherwise have none.
• Complex numbers 复数 extend the number line into a plane, and they're essential for electrical engineering, quantum mechanics, and signal processing.

Complex numbers

• A complex number is $z = x + iy$, where $i^2 = -1$ — $x$ is the real part 实部, $y$ the imaginary part 虚部.
• The conjugate 共轭 $z^* = x - iy$. Real-coefficient polynomials have non-real roots in conjugate pairs.
• The modulus 绝对值 $|z| = \sqrt{x^2 + y^2}$; the argument 辐角 is the angle from the positive real axis.

Self-similar patterns like Romanesco arise from iterating in the complex plane

Argand diagram 阿干图 and polar form 极坐标形式

• Plot $z$ as a point on an Argand diagram (real axis horizontal, imaginary axis vertical).

On an Argand diagram, the modulus |z| is the distance from the origin and the argument is the angle θ.

• Polar form: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ — multiplying multiplies moduli and adds arguments.

Division using the conjugate

• To divide, multiply top and bottom by the conjugate of the bottom.
• Example: $\dfrac{3+i}{1-i} = \dfrac{(3+i)(1+i)}{(1-i)(1+i)} = \dfrac{3+3i+i+i^2}{1+1} = \dfrac{2+4i}{2} = 1 + 2i$.

Solving equations with complex roots

• A quadratic with negative discriminant 判别式 has complex roots: $x^2 + 4 = 0 \Rightarrow x = \pm 2i$.
• Real-coefficient polynomials always have non-real roots in conjugate pairs: if $a + bi$ is a root, so is $a - bi$.

allow_no_figure: Complex number arithmetic and the Argand diagram are algebraic/geometric concepts that are effectively conveyed through notation and worked examples.

• Besides Cartesian and polar forms, find square roots of a complex number and sketch loci (e.g. $|z-a|=r$).

You've got it

Write from a modulus of 17

• z = 8 plus 15 i.
• Find the modulus and the conjugate.