Complex Arithmetic in the Plane

Complex numbers exist because polynomial equations refused to be solved. The equation $x^2 + 1 = 0$ has no real solution, squaring any real number gives something non-negative. Sixteenth-century algebraists chasing cubic formulas had to introduce a symbol $i$ with $i^2 = -1$ just to make their arithmetic close up. The strange thing is that once you accept that single symbol, every polynomial equation acquires a full set of solutions, and the geometry of the plane suddenly becomes algebraic.

Definition

A complex number is an expression $z = a + bi$ where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined by $i^2 = -1$. The number $a$ is the real part, written $\text{Re}(z)$. The number $b$ is the imaginary part, written $\text{Im}(z)$. The set of all complex numbers is denoted $\mathbb{C}$.

Every real number $a$ is the complex number $a + 0i$, so $\mathbb{R} \subset \mathbb{C}$. Pure imaginary numbers like $7i$ are the complex numbers with real part zero.

Arithmetic

Addition is componentwise:

$(a + bi) + (c + di) = (a + c) + (b + d) i.$

Multiplication is FOIL plus the rule $i^2 = -1$:

$(a + bi)(c + di) = ac + adi + bci + bd i^2 = (ac - bd) + (ad + bc) i.$

The minus sign in $ac - bd$ is where $i^2 = -1$ entered.

Conjugate and modulus

The conjugate of $z = a + bi$ is $\bar{z} = a - bi$, flip the sign of the imaginary part. The modulus (or absolute value) is

$|z| = \sqrt{a^2 + b^2}.$

Conjugate and modulus are linked by the identity $z \bar{z} = a^2 + b^2 = |z|^2$. This is the trick that makes division work: to compute $\dfrac{z}{w}$, multiply top and bottom by $\bar{w}$; the denominator $w \bar{w} = |w|^2$ becomes real, and the rest is easy.

The complex plane (Argand diagram)

Plot $z = a + bi$ as the point $(a, b)$ in the plane: the horizontal axis carries real parts, the vertical axis carries imaginary parts. This picture, the complex plane or Argand diagram, turns complex arithmetic into geometry. The modulus $|z|$ is the distance from $z$ to the origin. The conjugate is the reflection of $z$ across the real axis.

• Electrical Engineering: AC circuit analysis uses complex numbers to represent voltage and current phasors, impedance is a complex number $Z = R + jX$.
• Quantum Mechanics: Wavefunctions are complex-valued; the probability density is $|\psi|^2$, which is why complex numbers are not optional in QM.
• Computer Graphics: Rotations in 2D are multiplications by complex numbers $e^{i\theta}$, and that fact generalises to quaternions for 3D.

(Drag the point $z_1$