Complex Multiplication¶

Definition¶

Given two complex numbers $z=a+\mathrm{i}b$ and $w=c+\mathrm{i}d$ where $a,b,c,d\in\mathbb{R}$$\mathbb{R}$, the product of complex numbers $z$ and $w$ is defined as $zw = (a+\mathrm{i}b)(c+\mathrm{i}d) = (ac-bd) + \mathrm{i}(ad+bc).$

Interpretation¶

Multiplying two complex numbers works like multiplying two binomials: expand $(a+\mathrm{i}b)(c+\mathrm{i}d)$ term by term and replace $\mathrm{i}^2$ by $-1$. The term $\mathrm{i}b\cdot\mathrm{i}d=-bd$ moves into the real part, which is why the real part of the product is $ac-bd$$bd$.

In the figure $z$ has length $2$ and angle $30^\circ$, and $w$ has length $1.5$ and angle $50^\circ$. The product $zw$ has length $2\cdot1.5=3$ and angle $30^\circ+50^\circ=80^\circ$$80^{\circ }$: the arc $\varphi$ is the angle of $z$, and the arc $\psi$ on top of it is the angle of $w$, added on. In general complex multiplication multiplies the moduli, $|zw|=|z|\,|w|$$|z||w|$, and for nonzero $z$ and $w$ the arguments add modulo $2\pi$. So multiplying by $w$ stretches by the factor $|w|$ and rotates by the angle of $w$.

The two descriptions suit different tasks. The Cartesian formula is the quickest way to compute a product such as $(1+2\mathrm{i})(2-\mathrm{i})=4+3\mathrm{i}$$3\mathrm{i}$; the polar form shows what multiplication does to the picture and leads to De Moivre's formula for powers. Multiplying by the imaginary unit, which has length $1$ and angle $90^\circ$, is a quarter turn; multiplying by a real number $c>0$ is a pure stretch.

A common slip is to multiply part by part, as in complex addition: $(a+\mathrm{i}b)(c+\mathrm{i}d)$ is not $ac+\mathrm{i}bd$$\mathrm{i}bd$. For $z=w=1+\mathrm{i}$ that rule would give $1+\mathrm{i}$$\mathrm{i}$, but $(1+\mathrm{i})^2=1+2\mathrm{i}+\mathrm{i}^2=2\mathrm{i}$$2\mathrm{i}$.

Key Results¶

• The multiplicative identity for complex numbers is $1_\mathbb{C} = 1 + 0\mathrm{i},$ which is the real number $1.$
• Complex multiplication has commutativity, associativity, and distributivity over complex addition.
• In polar form, complex multiplication combines stretching, because $|zw|=|z|\,|w|$$|z||w|$, with rotation, because arguments add modulo $2\pi$.

Examples¶

• $(1+2\mathrm{i})(2-\mathrm{i})=4+3\mathrm{i}=(2-\mathrm{i})(1+2\mathrm{i})$$2\mathrm{i})$.
• $(3+4\mathrm{i})1_\mathbb{C}=(3+4\mathrm{i})(1+0\mathrm{i})=3+4\mathrm{i}$$4\mathrm{i}$.

Prerequisites¶

• Complex Numbers

Related Concepts¶

• Complex Addition
• Complex Reciprocal
• Polar Representation of Complex Numbers
• Imaginary Unit
• Distributivity

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: multiplication of complex numbers, product of complex numbers Symbols: complex numbers, element of, phi, pi, psi, real numbers

Last updated: 20 October 2026 03:02