Complex Arithmetic

A complex number is a pair of real numbers written $z = a + bi$, where $a$ is the real part $\operatorname{Re}(z)$ and $b$ is the imaginary part $\operatorname{Im}(z)$. All three arithmetic operations — addition, subtraction, and multiplication — follow directly from ordinary algebra together with the single rule $i^2 = -1$.

Addition and subtraction

Add or subtract componentwise: real parts with real parts, imaginary parts with imaginary parts.

Example. $(3 + 2i) + (1 - 5i) = (3+1) + (2-5)i = 4 - 3i$.

Multiplication

Multiply as if expanding a polynomial, then replace every $i^2$ with $-1$:

The key step is $i^2 = -1$: the term $bdi^2$ becomes $-bd$, which shifts into the real part.

Example. $(1 + 2i)(3 - i)$:

Why this matters for quantum mechanics

In quantum mechanics a system's state is described by complex amplitudes. When two quantum paths lead to the same outcome, their amplitudes are added (interference). When a gate acts on a state, the amplitude for each output is a sum of products of matrix entries and input amplitudes. Both operations reduce to complex addition and multiplication, so mastering them is the entry point to every quantum calculation that follows.

Try it

This is a numerical exercise — return a number. Let $z_1 = 2 + 3i$ and $z_2 = 1 - 2i$. Compute the real part of their product $z_1 z_2$.

Expand $(2 + 3i)(1 - 2i)$ by hand, replace $i^2 = -1$, collect the real terms, and return that number.

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