Operations with Complex Numbers in Algebraic Form
The Big Picture: Complex numbers are the plane equipped with multiplication. They look like vectors in $\mathbb{R}^2$ when you add them, but they become something richer — a full number system — once you can multiply, divide, and take conjugates. This lesson builds every operation from scratch, derives every formula, and ends with the complex numbers standing as a field: a number system in full standing.
1Addition and Subtraction — Componentwise Rules
Let $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$. Addition and subtraction work component by component:
$$z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)\,i$$ $$z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)\,i$$The Vector Parallel
If you identify $z = a + bi$ with the vector $(a, b)$ in $\mathbb{R}^2$, then complex addition is exactly vector addition: add the first components, add the second components. This is not a coincidence — the complex plane is $\mathbb{R}^2$ as a vector space. What makes it more than $\mathbb{R}^2$ is the multiplication we define next.
Properties (inherited directly from real arithmetic on each component)
| Property | Statement |
|---|---|
| Commutativity | $z_1 + z_2 = z_2 + z_1$ |
| Associativity | $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$ |
| Additive Identity | $0 = 0 + 0i$ satisfies $z + 0 = z$ |
| Additive Inverse | $-z = -a - bi$ satisfies $z + (-z) = 0$ |
Each property holds because it holds for the real and imaginary parts separately. No new proof is needed.
Interactive Visualization: Complex Addition as Vector Addition
Interactive visualization — available in the full lesson
Drag the blue and green points. The red point is their sum — the diagonal of the parallelogram.