Addition of Complex Numbers

Once we define complex numbers in the form $z = x + iy$, it becomes natural to ask: how do we perform arithmetic with them?

Adding Complex Numbers

To add two complex numbers, simply add their real parts together, and their imaginary parts together.

Let $z_1 = a + bi$ and $z_2 = c + di$. Then:

Example

Simplify $(3 + 4i) + (2-7i)$.

That's it! Complex addition works just like vector addition—component by component.

Geometric Interpretation

You can think of a complex number as a point or vector on the complex plane: the horizontal axis is the real part, and the vertical axis is the imaginary part. Adding complex numbers is equivalent to adding their position vectors.

If you've done vector addition before, this will feel very familiar. We will look at this at a later lesson when talking about complex numbers graphically.

Summary

Adding complex numbers is simple: just add real to real and imaginary to imaginary. This operation respects the structure of the complex plane and mirrors how vectors work. In the next section, we'll explore subtraction, which works just as intuitively.