Adding and Subtracting Complex Numbers

A complex number carries two parts that never mix. Adding and subtracting them is easier than it looks: the real parts combine with the real parts, the imaginary with the imaginary, and nothing crosses over.

Concept A quick reminder of i

For any positive

Concept The parts of a complex number

A complex number is written

, where

is the real part and

is the coefficient of the imaginary part. The whole term

is the imaginary part.

Theorem Combine like with like

This is the same idea as collecting like terms in algebra: real terms group together, and terms group together. A real part is never added to an imaginary part.

Example Adding two complex numbers

Work out .

Real parts:

Imaginary parts:

⟹ 8 + 4i

Note that on its own counts as , so its coefficient is 1.

Example Subtracting two complex numbers

Work out .

Real parts:

Imaginary parts:

⟹ 2 − 2i

The subtraction sign applies to both parts of the second number, so watch the signs carefully.

Note A look ahead to multiplying

Multiplication works differently — expand as usual, then replace with :

⟹ 2, a purely real number

Multiplying a complex number by its conjugate always removes entirely, leaving a real result.

Summary

A complex number has the form a + bi.
Add or subtract real parts with real, imaginary with imaginary.
(5 + i) + (3 + 3i) = 8 + 4i.
(5 + i) − (3 + 3i) = 2 − 2i; the minus applies to both parts.
Multiplying by the conjugate gives a purely real answer.