Adding and Subtracting Complex Numbers

The a + bi form, why real and imaginary parts never mix, worked addition and subtraction, and a first look at why a conjugate product is purely real.

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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
i = \sqrt{-1} \qquad i^2 = -1
For any positive a : \sqrt{-a} = i\sqrt{a} .
So \sqrt{-9} = 3i .
Concept The parts of a complex number
1 + 5i real part 1 imaginary part 5i
A complex number is written a + bi , where a is the real part and b is the coefficient of the imaginary part. The whole term bi is the imaginary part.
Theorem Combine like with like
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) - (c + di) = (a - c) + (b - d)i

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

Example Adding two complex numbers

Work out (5 + i) + (3 + 3i) .

Real parts: 5 + 3 = 8
Imaginary parts: 1 + 3 = 4
⟹ 8 + 4i

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

Example Subtracting two complex numbers

Work out (5 + i) - (3 + 3i) .

Real parts: 5 - 3 = 2
Imaginary parts: 1 - 3 = -2
⟹ 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 i^2 with -1 :

(1 + i)(1 - i) = 1 - i + i - i^2
= 1 - i^2 = 1 - (-1) = 2
⟹ 2, a purely real number

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

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