Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.

COMPLEX NUMBERS: ADDITION AND SUBTRACTION

Adding complex numbers:

$(a+b i)+(c+d i)=(a+c)+(b+d) i$

Subtracting complex numbers:

$(a+b i)-(c+d i)=(a-c)+(b-d) i$

HOW TO

Given two complex numbers, find the sum or difference.
1. Identify the real and imaginary parts of each number.
2. Add or subtract the real parts.
3. Add or subtract the imaginary parts.

EXAMPLE 3

Adding and Subtracting Complex Numbers

Solution

We add the real parts and add the imaginary parts.
(a)

$\begin{aligned} (3-4 i)+(2+5 i) &=3-4 i+2+5 i \\ &=3+2+(-4 i)+5 i \\ &=(3+2)+(-4+5) i \\ &=5+i \end{aligned}$

(b)

$\begin{aligned} (-5+7 i)-(-11+2 i) &=-5+7 i+11-2 i \\ &=-5+11+7 i-2 i \\ &=(-5+11)+(7-2) i \\ &=6+5 i \end{aligned}$

TRY IT #3

Subtract $2+5 i$ from $3-4 i$.