Add and subtract 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 combine the imaginary parts.

A General Note: Addition and Subtraction of Complex Numbers

Adding complex numbers:

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

$\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i$

Subtracting complex numbers:

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

$\left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i$

How To: Given two complex numbers, find the sum or difference.

Identify the real and imaginary parts of each number.
Add or subtract the real parts.
Add or subtract the imaginary parts.

Example 3: Adding Complex Numbers

Add $3 - 4i$ and $2+5i$ .

Solution

We add the real parts and add the imaginary parts.

{\begin{cases} (a + b i) + (c + d i) = (a + c) + (b + d) i \\ (3 - 4 i) + (2 + 5 i) = (3 + 2) + (- 4 + 5) i \\ = 5 + i \end{cases}

$\begin{cases}\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i\hfill \\ \left(3 - 4i\right)+\left(2+5i\right)=\left(3+2\right)+\left(-4+5\right)i\hfill \\ \text{ }=5+i\hfill \end{cases}$

Try It 3

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

Solution