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:
[latex]\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i[/latex]
Subtracting complex numbers:
[latex]\left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i[/latex]
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 [latex]3 - 4i[/latex] and [latex]2+5i[/latex].
Solution
We add the real parts and add the imaginary parts.
[latex]\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}[/latex]