Adding and Subtracting Complex Numbers
Adding or subtracting complex numbers is like adding or subtracting
polynomials.
| |
Adding
Polynomials
(5 + 3x) + (6 - 7x) |
Adding
Complex Numbers (5 + 3i) + (6 - 7i) |
| Remove parentheses. Combine like terms. |
= 5 + 3x + 6 - 7x
= 11 - 4x |
= 5 + 3i + 6 - 7i
= 11 - 4i |
Procedure —
To Add or Subtract Complex Numbers
To add two complex numbers, add their real parts and add their
imaginary parts.
(a + bi) + (c + di) = (a + c) + (b + d)i
To subtract one complex number from another, subtract their real
parts and subtract their imaginary parts.
(a + bi) - (c + di) = (a - c) + (b - d)i
Note:
When we add or subtract complex
numbers, we usually write the result
in the form a + bi.
Example 1
Find: (10 - 7i) + (-2 + 5i).
| Solution
Remove the parentheses.
Combine like terms.
So, (10 - 7i) + (-2 + 5i) = 8 - 2i. |
(10 - 7i) + (-2 + 5i) = 10 - 7i - 2 + 5i
= 8 - 2i |
We write 8 - 2i in the form a + bi like
this:
8 + (-2)i.
Example 2
Find: (6 + 12i) - (15 - 3i).
| Solution
Remove the parentheses. Be sure to
subtract each term of (15 - 3i).
Combine like terms.
So, (6 + 12i) - (15 - 3i) = -9 + 15i. |
(6 + 12i) - (15 - 3i) = 6 + 12i - 15 + 3i
= -9 + 15i |
Example 3
Find:
| Solution
Rewrite each square root
using 
Simplify each square root.
Remove the parentheses.
Be sure to subtract each
term of (-15 + 12i).
Combine like terms. |
= (6 + 8i) - (-15 + 12i) = 6 + 8i + 15 - 12i = 21 - 4i |
So,
Note:
We write 21 - 4i in the form a + bi
like this:
21 + (-4)i.
|