Remember the commutative property? The commutative property is about ordering. When adding or multiplying, changing the order does not change the result. For example, 3 + 6i is the same as 6i + 3. How about the associative property? The associative property is about grouping. When adding or multiplying, we can group the terms in any way without changing the result. For example, (2 + 3i) + (3 – 4i) is the same as (2 + 3) + (3i – 4i) which gives us 5 – i. Lastly, do you remember the distributive property? The distributive property is about distributing a multiplication over an addition. When multiplying a number times a parenthesis containing the sum of two or more numbers, the multiplication applies to every number in the parenthesis. For example, 2(3 – 5i) is the same as 2(3) + 2(-5i) which gives us 6 – 10i.
In the following examples we will use these four complex numbers:
- z1 = 2 + 3i
- z2 = -3 + 2i
- z3 = 4 – 2i
- z4 = -2 – 4i
Adding Complex Numbers
Example: Add z1 to itself
Start by substituting:
The associative property allows any grouping, so we can eliminate the parentheses:
The commutative property allows any ordering for adding. The goal is to add the real parts and imaginary parts separately:
Adding the real parts gives us 2 + 2 = 4. Adding the imaginary parts gives 3 + 3 = 6. The answer is 4 + 6i.
Example: Calculate z2 + z3 + z4.
Then, we can remove the parentheses (associative property):
Did you see where the +(-2 – 4i) became -2 – 4i ? We can think of +(-2 – 4i) as +1 times (-2 – 4i). The +1 times the (-2 – 4i) gives -2 – 4i.
Now, we can reorder the addition (commutative property):
Adding the real parts gives -3 + 4 – 2 = -1. Adding the imaginary parts, 2 – 2 – 4 = -4. The answer is -1 –