Step by step
The question
Question: work out (6 + 2i)/(1 + 3i).
Remember: i is the square root of minus 1.
So i squared is minus 1.
1 Multiply by the conjugate
The bottom has an i in it.
To remove it, we use the conjugate of 1 + 3i: the same number with the sign of the i part changed.
It is 1 − 3i.
Careful!
We can't divide part by part.
The bottom 1 + 3i is one number.
First we make it a plain number.
We multiply the top and the bottom by 1 − 3i.
(1 − 3i)/(1 − 3i) is 1, so the value stays the same.
2 Multiply out
On top, each part times each part.
On the bottom too.
3 Change i² to −1
i² is minus 1.
So −6i² and −9i² change sign.
On the bottom, −3i + 3i make zero.
Cross them out.
That is why we use the conjugate.
4 Group like parts
Real parts together, i parts together.
Top: 12 − 16i.
Bottom: 10, a plain number.
No i at the bottom now.
5 Split the fraction
Each part on top is divided by 10.
12/10 is 6/5.
16/10 is 8/5.
So the answer is 6/5 − (8/5)i.
6 Check
Let's check: times 1 + 3i undoes divided by 1 + 3i.
Our top 12 − 16i times 1 + 3i is 60 + 20i.
That is 10 times the question's top, 6 + 2i, and the bottom is 10.
So the answer is right!
Let's look back.
First: multiply by the conjugate.
Then: multiply out.
Then: change i² to −1.
Then: group like parts.
Then: split the fraction.
Last: check.