Step by step
The question
Question: work out (5 + 3i)/4i.
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 4i: the same number with the sign of the i part changed.
It is −4i.
We multiply the top and the bottom by −4i.
−4i over −4i is 1, so the value stays the same.
2 Multiply out
On top, times each part.
On the bottom too.
3 Change i² to −1
i² is minus 1.
So −12i² and −16i² change sign.
4 Group like parts
Real parts together, i parts together.
Top: 12 − 20i.
Bottom: 16, a plain number.
No i at the bottom now.
5 Split the fraction
Each part on top is divided by 16.
12/16 is 3/4.
20/16 is 5/4.
So the answer is 3/4 − (5/4)i.
6 Check
Let's check: times 4i undoes divided by 4i.
Our top 12 − 20i times 4i is 80 + 48i.
That is 16 times the question's top, 5 + 3i, and the bottom is 16.
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.