Step by step
The question
Question: work out (1 + i)/(1 − i).
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 − i: the same number with the sign of the i part changed.
It is 1 + i.
Careful!
We can't divide part by part.
The bottom 1 − i is one number.
First we make it a plain number.
We multiply the top and the bottom by 1 + i.
(1 + i)/(1 + i) 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 i² and −i² change sign.
On the bottom, i − i make zero.
Cross them out.
That is why we use the conjugate.
4 Group like parts
Real parts together, i parts together.
Top: 2i.
Bottom: 2, a plain number.
No i at the bottom now.
5 Split the fraction
Each part on top is divided by 2.
2/2 is 1.
So the answer is i.
6 Check
Let's check: times 1 − i undoes divided by 1 − i.
Our top 2i times 1 − i is 2 + 2i.
That is 2 times the question's top, 1 + i, and the bottom is 2.
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.