For i = √-1, which of the following equals (3 + i)/(1

Step by step

The question

Question: work out (3 + i)/(1 − i).

3 + i1 − i

Remember: i is the square root of minus 1.

A square root times itself gives back what is under it.

So i squared is minus 1.

i = √(−1)

i² = √(−1) × √(−1)

= −1

There are four choices.

First we solve it, the normal way.

Then we pick.

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.

1 − i → 1 + i

Why the conjugate?

A number times its conjugate has no i left.

The middle parts make zero.

And i² is minus 1, so minus b squared i squared is plus b squared.

Here a is 1 and b is 1 (i alone means 1i).

So 1 squared plus 1 squared is 2.

We will see it on the bottom.

(a + bi)(a − bi) = a² − abi + abi − b²i²

= a² − b² × (−1)

= a² + b²

here: 1² + 1² = 2

Careful!

We can't divide part by part.

The bottom 1 − i is one number.

First we make it a plain number.

Careful: not 31 + i−i ✗

We multiply the top and the bottom by 1 + i.

(1 + i)/(1 + i) is 1, so the value stays the same.

= 3 + i1 − i × (1 + i)(1 + i)

2 Multiply out

Each part times each part: first the top, then the bottom.

3 × 1 = 3

3 × i = 3i

i × 1 = i

i × i = i²

1 × 1 = 1

1 × i = i

−i × 1 = −i

−i × i = −i²

We write all the parts: the top over the bottom.

= 3 + 3i + i + i²1 + i − i − i²

3 Change i² to −1

i² is minus 1.

Each i² becomes minus 1.

i² = −1

−i² = −(−1)

= 1

So i² becomes −1, −i² becomes 1.

= 3 + 3i + i − 11 + i − i + 1

On the bottom, i − i make zero.

Cross them out.

That is why we use the conjugate.

4 Group like parts

We add the like parts.

3 − 1 = 2

3i + i = 4i

1 + 1 = 2

Real parts are the plain numbers, with no i.

Real parts together, i parts together.

Top: 2 + 4i.

Bottom: 2, a plain number.

No i at the bottom now.

= 2 + 4i2

5 Split the fraction

Each part on top is divided by 2.

= 22 + 4i2

2/2 is 1.

4/2 is 2.

So the answer is 1 + 2i.

= 1 + 2i

6 Check

Let's check: the answer times the bottom must give the question's top.

Our answer is (2 + 4i)/2.

First, (2 + 4i) times (1 − i): each part times each part, and i² is minus 1.

We get 6 + 2i.

Check: (2 + 4i) × (1 − i) = 2 − 2i + 4i − 4i²

= 2 − 2i + 4i + 4 = 6 + 2i

Real parts together, i parts together.

2 + 4 = 6, −2i + 4i = 2i

Then over 2: (6 + 2i)/2 is 3 + i.

That is the question's top.

So the answer is right!

6 + 2i2 = 3 + i

7 Pick the choice

Our answer is 1 + 2i.

That is choice A.

A) 1 + 2i

8 Why not the others

B is 2 + 2i.

Wrong: i² is −1, not 1.

B) 2 + 2i ✗ i² is −1, not 1

C is 2 + 4i.

Wrong: forgot to divide by 2.

C) 2 + 4i ✗ forgot to divide by 2

D is 3 − i.

Wrong: divided 3 by 1 and i by −i.

D) 3 − i ✗ divided 3 by 1 and i by −i

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.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer