Simplify (3 - 4i)/(2i) Step by Step | Dividing complex numbers

Step by step

The question

Question: work out (3 − 4i)/2i.

3 − 4i2i

Remember: i is the square root of minus 1.

So i squared is minus 1.

i = √(−1)

i² = −1

1 Multiply by the conjugate

The bottom has an i in it.

To remove it, we use the conjugate of 2i: the same number with the sign of the i part changed.

It is −2i.

2i → −2i

We multiply the top and the bottom by −2i.

−2i over −2i is 1, so the value stays the same.

= 3 − 4i2i × −2i−2i

2 Multiply out

On top, times each part.

On the bottom too.

= −6i + 8i²−4i²

3 Change i² to −1

i² is minus 1.

So 8i² and −4i² change sign.

= −6i − 84

4 Group like parts

Real parts together, i parts together.

Top: −8 − 6i.

Bottom: 4, a plain number.

No i at the bottom now.

= −8 − 6i4

5 Split the fraction

Each part on top is divided by 4.

= −84 − 6i4

8/4 is 2.

6/4 is 3/2.

So the answer is −2 − (3/2)i.

= −2 − (32)i

6 Check

Let's check: times 2i undoes divided by 2i.

Our top −8 − 6i times 2i is 12 − 16i.

Check: (−8 − 6i) × 2i = 12 − 16i

That is 4 times the question's top, 3 − 4i, and the bottom is 4.

So the answer is right!

12 − 16i = 4 × (3 − 4i)

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.

Answer