The question
Question: work out (6 + 4i)/2.
6 + 4i2
Remember: i is the square root of minus 1.
So i squared is minus 1.
i = √(−1)
i² = −1
1 Split the fraction
The bottom is a plain number: 2.
So each part on top is divided by it.
a + bc = ac + bc
6/2, and 4i/2.
= 62 + 4i2
6/2 is 3.
4/2 is 2.
So the answer is 3 + 2i.
= 3 + 2i
2 Check
Let's check: times 2 undoes divided by 2.
We get 6 + 4i again.
It works!
Check: (3 + 2i) × 2 = 6 + 4i
Let's look back.
First: split the fraction.
Last: check.
▶ Watch it solved
The question
Question: work out (2 + 3i)/(1 + i).
2 + 3i1 + i
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 1 + i: the same number with the sign of the i part changed.
It is 1 − i.
1 + i → 1 − i
Careful!
We can't divide part by part.
The bottom 1 + i is one number.
First we make it a plain number.
Careful: not 21 + 3ii ✗
We multiply the top and the bottom by 1 − i.
(1 − i)/(1 − i) is 1, so the value stays the same.
= 2 + 3i1 + i × (1 − i)(1 − i)
2 Multiply out
On top, each part times each part.
On the bottom too.
= 2 − 2i + 3i − 3i21 − i + i − i2
3 Change i² to −1
i² is minus 1.
So −3i² and −i² change sign.
= 2 − 2i + 3i + 31 − i + i + 1
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: 5 + i.
Bottom: 2, a plain number.
No i at the bottom now.
= 5 + i2
5 Split the fraction
Each part on top is divided by 2.
= 52 + i2
So the answer is 5/2 + (1/2)i.
= 52 + (12)i
6 Check
Let's check: times 1 + i undoes divided by 1 + i.
Our top 5 + i times 1 + i is 4 + 6i.
Check: (5 + i) × (1 + i) = 4 + 6i
That is 2 times the question's top, 2 + 3i, and the bottom is 2.
So the answer is right!
4 + 6i = 2 × (2 + 3i)
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.
▶ Watch it solved
The question
Question: work out (5 + 3i)/4i.
5 + 3i4i
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 4i: the same number with the sign of the i part changed.
It is −4i.
4i → −4i
We multiply the top and the bottom by −4i.
−4i over −4i is 1, so the value stays the same.
= 5 + 3i4i × −4i−4i
2 Multiply out
On top, times each part.
On the bottom too.
= −20i − 12i2−16i2
3 Change i² to −1
i² is minus 1.
So −12i² and −16i² change sign.
= −20i + 1216
4 Group like parts
Real parts together, i parts together.
Top: 12 − 20i.
Bottom: 16, a plain number.
No i at the bottom now.
= 12 − 20i16
5 Split the fraction
Each part on top is divided by 16.
= 1216 − 20i16
12/16 is 3/4.
20/16 is 5/4.
So the answer is 3/4 − (5/4)i.
= 34 − (54)i
6 Check
Let's check: times 4i undoes divided by 4i.
Our top 12 − 20i times 4i is 80 + 48i.
Check: (12 − 20i) × 4i = 80 + 48i
That is 16 times the question's top, 5 + 3i, and the bottom is 16.
So the answer is right!
80 + 48i = 16 × (5 + 3i)
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.
▶ Watch it solved
