A brief introduction to complex numbers
Watch the video below to see how complex numbers can be defined.
If you can't see the video, reveal the hidden text which describes the video.
A complex number $z$ can be written as $x+iy$ where $x$ and $y$ are real numbers.
Suppose we had two complex numbers $z_1=2+2i$ and $z_2=3-i$.
Then we can perform arithmetical operations with them.
The sum $z_1+z_2=2+2i+3-i=5+i$
The product $z_1z_2=(2+2i)(3-i)$.
Expanding the brackets gives $(6+6i-2i-2i^2)$
Since $i^2=-1$, this simplifies to $8+4i$.
Choose some complex numbers of your own and practise adding, subtracting and multiplying them.
A real number is of the form $x+0i$. An imaginary number is of the form $0+iy$.
Here are some questions you might like to consider:
- Can you find some pairs of complex numbers whose sum is a real number?
- Can you find some pairs of complex numbers whose sum is an imaginary number?
In general, what would you need to add to $a+ib$ to get a real number? Or an imaginary number?
- Can you find some pairs of complex numbers whose product is a real number?
- Can you find some pairs of complex numbers whose product is an imaginary number?
In general, what would you need to multiply by $a+ib$ to get a real number? Or an imaginary number?
Now have a look at A Brief Introduction to the Argand Diagram.
The next set of numbers you met when you were younger might have been the integers, $\mathbb{Z}$, the positive and negative whole numbers.
You will also have met the rationals, $\mathbb{Q}$, numbers that can be written in the form $\frac{a}{b}$ where $a$ and $b$ are whole numbers which are coprime.
Finally, you will have come across irrational numbers such as $\sqrt2$ and $\pi$; these, together with the rationals, form the set of real numbers $\mathbb{R}$.
This problem introduces the set of complex numbers, $\mathbb{C}$
When you add together $2+2i$ and $3-i$, where does the real part of the answer come from? Where does the imaginary part of the answer come from?
What about when you multiply?
The students from St Stephens School, Australia, found the following examples:
- Pairs of complex numbers whose sum is a real number:
- Pairs of complex numbers whose sum is an imaginary number:
- Pairs of complex numbers whose product is a real number:
- Pairs of complex numbers whose product is an imaginary number:
Sina Sanaizadeh from Hinde House Secondary School, Sheffield, sent in the following explanations:
- In general, what would you need to add to a+bi to get a real number?
- Or an imaginary number?
- In general, what would you need to multiply by a+bi to get a real number?
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
For this product to be real, the imaginary part must be 0, so ad+bc = 0
As a and b are fixed, must have $\frac{c}{d}$ = $\frac{-a}{b}$.
So, in general, for the product of two complex numbers to be real, the ratio of the real to imaginary parts of each complex number must be equal up to a minus sign.
- Or an imaginary number?
For this product to be imaginary, the real part must be 0, so ac-bd = 0
As a and b are fixed, must have $\frac{d}{c}$ = $\frac{a}{b}$.
So, in general, for the product of two complex numbers to be imaginary, the ratios of the real to imaginary parts of each complex number must be the reciprocal of the other.
Why do this problem?
This problem can be used with students who have never met complex numbers. The challenges posed offer practice in manipulating complex numbers while at the same time offering insight into their structure.
Possible approach
To set the scene for the lesson, you may wish to remind students of the different sets of numbers they have met before (Naturals, Integers, Rationals, Reals).
Show students the video below, or recreate it on the board.
Then pose the following questions:
"Choose some pairs of complex numbers and add them together.
Can you find some pairs of complex numbers whose sum is a real number?
Can you find some pairs of complex numbers whose sum is an imaginary number?"
"In general, what would you need to add to $a+ib$ to get a real number? Or an imaginary number?"
Next, invite students to multiply some pairs of complex numbers:
"Can you find some pairs of complex numbers whose product is a real number?
Can you find some pairs of complex numbers whose product is an imaginary number?"
Give students some time to work on their own or in pairs before bringing the class together to discuss the question:
"In general, what would you need to multiply by $a+ib$ to get a real number? Or an imaginary number?"
Students could tackle A Brief Introduction to the Argand Diagram next.
Key questions
When you add two complex numbers, what contributes to the real part of the answer? What contributes to the imaginary part?
When you multiply two complex numbers, what contributes to the real part of the answer? What contributes to the imaginary part.
Possible extension
After working on this problem and A Brief Introduction to the Argand Diagram, students could try Complex Squares.
Possible support
Finding Factors offers practise in expanding and factorising brackets, and might offer useful preparation for multiplying pairs of complex numbers.