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A Brief Introduction to Complex Numbers

Age 14 to 18
Challenge Level

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.

Define $i$ to be the number such that $i \times i = -1$. Since no real number squares to give a negative number, $i$ is called an imaginary number.
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.