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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.