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 $8+4i$.

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.