### Stage 4 & 5 Toughnuts

These Stage 4 and 5 problems haven't been solved yet. Can you be the first?

### Strolling Along

What happens when we multiply a complex number by a real or an imaginary number?

# Opening the Door

##### Stage: 4 and 5 Challenge Level:

In the article Vanishing Roots, we posed the question:
"What if we could have the square root of a negative number?"
and introduced $i$ as the $\sqrt{-1}$, so $i^2=-1$.

A Complex Number has a real part and an imaginary part, for example:

$4+ 2i$,
or $\frac{7}{5} - 3i$,
or $-\sqrt{3} + \pi i$

The complex number $a + bi$ has real part $a$ and imaginary part $b$.

In the same way that we use the Number Line to represent Real numbers, we can use a coordinate plane to represent Complex numbers. This plane is called the Argand Diagram.

Adding and subtracting complex numbers is just the same as collecting like terms in algebra. For example,

$(4 + 2i) + \left(\frac{7}{5} - 3i \right) = \frac{27}{5} - i$
and
$(4 + 2i) - \left(\frac{7}{5} - 3i \right) = \frac{13}{5} + 5i$.

We have created the GeoGebra interactivity below for you to explore the questions that follow.

Use the Geogebra interactivity to find some pairs of complex numbers whose sum is a real number. What do you notice?
Can you explain it algebraically?

Use the Geogebra interactivity to find some pairs of complex numbers whose sum is an imaginary number. What do you notice?
Can you explain it algebraically?

In general, what would you need to add to $a + bi$ to get a real number?  Or an imaginary number?

Now that you've been introduced to the world of complex numbers, you might like to start Strolling along.