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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Into the Wilderness

*This activity follows on from Strolling Along.*

**To multiply two complex numbers** we can expand brackets, taking care to remember that $i^2 = -1$.

For example:

$$\eqalign{\left(4+2i\right)\left(\frac{7}{5}- 3i\right) &= \frac{28}{5} + \frac{14}{5}i - 12i - 6i^2 \\ &=

\left(\frac{28}{5} + 6\right) + \left(\frac{14}{5} - 12\right)i \\ &= \frac{58}{5} - \frac{46}{5}i.}$$

But what's happening geometrically when we multiply complex numbers?

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

$z_3$ is the product $z_1z_2$.

You can move the complex number $z_1$ around the circle, and you can move $z_2$ anywhere on the grid. Use the slider $a$ to change the radius of the circle.

Choose a radius, and a position for $z_2$.

Move $z_1$ around the circle.

**What do you notice?**

Now explore what happens when you fix $z_1$ and move $z_2$.

*You may find it helpful to "Show Lines" joining each point to the origin.*

For which values of $z_2$ does the line through $z_1$ and $z_3$ pass through the origin?

What is special about the positions of $z_1$ for which $z_2$ and $z_3$ are equidistant from the origin?

*Now that you have explored multiplication of numbers, you might like to try the Complex Puzzle.*

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Age 14 to 18

Challenge Level

- Problem
- Student Solutions

*This resource is part of our Adventures with Complex Numbers collection*

For example:

$$\eqalign{\left(4+2i\right)\left(\frac{7}{5}- 3i\right) &= \frac{28}{5} + \frac{14}{5}i - 12i - 6i^2 \\ &=

\left(\frac{28}{5} + 6\right) + \left(\frac{14}{5} - 12\right)i \\ &= \frac{58}{5} - \frac{46}{5}i.}$$

But what's happening geometrically when we multiply complex numbers?

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

$z_3$ is the product $z_1z_2$.

You can move the complex number $z_1$ around the circle, and you can move $z_2$ anywhere on the grid. Use the slider $a$ to change the radius of the circle.

Choose a radius, and a position for $z_2$.

Move $z_1$ around the circle.

Now explore what happens when you fix $z_1$ and move $z_2$.

For which values of $z_2$ does the line through $z_1$ and $z_3$ pass through the origin?

What is special about the positions of $z_1$ for which $z_2$ and $z_3$ are equidistant from the origin?

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.