Geometric Intuition of Multiplying Numbers
Multiplying Real Numbers
We are familiar with multiplying real numbers together, but in order for the next couple of sections to make sense we must visualise what is happening on the real number line.
In the activity below, consider a starting value and then choose a multiplier. Notice that the activity has a choice as to whether to multiply by the positive or negative multiplier. This is key since the idea is to become familiar with the direction in which the point will travel.
Notice that on the real number line the starting point (red) will move in either the left or right direction once multiplication has taken place.
Using Vectors
To make things easier to understand, we can express every point on the real number line as a movement from the origin.
For instance, the point 4 is 4 units to the right of the origin and the point -13 is 13 units to the left of the origin.
We can express this movement as an arrow starting at the origin and ending at the point. This arrow is called a vector.
Now we can think of multiplication as growing (or shrinking) our vector and also (in some cases) changing its direction. Notice that the multiplier is the factor which tells us how much our vector grows by.
Notice as well that the only possible rotations on the real number line are (i.e. the vector doesn't change direction) or (i.e. the vector completely switches direction).
So in summary,when we multiply by a number on the real number line:
- Our vector grows by the number we're multiplying by
- Our vector rotates by either or
Multiplying by i
So we've seen on the real number line that when we multiply by a number the vector grows by the number we're multiplying by and rotates by either or . Now let's see what happens on the complex plane.
In order to understand it, let's first start with plotting points and multiplying by .
Notice that the pattern repeats:
This is useful to keep in mind...
Multiplying by i Using Vectors
If we use vectors to represent the movement from the origin to the point of interest we can see that there is another rotation available to us.
Whereas with real numbers, the only rotations were or , with imaginary numbers we can now also rotate by as well.