The real number line had just one dimension, which made the idea of direction simple - two of them: left and right. These were commonly known as the positive and negative directions. Each number was described with two pieces of information. Each number came with a direction and a distance. We usually give the direction information first and then the distance.

• For example \(-3\). The hyphen out front means the left (negative) direction and the \(3\) gives the distance.
• For example \(+5\). The plus sign out front means the right (positive) direction and the \(5\) gives the distance.

Note: We usually do not write the \(+\), \(+3 = 3\).

We represent the direction first. A hyphen appears to the left of the distance number for a negative number. Or, the a hyphen is absent for a positive number.

The Complex Numbers have many more directions, but the idea is the same. Each complex number can be described with a direction and a distance. This type of representation is known as polar form.

Direction vectors or unit vectors are vectors of length \(1\), which makes the unit circle the key to understanding the Complex Numbers.

We can describe every complex number by providing a complex number on the unit circle to give the direction and then a distance to move in that direction.

We have already described points on the unit circle with sine and cosine: \((\cos (\theta ), \sin (\theta ))\).

Therefore, we have already described complex numbers on the unit circle with sine and cosine: \(\cos (\theta ) + \sin (\theta ) \, i\).

All of this came from investigating the right triangle formed from points on the unit circle.

Learning Outcomes

In this section, students will

• represent real numbers on the 2D number line in polar form.
• examine the unit circle via right triangles.
• decompose vectors into perpendicular components.
• describe components with sine and cosine.
• describe complex numbers with this information