Why We Need Complex Numbers We all know that there are certain real life situations that can be modeled by mathematical equations. For a simple example, if someone has a certain amount of apples, a, and you give them 6 more apples and they now realize they have 10 apples, we can use the equation a+6=10 to find that they originally had 4 apples. In order to solve an equation like this, we need to be able to use the integers. These are zero and positive and negative whole numbers. Let’s say you and two friends order a pizza and you want to decide how to evenly divide up the 8 slices. We can model this situation with the equation 3p=8 and find that you should each receive 8/3, or 2 and 2/3, slices. Splitting individual slices into thirds may be impractical, but nevertheless we are given a solution to the problem. The only way we can solve this is by dividing an integer by another integer. These are the rational numbers. Now let us say that you throw a ball in a parabolic trajectory to a maximum height of 98 feet. The ball’s path can be described by the equation -x2+98=0. By solving this equation, we find that x=±7√2. The number 7√2 cannot be represented by one integer divided by another integer because it is an irrational number. This forces us to expand to the real number system. What if we are given the equation z2+1=0? Rearranging this equation tells us that z2=-1. Since we know that squaring any number will give us a positive number, we know that no negative or positive number or 0 will be a solution to this equation. So how can we make sense of this? We again need to expand our number system to include what we call the complex numbers. By letting i=√-1, we are now able to work with a whole new range of equations. While i is called an imaginary number, it is essential to some real world fields, such as electrical engineering, quantum mechanics, cartography, and many others. Working in the Complex Plane When working with equations that force us to expand to the complex numbers, we may recognize that some equations require us to have not only the imaginary component, but also a real component. For example, when we solve the equation (z-3)2=-25, we get z=3+5i with 3 being the real part and 5i being the imaginary part. So that we can effectively work with complex numbers, it can be helpful to geometrically visualize them and establish the Complex plane. Similar to the Cartesian plane, we set up a vertical axis that is perpendicular to a horizontal axis. However, in the Complex plane we say that the

horizontal axis is the real component and the vertical axis is the imaginary component. So using this example complex coordinate, z=3+5i, we would graph the point like this:

Multiplication by i Now we want to investigate what happens when we multiply a complex coordinate by a scalar. Let’s continue with our example point, z=3+5i. If we multiply this by -1 we find that -1z=-1(3+5i)=-3-5i. After graphing this we see that z has been rotated 180° about the origin (we can easily see this is a 180° rotation because the arc forms a semicircle):

Since we know that i=√-1, we can substitute (i*i) in for -1 and so (i*i)z=(i*i)(3+5i)=i*(-5+3i)=-3-5i. So we can see that after distributing i into the coordinate (the second equality) that the point is rotated by 90°. Then we see that distributing the second i we get another rotation by 90° giving us a total rotation of 180°.

So we can see that each time we multiply the complex coordinate by i, we will get a rotation of 90°. What inferences can be made about subsequent powers of i? Moreover, can you conjecture general formulas for all subsequent powers of i? After thinking about these questions, you can investigate them with the interactive tool below. Interactive tool for Multiplication by powers of i Link to jsp file Link to Valerie’s project Link to Travis’ project

In order to see what happens when a complex coordinate is multiplied by i, you can use this interactive tool. Follow the directions listed below.

Explanation of Process Let the green line be our original vector. 1. Drag the end point of this vector to any point on the complex plane. You can see that the red, blue, and pink vectors will vary when you change the direction and magnitude of the vector. 2. Notice the coordinates of the endpoints of each vector. What is the relationship between each vector and the green vector?

You should notice that the red vector represents the green vector multiplied by i. The blue vector represents the green vector multiplied by i^2. The pink vector represents the green vector multiplied by i^3.

3. Line the green vector along the x-axis. You should notice that the red and pink vectors are along the y-axis and the blue vector is along the x-axis. This means the red and pink vectors are perpendicular to the green vector and the blue vector is on the same line as the green vector.

So it is obvious, since the relationship between the 4 vectors does not change, that the red vector will always be a 90 degree rotation of the original vector, that the blue vector will always be a 180 degree rotation, and that the pink vector will always be a 270 degree rotation in relationship to the originally given vector.