A.1. Explain how the complex number system is an extension of the real number system

A complex number is an extension of real numbers because it can be a combination of real numbers and imaginary numbers. Examples of real numbers are 1, 34.67, -5; pretty much any number is a real number. What makes imaginary numbers unique is when they are squared, they yield a negative result. It’s difficult to imagine this because when you square a positive number, you get a positive result. When you square a negative number, you also get a positive result because a negative times a negatives gives a positive. We use the letter i to denote the imaginary number i is equal to the square root of negative one. This comes in handy when we get an equation like + 1 = 0 where we end up with = -1 and we have to square root each side we end up with x = so x is i. Another cool thing about imaginary numbers is that the value of letter i is cyclical. For example, i = ; = -1; = -; = 1; and then is starts over so that = because and 1 = and this continues in cycles of four

2. Describe the individual parts of a complex number.

So a complex number is any real number added or subtracted by a real number multiplied by i. An example of this is 2 + 3i. So the real number part is the 2 and the imaginary number is 3i and, therefore, making it a complex number. 3. Explain how complex numbers combine under the following operations: Addition and division Combining complex numbers through addition is pretty simple. Let’s take the equation + Let’s put the two parts over each other so it looks like an elementary math problem

We get

Dividing

One way to approach this is to remember that a simplified version of this equation looks like this We have to multiply by the conjugate × Next we have to multiply this through and we’ll use the foil method to do that. We get But we remember that i squared is -1 so simplifying this further we get = and that doesn’t simplify any further, unless you want to say (Burger, 2014)

Use one supporting example for each operation; include both algebraic and graphical interpretations in your responses

4. Verify De Moivre’s theorem for n = 2

When we write this out to simplify it, keeping in mind and substituting values that and , it becomes z z and r Which then becomes

And that turns into

Which is

(Blitzer, 2010)

xy = r (cos u + i sin u) • t (cos v + i sin v) we distribute the r and the t through = (r cos u + ir sin u) • (t cos v + it sin v) and then we use the foil method to get rt cos u cos v + irt cos u sin v + irt sin u cos v + iirt sin u sin v then factor in the imaginary values to get rt cos u cos v + irt cos u sin v + irt sin u cos v + (-1)rt sin u sin v i • i = ✁ 1 and after swapping the terms we get

rt cos u cos v ✁ rt sin u sin v + irt sin u cos v + irt cos u sin v then carry the terms through to get rt cos u cos v ✁ rt sin u sin v + i rt (sin u cos v + cos u sin v ) Then multiply everything through to get

rt [cos u cos v ✁ sin u sin v + i (sin u cos v + cos u sin v )] which then becomes rt [ ( cos u cos v ✁ sin u sin v ) + i (sin u cos v + cos u sin v )] and now we can insert trig identities to get rt [ ( cos ( u + v )) + i (sin u cos v + cos u sin v )] and then rt [ ( cos ( u + v )) + i (sin ( u + v )] and it its polar form we get rt [ cos ( u + v ) + i sin ( u + v )]

Therefore the modulus of xy = rt which is the product of moduli. This proves the statement about amplitudes. Amplitude of x = u. Amplitude of y = v. Amplitude of xy = (u + v). Therefore the amplitude of (xy) is the sum of the amplitudes of x and y. (Blitzer, 2010)

Burger, E. (Performer) (2014). Dividing complex numbers[Web]. Retrieved from

http://wgu.thinkwell.com/cf/play.cfm

Blitzer, R. (2010). Algebra and trigonometry, fourth edition. (fourth ed., p. 712). Upper Saddle River: Prentice Hall. Retrieved from http://media.pearsoncmg.com/ph/esm/esm_blitzer_bzat4e_10/ebook/bzat4e_flash_main.html?chapter=null&page=706&anchory=null&pstart=null&pend=null Blitzer, R. (2010). Algebra and trigonometry, fourth edition. (fourth ed., p. 695). Upper Saddle River: Prentice Hall. Retrieved from http://media.pearsoncmg.com/ph/esm/esm_blitzer_bzat4e_10/ebook/bzat4e_flash_main.html chapter=null&page=616&anchory=350&pstart=null&pend=null