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
The Dissertation on Imaginary Numbers Number Root Complex
... complex number equation = a+bi. It is a real number with an imaginary number. Quadratic Formula and Imaginary numbers Throughout our lifetime, teachers have informed students that negative numbers cannot be squared. ...
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).
The Coursework on British People Blitz Shows Source
History Coursework Britain in the age of total war 1939-1945 The statement "The impression that the British faced the blitz with courage and unity is a myth", can be argued for and against. The British people were supposed to have been showing the British "grit" but this seemed to deteriorate towards the end of the blitz. People lost interest and just wanted it to stop. The people became very ...
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
