Although I didn’t get as far as I wanted to this week, I was happy with the amount of work I completed and, even better, actually enjoyed doing it (loosely speaking). I began this week halfway through the unit Complex Numbers and spent the entire week using trigonometry to find the coordinates of complex numbers on the complex plane. I was happily surprised that I remembered ~90-95% of what came up that required trigonometry and the few things that I didn’t initially understand didn’t take me long to figure out. As always seems to happen, for the most part trigonometry seemed fairly straightforward and unambiguous having taken half a year off from it and having time for it to settle in my mind. Even the things I didn’t immediately understand didn’t seem too difficult to quickly wrap my head around. Considering how tough I found trig as I was going through it, it was incredibly satisfying being able to get through it all relatively easily.

I started the week off by going through the meaning of the Absolute Value of a complex number which is the total distance the complex number is away from the origin, a.k.a. (0, 0i), regardless of which direction. 

This is why trigonometry is necessary as the absolute value of the complex number, r, can be thought of as the hypotenuse of a right triangle where the real part of the number, a.k.a. the X-coordinate, is the adjacent side and the imaginary part of the number, a.k.a. the Y-coordinate, is the opposite side. As shown in the photo above, you can find the complex number’s absolute value by using Pythagoreans Theorem.

After that, I was then shown a few more variables used to denote parts of complex numbers and the complex plane:

• Notation used in Complex Numbers:
  
  • Complex number = z
    • Ex. z = (3 – 4i)
  • Real part of Complex number = Re(z) = a
    • Re(z) = 3
  • Imaginary part of complex number = Im(z) = b
    • Im(z) = (-4)
  • Absolute Value = “Magnitude” = r
    • r(z) = √(3^2 + 4^2) = √25 = 5
  •  Angle of complex number = “Phi” = Φ
    • For some reason, instead of using Theta, θ, to denote the angle of a complex number you more often use Phi.
    • The following is an example of how Phi would be labelled on the complex plane:

It was then explained to me that when a complex number is written as (a + bi) this is called the Rectangular Form (no idea why it uses the word rectangle) but complex numbers can also be written in another form called the Polar Form which is (r)(cos(Φ) + (i) * sin(Φ)). Here’s a photo from my notes to help explain how you derive the polar form from the rectangular form:

In the top left corner of the photo you can see a complex number, z, marked down on a complex plane. You can also see how a right triangle is created with z where the opposite side equals b, the adjacent side equals a, and the hypotenuse equals the magnitude of the complex number, r. Phi, Φ, is the angle from the X-axis to the hypotenuse, a.k.a. r. 

As shown in the middle of the photo, you can use SOH CAH TOA to define the opposite and adjacent sides:

• SOH
  • Sin(Φ) = Opp/Hyp = b/r
    
  • (r)sin(Φ) = b
    
• CAH
  • Cos(Φ) = Adj/Hyp = a/r
    
  • (r)cos(Φ) = a

You can then substitute the derived trigonometric equations for a and b into the rectangular form, then pull out r from both terms to derive the polar form:

• Rectangular Form = (a + bi)
  
  • a = (r)cos(Φ)
  • b = (r)sin(Φ)
    
  • ((a) + (b)i) = ((r)cos(Φ) + (r)sin(Φ)(i))
    • (r)(Cos(Φ) + sin(Φ)(i))
      
• Polar Form = (r)(Cos(Φ) + sin(Φ)(i))

After learning the differences between these two forms of complex numbers, I spent most of the week working through exercises that had me convert one form to the other, come up with the values for a and b or (r)cos(Φ) and (r)sin(Φ), and/or determine the angle of a complex number. Half of the time I was asked to determine the angle of a complex number I was asked to use radians which I always used to struggle to understand but didn’t seem too difficult this week. WOO!!

Lastly, many of the questions I worked through had complex numbers that created 30-60-90 triangles and 45-45-90 triangles. I would be given the length of one of the sides and asked to determine the length of the other two sides. I had to review what the ratio of the side lengths were for each type of triangle which are:

In the past, I never fully wrapped my head around how the ratios of the side lengths of each triangle were derived. This week I realized you find the ratio of the sides of a 30-60-90 triangle by splitting an equilateral in half and using Pythag’s theorem to come up with the ratio of each side. Using the ratios, if you know the length of one of the sides you can then multiply that value by the other side’s ratios to come up with each side’s value. The same principle is applied to 45-45-90 triangles. When I went through these questions in the past, I always found them very difficult to understand but going through these questions this week it all seemed fairly straightforward and clear. Again, WOO!!

I’m now 80% of the way through this unit Complex Numbers (1600/2000 M.P.) and only have one section remaining which deals with multiplying and dividing complex numbers in polar form. I watched a few videos from this section and they seemed pretty tough to understand. It seems like the type of thing that I’ll eventually be able to figure out but at the moment seems like a complete enigma. I think it’s possible for me to get through this unit by Wednesday which would give me 3 days to work on the following unit, Polynomials (0/800 M.P.). My goal this week is to get 50% of the way through the ladder unit. I feel like that’s setting he bar at a reasonable level so hopefully I’ll get it done. 🤞🏼