Well, so far calculus is much more enjoyable to learn than statistics. I spent the entire week working on the unit Complex Numbers and only got halfway through but, even still, got through a lot of material. The majority of my time was spent working through algebraic equations which I found more compelling than nearly everything I worked on in statistics. Although I was a bit rusty, for the most part I remembered how to solve the equations which I was happy about. I also got to work on basic (x, y) graphs which wasn’t too difficult and was, again, quite satisfying remembering how to solve the equations. I can already tell I’m going to enjoy calculus more than statistics.

My week began by reviewing the imaginary unit, “i”. The most important thing to remember when thinking about this unit is that i^2 = -2. From what I’ve come to understand, the purpose of this value is to make it possible to solve equations that have a -1 inside a square root. At this point, I don’t know too much more about this unit other than that but I have a feeling it will be very important to know as I get further into calculus. Here’s a page from my notes that covers the pattern of values you get when taking i to different powers:

It may be hard to tell from the photo, but there are four different values you can get by taking i to different powers. To understand the pattern, it’s easiest for me to remember what i equals when taking it to the powers of zero, one, two, and three:

• i^0 = 1
• i^1 = 1 * i = i
• i^2 = -1
  • (Note: you must take this as a given.)
• i^3 = i^2 * i = (-1) * i = -i

Knowing these four powers of i, it’s not too difficult to figure out the value of any other power of i. for example:

• i^8 = (i^2)^4
  • = (-1)^4
  • = (-1) * (-1) * (-1) * (-1)
  • = 1 * 1
  • = 1

The imaginary unit is necessary to know in order to understand complex numbers. To be honest, I don’t know much about complex numbers at this point or why they’re important. As far as I know, the definition of a complex number is that it’s a single number that is comprised of a real and imaginary part. Here’s another page from my notes that shows a few examples:

As you can see, any real or imaginary number is also a complex number since you can always put a 0 in the place of the missing part. I was then shown the Complex Plane which made the concept of complex numbers having a real and imaginary part to them become easier to understand:

As you can see from the photo, the X-axis is used to identify the real part and the Y-axis is used to identify the imaginary part. When writing a complex number, you always write the real part first followed by the imaginary part just like you’d always write the X-coordinate first followed by the Y-coordinate (as in (x, y)) when describing (x, y) coordinates. In the photo I used the example (-2 + 4i) which lands in the second quadrant of the (x, y) graph. It’s worth noting that the conventional letter used to denote complex numbers on a graph is the letter z, using z_1, z_2, etc. when marking down more than one complex number on the graph.

Next I moved on to adding and subtracting complex numbers. To do so, you use simple arithmetic. Here’s an example of both addition and subtraction:

(Note: for both examples, z_1 = (2 + 4i), and z_2 = (-3 – 6i).)

• Addition:
  • z_1 + z_2 = (2 + 4i) + (-3 – 6i)
    • = 2 + 4i + (-3) – 6i
    • = 2 + (-3) + 4i – 6i
    • = (-1 – 2i)
• Subtraction:
  • z_1 + z_2 = (2 + 4i) – (-3 – 6i)
    • = (2 + 4i) + (-1)(-3 – 6i)
    • = (2 + 4i) + (3 + 6i)
    • = 2 + 4i + 3 + 6i
    • = 2 + 3 + 4i + 6i
    • = (5 + 10i)

I then reviewed multiplying complex numbers which was fairly straightforward. I remembered how to do it, but was shown a video on the the acronym used to describe the process of multiplying binomials which is FOIL:

• FOIL – First, Outer, Inner, Last
  • Example: (a * b)(c * d)
    • First: a * c
    • Outer: a * d
    • Inner: b * c
    • Last: b * d
      • All together: (a * c) + (a * d) + (b * c) + (b * d)

The key when multiplying complex numbers is to remember that i^2 = (-1). This is important because inevitably at some point you’ll be left with a term containing i^2 at which point you need to substitute (-1) in for that variable and then solve the remainder of the equation. Here’s an example:

• (1 + 4i)(5 + i) = (5 + i + 20i + 4i^2)
  • = 5 + 21i^2 + 4(-1)
  • = 5 + 21i^2 + (-4)
  • = 5 + (-4) + 21i^2
  • = (1 + 21i)

After reviewing how to multiply complex numbers, I then learned something new which was how to divide complex numbers. In order to divide a complex number, I first had to learn what the Conjugate of a Complex Number is:

A complex number’s conjugate is, simply put, the exact same number as the given complex number except the (+/-) sign in front of the imaginary part of the conjugate is changed to the opposite sign of the given complex number. (It’s much more simple than that description made it seem.) Here are three examples:

Complex Number	Conjugate
(3 + 4i)	(3 – 4i)
(-18 – 23i)	(-18 + 23i)
(0 + 44.5i)	(0 – 44.5i)

The reason why I had to learn about the conjugate of complex numbers is that, when you multiply any complex number by its conjugate, you’re left with a real number which is helpful for dividing complex numbers. The idea is that when you’re dividing two complex numbers by each other, you multiply both the numerator and denominator by the conjugate of the denominator (it’s helpful to remember that you can dot his since when you multiply the numerator and denominator by the same value it’s the same thing as multiplying the entire equation by 1 so you’re not altering the equation). This leaves you with a real number in the denominator allowing you to solve the equation. For example:

• (6 + 3i) / (7 – 5i) = (6 + 3i) * (7 + 5i) / (7 – 5i) * (7 + 5i)
  • = (42 + 30i + 21i + 15i^2) / (49 + 35i – 35i -25i^2)
  • = (42 + 51i + 15(-1)) / (49 -25(-1))
  • = (42 + 51i + (-15)) / (49 + 25)
  • = (42 + (-15) + 51i) / 74
  • = (27 + 51i) / 74
  • = ((27/74) + (51/74)i)

Lastly, I learned how to factor the type of equations known as Sum-of-Squares equations. This also helped me figure out why the type of equations know as Difference-of-Squares, which I learned about a long time ago, are given that name. The basic equations of each would be (x^2 + y^2) and (x^2 – y^2) for the sum-of-squares and difference-of-squares equations, respectively. I had a “duuuh” moment when I realized that sum-of-squares simply means adding two numbers/variables that are both being squared and difference-of-squares means the same thing but subtracting them. (I definitely felt like an idiot for not understanding why difference-of-squares equations used that name before this.) Here’s a photo from my notes that goes over how to factor sum-of-squares equations:

From the photo, the key thing to remember is that in step (1) you can rewrite the equation from x^2 + y^2 to x^2 – (-1y^)because adding y^2 is the same thing as subtracting (-1y^2). From there, you can substitute i^2 in for the (-1) and then factor the remainder of the equation using FOIL. As I write this post, I just realized that this is the first type of equation I’ve learned where knowing the imaginary unit i comes in handy to solve the equation which otherwise wouldn’t have been solvable (I think).

As I’ve mentioned a few times, I don’t really understand why complex numbers and the imaginary unit are important to know about but I’m pretty sure they’ll both be a big part of calculus going forward. This is always the tricky part for me when learning a new subject. It feels like I’m walking through a dark room trying to figure out where I am and what’s in the room using only my hands (a weird metaphor, I know). The good thing, however, is that after 69 weeks of KA I’m starting to feel comfortable feeling uncomfortable, so to speak. I’m hoping to get through this unit, Complex Numbers (1360/2000 M.P.), by the middle of this week and then get through a good chunk of the following unit, Polynomials (0/800 M.P.), before the week’s over. Once I get through both of these units, I’ll be ~55% of the way through the course. My goal is to get through the course by the end of January but that might be a bit over ambitious. Always best to aim high though!