UGH!! For the third week in a row, I once again didn’t get through the unit Complex Numbers. 🙁 BUT, apart from that, I was happy with how my week went. I now feel like I have a solid understanding of the different forms of complex numbers (except for exponential form which I’ll explain why) and how to multiply, divide complex numbers as well as raise them to different powers. This week I got a lot of practice using SOH CAH TOA, 30-60-90 triangles, adding, subtracting, and multiplying exponents, calculating angles in both degrees and radians, and using trigonometry on the complex plane. Although my progress has been much slower than I wanted it to be, it’s nice feeling like I truly understand everything I worked through this week for the most part.

Throughout the week I worked on manipulating complex numbers in polar form which was helpful for me to strengthen my grasp of the different components of the polar form and how trigonometry is involved. I always found trig difficult to understand so spending much of the week working with trig helped simplify the formulas in my mind. Here’s a page from my notes that goes over the components of the polar form:

Moving forward from there, the most difficult and frustrating part of the week was understanding what the exponential form of a complex number is and learning about Euler’s Formula. Before I explain, here’s a photo of my notes that shows how the exponential form of a complex number relates to the polar form:

The way Euler’s Formula and the exponential form were brought up this week, it was as if I should have already learned about both but I’m sure I’ve never been shown either. I had to spend a day and a half Googling videos to try and understand how the exponential form works which was very frustrating. I found a video from another website that helped me understand the gist of how to interpret the exponential form, but it didn’t help me understand how it’s derived from the polar form via Euler’s Formula. Although I don’t fully understand exponential form, I resigned myself to simply accepting it as a given and moving forward.

I realized that taking the exponential form of a complex number is useful because it helps you to multiply and divide complex numbers by taking them to the (i)x^th power where x equals the angle. Before I explain, to reiterate the photo from above, the two forms are:

• Polar Form:
  • (r)(cos(θ) + (i)sin(θ))
• Exponential Form:
  • (r)e^θi
    
• (Note: I don’t know why it works, but all you do to convert a complex number from polar form to exponential form is take the angle, θ, out of polar form and drop it into the formula (r)e^θi.)

Using the complex form makes it easier to understand how to multiply and divide complex numbers by using exponent rules. As a quick recap, here’s an example of how/why adding and subtracting exponents with the same base is acceptable when multiplying and dividing them, respectively:

• 2⁴ * 2² = (2 * 2 * 2 * 2) * (2 * 2)
  • = 2 * 2 * 2 * 2 * 2 * 2
  • = 2⁶
  • = 2^{(4 + 2)}
• 2⁴ / 2² = (2 * 2 * 2 * 2) / (2 * 2)
  • = 2 * 2 * 2 * 2 / 2 * 2
    • (Note: Two 2’s in the numerator cancel out with the two 2’s in the denominator.)
  • = 2²
  • = 2^{(4 – 2)}

So, knowing how exponents work when multiplying and dividing number with the same base, here’s an example of how multiplying and dividing complex numbers in exponential from would work:

(Note: The angles are in degrees but the degree symbol, °, is in a weird spot.)

• e⁶⁰°ⁱ * e^-45°ⁱ = e⁽⁽⁶⁰° ^{+ 45}°⁾ⁱ⁾
  • = e¹⁰⁵°ⁱ
    
• e⁶⁰°ⁱ / e^-45°ⁱ = e⁽⁽⁶⁰° ^{– 45}°⁾ⁱ⁾
  • = e¹⁵°ⁱ

By understanding both of these concepts, you can then understand why the following formulas work when multiplying and dividing complex numbers in polar form works:

I still find the idea of adding and subtracting angles in polar form a bit abstract and can’t visualize in my head how it would work on a complex plane. I’m hoping that, if/when I understand how exponential form is derived by using Euler’s Formula, I’ll be able to wrap my head around this concept.

To understand how to raise complex numbers to certain powers, I find it useful to think of simple examples of how it works in general. A quick recap/example would be:

• (2²)³ = (2²) * (2²) * (2²)
  • = (2 * 2) * (2 * 2) * (2 * 2)
  • = 2 * 2 * 2 * 2 * 2 * 2
  • = 2⁶
  • = 2^{(2 * 3)}

Here’s a page from my notes that goes through an example of how taking a complex number in exponential form to a power relates to taking that same complex number to the same power in polar form:

(FYI, in the example the angle on left side of the equation is in radians and the angle on the right side is in degrees. I did this for my own sake to practice understanding how radians and degrees relate to each other but I’m not sure if this would normally be acceptable to do.)

Things to note form this example are:

• The magnitude of the complex number before being raised to the power of 6 is r = 2. You can see that both sides of the equation separate r from the rest of the equations and r is raised to the power of 6 separately.
• In polar form, when you raise the complex number to the power of 6 you simply multiply the angle by 6. You can see how this relates to multiplying the angle on the left side of the equation in exponential form.
• In polar form where it states 64(cos(360°) + (i)sin(360°)) = 64(1 + (i)0), this is because if you go 360°around the unit circle you’re back to 1 on the X-axis and 0 on the Y-axis. The four notable cosine and sine values to remember when using the complex plane are:
  • Cos(k * 360°) = 1
  • Cos(180° + (k * 360°)) = -1
  • Sin(90° + (k * 360°) = i
  • Sin(270° + (k * 360°) = -i  
    
  • (Note: k can only be an integer.)

Lastly, here’s a photo from my notes shows the formula for taking a complex number, z₁, to the power of n:

It’s also worth noting that if you take a complex number to a fractional power, ex. z^1/3, that’s the same thing as finding the cubed root of that number so when working in polar form you would divide the angle by 3.

This coming week, I’m 99% sure I’ll be able to get through the unit test of Complex Numbers (1950/2000 M.P.) on Tuesday which will give me most of the week to get through unit Polynomials (0/800 M.P.). There are only 5 sections within Polynomials so I’m hoping it won’t be as difficult to get through as Complex Numbers has been. Looking back in my notes, the first time I worked on polynomials was in Week 7 so hopefully I’ll still remember how they work. Compared to statistics, I’ve really enjoyed working on complex numbers. I view them as little puzzles which I need to solve and I’m hoping working through Polynomials will feel similar. I still wouldn’t say learning math has been “fun” but, as I’ve said before, working through these puzzle-like problems can be pretty satisfying and even enjoyable in a way. Especially when compared to stats. 🤮