In the first week of the second half of my first year (pretty sure that’s right), Week 27, I got through the first two units of the course Algebra 2, Polynomial Arithmetic and Complex Numbers. To start this course, I was happy that I was getting back into Algebra and, also, that it seemed likely that I’d finally be learning about things I had never worked on before. Up until this point, most, if not all, of what I’ve covered have been things that I remembered learning about in high school (albeit vaguely). This week, learning about the imaginary unit i and complex numbers, I definitely was introduced to things I was never taught in high school.
The week started off with polynomial arithmetic which was all review of things I learned in Algebra 1. It was nice going through this unit as a) I find this type of math very satisfying to work through (as I mentioned in other posts, the equations are like puzzles to me), and b) I remembered nearly everything I needed to know to get through the unit which was hugely gratifying. One thing I made a note of in my workbook to mention here is that when taking the product of two polynomials where you’re subtracting one from the other, you need to remember to multiply the terms in the second polynomial by negative one.
- (-w^3 + 8w^2 – 3w) – (4w^2 + 5w – 7)
- = (-w^3 + 8w^2 – 3w) + (-1)(4w^2 + 5w – 7)
- = (-w^3 + 8w^2 – 3w) + (-4w^2 – 5w + 7)
- = -w^3 + 8w^2 – 3w + -4w^2 – 5w + 7
- = -w^3 + 8w^2 – 3w + -4w^2 – 5w + 7
- = -w^3 + 8w^2 – 4w^2 – 3w – 5w + 7
- = -w^3 + 4w^2 – 8w + 7
The unit briefly touched on “special products of polynomials” which covered the following equations:
- (a + b)(a – b)
- = a^2 – b^2, and
- (a + b)^2
- = a^2 + 2(a)(b) + b^2
Going through some of the unit’s exercises made it clear that it can be much easier to simply remember the product of each of the “special” polynomials and simply insert the values of a and b into the resulting product equations. When I did the practice questions, I first used the “shortcut” method by simply taking the a and b values and putting them into the second equation, but I also went back and did the math to verify that it worked. The unit seemed to emphasize that knowing these “shortcuts” will make things a lot easier going forward.
It was the second unit, Complex Numbers, that introduced me to things I’d never seen before, the main thing being the imaginary unit i. As far as I have surmised, the imaginary unit i was created to be used in place of a -1 inside of a square root. For example:
- {-9}
- ={(i * i)(3 * 3)}
- =3i
I’d learned in past units that there are times in math where you get a negative number inside a square root. Since any number squared equals a positive number however, it is impossible for a square root to naturally have a negative number inside of it. It seems that this is why the imaginary unit i was created, to be able to substitute itself into a square root to take the place of a -1.
An interesting pattern that is created when raising i to successive powers is as follows:
- i to the power of {…, -8, -4, 0, 4, 8, …} equals 1,
- i to the power of {…, -7, -3, 1, 5, 9, …} equals i,
- i to the power of {…, -6, -2, 2, 6, 10, …} equals -1,
- i to the power of {…, -5, -1, 3, 7, 11, …} equals –i,
After learning about the imaginary unit i, I was then taught about the number set known as Complex Numbers. These are numbers in which you have both a Real Number and an Imaginary Number together in what looks like an equation (ex. “2 – 6i”), but which is simplified as much as it can be and is therefore considered a single number. It’s impossible to combine a Real Number and an Imaginary Number so, though they look like an equation, Complex Numbers are considered a single number just like any other more typical number.
Going through this part of the unit, it seemed apparent to me that I should go back and relearn the different number sets and try to memorize them. I made a chart in my workbook to help me.
- Natural Numbers
- The whole numbers from 1 and up.
- {1, 2, 3, …}
- (Some mathematicians include 0 in this set of numbers.)
- Integers
- All whole numbers in the positive and negative directions (i.e. no decimals).
- {…, -3, -2, -1, 0, 1, 2, 3, …}
- Rational Numbers
- Any set of integer numbers that can be used in a fraction (i.e. numbers including decimals).
- 1/1, 3/2, -1/1000
- Irrational Numbers
- A number that goes on forever but does not repeat.
- Pi, {2}, e, the Golden Ratio
- Real Numbers
- All rational and irrational numbers.
- (All numbers that don’t include imaginary numbers.)
- Imaginary Numbers
- Numbers that, when squared, produce an imaginary product.
- Any negative number inside a square root.
- {-9}, 3i, {-16}, {-144}
- Complex Numbers
- A number that includes a Real Number and Imaginary value.
- 1 + i, 2 – 6i, -14 + 0i, 0 + -3i
- (Because 0 can be used in place of a Real Number or an Imaginary Number, all numbers are considered Complex Numbers.)
I’m happy that I managed to get through two units this week. As I mentioned in my last post, it looks like I’m going to have to get through two units per week on average to get into the calculus courses by the end of the year. The next two units in this course which I need to get through are Polynomial Factorization (100/1000 M.P.) and Polynomial Division (0/800 M.P.). My guess if that the former will mostly be review and shouldn’t be too difficult. I can’t remember if I did any polynomial division in Algebra 1 so I think that may be something brand new to me and potentially difficult to get through. If I can manage to keep up the pace of two units per week, I’ll be finished this course in 6 weeks by the middle of April. I think that will be tough but I always find setting up a goal-timeline to be useful.