{ Michael Gale } | Web Developer - Melbourne, Victoria

First Week of Maths

The first couple weeks of my maths bridging course

Pre-preamble: I wrote about going back to school to study mathematics in Back to school: how much will I suck before census? I’m following that up here (mostly as a self-soothing exercise) but I will also attempt to answer that question in due time. Follow along with my learning if you like; but this is really just for my records, I guess.

💡 Note: This page uses the “mathjax” JavaScript library to convert markdown LaTeX formulas into nicely formatted mathematical expressions. If the math expressions look a little funky while you’re reading; try refreshing the browser.

Preamble: Real Numbers

It stands to reason that when starting a course called Foundations of Mathematics, the course coordinator would want to specifically state: this is what counts as far as our meta-language; or This is what we will be covering in the course. It sets useful parameters and expectations and is supposed to set everyone at ease. No fearing the unknown. And the same goes here - the very first thing we discuss is; what are “real” numbers.

Real numbers are made up of Rational and Irrational numbers.

Rational numbers CAN be expressed as a fraction. eg:

$$1.25 = \frac{5}{4} = \frac{20}{16} = \frac{500}{400}$$

Irrational numbers CANNOT be expressed as a ratio of two whole numbers, eg:

$$\pi = 3.1415927$$

An example of numbers that are not “real” are the square roots of negative numbers. The following doesn’t exist for our course:

$$\sqrt{-5}$$

Comforting. Forget about that!

However, a negative of square root is fine:

$$-\sqrt{5}$$

Cartisian plane / Coordinate system

Cartisian plane

Descartes, responsible for exposing relationships between algebra and geometry link.

I’ve quoted (purposefully misquoted, actually) Descartes in my music before, but never put it together that “Cartesian” meant something akin to “of Descartes”. The screen grab above is taken from a video from Khan Academy; an extremely useful (and free) resource for mathematics students; with my own scribbles (in red) over the top.

I was given a loose outline of what to expect from the course before it began, so I started to try and layout some important terms and concepts in my mind prior to week 1. Please keep in mind, I have absolutely no experience with any of these terms, beyond what I’ve picked up from programming in JavaScript, or positioning things with CSS. So my instincts in a lot of cases are to relate things back to that.

Important terms

Cartesian Coordinate system

Given “independent” foo you will always get “dependent” bar. With multiple expressions, this can be written in the table format, as per above ↑, or in a geometric graph. The straight line passing through the graph version will actually reveal the correct bar for any given foo. However, we’re really interested in the algebraic versions of those graphs - due to their superior accuracy!

Ordered Pair

$$(-2,-5)$$

An ordered pair is just a pair of objects. In our case, it will usually reference a pair of coordinate values.

Coordinates

Coordinates are a set of values that show an exact position, or point. A pair of coordinates represent a point on the cartesian coordinate system: Always x, then y. (←→, ↑ ↓). Hence the significance of the name ordered pair. Two of these points can be represented as a line, by joining the dots!

Coordinate Pairs

Parallel Lines

Lines running along side each other. It might seem simple to identify these visually - but can we be absolutely mathematically certain that the two lines are parallel, just by eye-balling a graph? If only there were some formula to compare the two lines…

Parallel lines example

Perpendicular Lines

The definition: Lines at a 90° angle to a given plane. As above though, we should be able to work this out mathematically / scientifically to ensure accuracy in our calculations.

Independent and Dependent variablesIndependent and Dependent variables

The independent variable influences change on the dependent variable.

Range and Domain

When I think of “domain” I think of the Zora’s Domain level from the 1997 Nintendo game “Ocarina of Time”.

Range really means something like “possible values”…?

The above statement says that we can’t have fewer or greater than ONE King Ruto, we can have 1-2 signs, a torch, and 2 staircases. but not 0. The domain of Zora’s domain is [1, 2, 2].

I know this is probably a terrible way to remember these definitions, but this is where my brain is going and it’s not stopping with this analogy anytime soon.

So, with all these definitions out of the way - we can finally start working on our actual expressions. Which I’ll save for another blog. When I have time. Which might not be for quite some time…

Peace, and good viebs everyone ~~~

😘 Michael

This will make sense later