There are very few principles in mathematics more important than the following: be as general as possible. This won't make sense off the bat, so let me explain. Expect me to go off on many different tangents (hopefully finitely many); I find it way too easy to talk about this stuff.
Since you're starting unit 1, I'll use the example of simultaneous linear equations. They're presented in this course after they're already been abstracted by centuries of mathematicians, so you see a lot of questions with the same structure: a pair of linear equations in variables
and
. Why this structure is important or special is not really addressed; at least, it wasn't when I was doing unit 1.
To attempt to explain this (and it will take me a while to get to the point, so bear with me), I will steal an example from A Mathematician's Lament (which I highly recommend reading by the way - google it). If we are given the sum and difference of two numbers, can we find those two numbers? If so, how?
The answer to the first question is yes, we can in fact always find two numbers given their sum and difference. But for my own example, let's say we had less information... just their difference, perhaps. Say we're given that the difference of two numbers is 2.
Well, let's just call the larger number
and the smaller number
. Their difference is 2 and
is the larger one, so we can conclude
, or in perhaps a more familiar form,
.
From your high school training, you'd know what this is. It can be represented by a straight line if you represent
and
in the standard Cartesian format. In this example, the interesting thing is that we don't get a unique pair of numbers (since we have a whole line of pairs). There are infinitely many numbers whose difference is 2. This agrees with common sense; pick any number (there are infinitely many of those!) and add 2 (there's always a larger number!), and you've got your pair.
But anyway, what I've hopefully illustrated here is that something fell out of the problem that you could recognise - a straight line equation. This is because we stumbled across a particular straight line, so it was easy to recognise if we already knew about them. Of course, a more general straight line equation can be used for more things than just this problem, and this is an example of why generality is useful.
However, I chose the symbols
and
because the equation
would be more easily recognised as a straight line equation than if I used, say,
and
. This is perhaps one of the pitfalls of convention; it's easy to miss things if they're not written in a certain way. This is why it's important to look for underlying structure.
Looking for structure ties into generality - sometimes seemingly unrelated objects can share certain features, and you can apply the same ideas in analysing them. In the context of the difference problem, would it matter if we had the equation
instead? It certainly wouldn't affect the underlying idea that there are infinitely many pairs of numbers with a difference of 2. So it's important to realise when things are exactly the same thing but for a few labels.
Let's go back to the initial problem, so we know the sum of the two numbers as well; we'll say the sum of two numbers is 10 and the difference of those two numbers is 2. For the sake of variety, we'll call the larger number
and the smaller number
. So we have
and
.
If you sketch these on
- and
-axes, you'll get two straight lines that intersect at some point. I noted before that the difference equation, on its own, provided you with infinitely many pairs of numbers with a difference of 2. Similarly, there are infinitely many pairs of numbers whose sum is 10. These pairs are represented here by points on a plane. There is exactly one intersection point, which tells us that there is exactly one pair of numbers satisfying both of these properties. With some work, you'll find this pair of numbers is 4 and 6.
The way I went about the problem just now used notions of straight line equations and Cartesian coordinates which are familiar to you. In my own time I've also solved the sum and difference problem in a way that didn't require knowledge of these things - just knowledge of what sums and differences are. That's all you really need. But when you're familiar enough with straight line equations, you use them just because they're at your disposal. You can solve the sum and difference problem with less sophisticated (although arguably more elegant) means, but it's a bit like using a quill instead of a pen. Of course, a pen is harder to make (and it's fair to say most people don't make them on their own), but once you have it you tend to use it. There are many parallels with high school math here, which this paragraph is too small to contain.
To recap a bit, the sum and difference problem gave us an opportunity to use knowledge of straight line equations. But mathematicians didn't always have these at their disposal - they needed to realise that straight lines had uses in all sorts of situations, so they figured out what the defining features of a straight line equation were, and then studied their properties. This comes from doing problems like these from scratch. You'd start of with the pair of equations
and
, without necessarily having seen straight line equations before. Then you might find a way to solve them, by substitution or whatever comes to mind. Then, as a mathematician, you think "Can I still do this if I'm given
and
? What about
and
?", and you suddenly have the structure of question I mentioned earlier. You find some pairs of equations work and some don't. You look for reasons why. You might find that, intuitively, it relates to straight lines intersecting. Parallel lines cannot intersect at a single point, so they either overlap or miss each other completely and share no points. So if your two equations are 'parallel', you won't get a unique pair of numbers as your answer.
But of course, you don't get to do much of this sort of mathematical thinking in Methods. You're told that some smart guys already figured this out, and you're given ways to recognise each of the situations, with matrix determinants and whatnot. You're just left to solve examples of the different types of simultaneous equations.
On some level, it's good that you're given a theory that can be applied to worded problems, even if there's no real talk about why the theory exists. The word problems are, in essence, what gave rise to the theory, so to tackle word problems, it's good to first try to imagine why the theories you need to apply came about. I personally have no idea where simultaneous equations started; presumably they were derived independently by a number of people. But it doesn't really matter - it's entertaining to think about where ideas might have come from, and it helps you recognise structures in worded problems so you know what you need to do.
I might continue this in another post sometime, this is getting quite long haha.
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