I wanted to talk about drawing graphs that represent something, and to get there have to say what kinds of things I mean to represent. The quick and expected answer is that I mean to represent some kind of equation, such as “y = 3*x – 2” or “x2 + y2 = 4”, and that probably does come up the most often. We might also be interested in representing an inequality, something like “x2 – 2 y2 ≤ 1”. On occasion we’re interested just in the region where something is not true, saying something like “y ≠ 3 – x”. (I’ve used nice small counting numbers here not out of any interest in these numbers, or because larger ones or non-whole numbers or even irrational numbers don’t work, but because there is something pleasantly reassuring about seeing a “1” or a “2” in an equation. We strongly believe we know what we mean by “1”.)
Anyway, what we’ve written down is something describing a relationship which we are willing to suppose is true. We might not know what x or y are, and we might not care, but at least for the length of the problem we will suppose that the number represented by y must be equal to three times whatever number is represented by x and minus two. There might be only a single value of x we find interesting; there might be several; there might be infinitely many such values. There’ll be a corresponding number of y’s, at least, so long as the equation is true.
Sometimes we’ll turn the description in terms of an equation into a description in terms of a graph right away. Some of these descriptions are like as those of a line — the “y = 3*x – 2” equation — or a simple shape — “x2 + y2 = 4” is a circle — in that we can turn them into graphs right away without having to process them, at least not once we’re familiar and comfortable with the idea of graphing. Some of these descriptions are going to be in awkward forms. “x + 2 = – y2 / x + 2 y /x” is really just an awkward way to describe a circle (more or less), but that shape is hidden in the writing.
To try un-obscuring things, we try manipulating whatever it is we have in ways that we know won’t change whether or not the thing we’ve said is true. There are a small set of manipulations we can do, but there’s an incredible variety of ways we can apply them. The basic techniques are to add the same number to both sides of the equation, or to add zero to one side of the equation, multiply both sides of the equation by some number other than zero, or to multiply one side of the equation by one, or to substitute one thing with another that has the same value.
Adding zero to one side sounds ridiculous, but that’s just because we add zero in an obscured way. We don’t just put “+0” into something; instead, we put, say, “+ 1 – 1” in or “-3 + 3”, or “+b – b”, or so on. Add something and subtract it right away. Similarly that multiplying by one idea is really multiplying by something and dividing by it right away. That again can be multiplying by 2 and dividing by 2, or multiplying by x and dividing by x, or so on, as long as x doesn’t equal zero. The idea is to turn part of the equation into something that’s recognizable as some nice, familiar, simple pattern rather than whenever it was to start.
That we don’t want to do things which could change whether the statement is true is one of the reasons we don’t multiply by zero in these manipulations. The statement that “2 = 2” is true — if it’s not, we’re in deeper trouble than we thought — and multiplying both sides by zero gives us “0 = 0”, which is true enough. However, “2 = 3” better not be true, yet multiplying both sides by zero gives us the same “0 = 0” and we know that’s changed the truth value.
But we supposed we started with something true, so, does it hurt us to change the equation in a way that makes it definitely true? Well, it might. Being willing to suppose something’s true doesn’t mean that it is true, or even that we want it to be true. A perfectly respectable, and fun, way of proving things is the reducto ad absurdum, where one starts off with a proposition the opposite of what one wants to prove, and we carry out manipulations that preserve whether it’s true or false, until, hopefully, we reach some statement that is obviously false. If we didn’t change the truth-value along the way, then we know we had to start with a false statement. And if the opposite of what we want to prove is false, then the thing we really wanted to prove is true.
At least that’s one way of looking at it.