Functions & Algebra

OK, before we can progress look at \(e\) (and to the joys of proving the irrationality of the square root of two) we need to stop for a minute to think about functions. This is yet another fairly basic element of mathematics that was glossed over completely in my formal maths education.

So what is a function? In a basic sense, a function is a list of mathematical operations that you carry out on a variable. Hmm. I think what this scenario requires is an ill thought out analogy. Tradition dictates that it should involve some sort of hot drink.

So lets have our hot drink function. It would state:

1. Boil Kettle
2. Put boiled water in mug
3. Put drink flavouring in mug.
4. Stir mug

Those instructions are good enough to make coffee, tea, hot orange, or a cup-a-soup. The only bit of the instructions that would have to be changed is the "drink flavouring" bit. That could be instant coffee for coffee, a tea bag for tea, some orange squash for hot orange, or a sachet of cup-a-soup powder for the soup. All we have to do is change that bit, and follow the rest of the instructions to the letter, and we get a different drink at the end. The "drink flavouring" bit is the VARIABLE, meaning that it is the bit which can change, if we want to change the hot drink we end up with.

Once we have listed all these instructions once, it would be dull to write them out every time. So instead we use a shorthand system. Lets give the group of instructions a name: "hotbeverage". To show that the outcome of the hotbeverage instructions depends on the drink flavouring variable, we put that in brackets after the name: hotbeverage(drink flavouring). If you were reading the notation aloud, you would say "hotbeverage of drink flavouring". Which makes some sense.

That notation means that you can change what goes in the brackets (tea bag, soup powder etc) and you STILL FOLLOW exactly the same instructions, and you end up with something else. If you decided that you were going to make hot orange, you would write that as hotbeverage(orangesquash). You have replaced the placeholder with the ingredient. Someone reading that would know that you meant "follow the list of instructions that I have called hotbeverage replacing the variable 'drink flavouring' with orangesquash".

Now, lets bring some algebra into this. First question, what the fuck is algebra? Answer, a very useful system of mathematical thinking that Muslims came up with in the middle ages, based on earlier work of Hindus, all at a time when western society was trying to work out how the Romans had built the bloody aqueducts. You can tell the word has Arabic origins because of the 'al' - much like alchemy, and algorithm.

So what actually is it? Well, in the example above we had a variable - something that we could change to change the outcome of the function. Algebra is a way of working with these variables. Sometimes that are referred to as 'unknowns' which is coming at the thing from a different angle. Before we had the concept of algebra, mathematical thinking was really done using geometry. So if you saw an ancient mathematician scribbling away on papyrus, or a sand board, they would be drawing lines and circles and so on, NOT the kind of symbols and operations that we use these days. The difference that algebra brought to the table, so to speak, was the ability to think more abstractly.

The Muslims made the leap from lines and circles to abstract ideas, but they could only express those ideas in long wordy sentences. Such as "if you take the third part of the first party raised to the second power and then find the root of the difference between that and the fifth part of the....". You get the drift. Much later than the Muslims, Western Europeans started using letters of the alphabet in place of the unknowns. They also started using the symbols we have already looked at to show what operations you were doing to numbers or these unknowns. So finally instead of drawings, or long wordy sentences, we finally had the numbers, letters and symbols that we now recognise as 'algebra'.

Tradition has it that we use letters from the end of the alphabet to represent these variables or unknowns, starting with \(x\). We have already used a letter from the Greek alphabet to represent a number - \(\pi\). Is \(x\) the same? No. \(\pi\) is always the same number a little bit more than 3, roughly \(\tfrac{22}{7}\). It does not vary. We just use the letter symbol, because we would never finish writing out the number otherwise. \(\pi\) does not vary - it remains constant, and so we call it a ... constant. It is like the symbol for two, '2', or five, '5'. Those symbols always means two or five. In the same way \(\pi\) always means just a little bit more than three.

So we normally use \(x\) to represent the first variable, and then \(y\) and \(z\) if there are more variables. In other subjects where algebra is bring used, you find different letters, or even other Greek letters. Physics is bloody littered with different letters ('s' usually stands for a variable which is the speed of something, 'd' for distance, 'v' for velocity (not the same as speed but never mind that now) and so on). So, a typical algebraic equation would look like this:

\[x+1=3\]

We are now supposed to follow some formal algebraic rules to get a statement that starts with:

\[x=\]

The bit on the right side of the = tells us what \(x\) actually is in this example. In this case, the rules we follow are to deduct 1 from each side of the equation:

\[x=2\]

We now know what \(x\) is. We can plug 2 into the place of \(x\) in the first equation:

\[2+1=3\]

Yep, that is correct. Richard Feynman explained algebra best when he said that it is just a puzzle game, where the goal is to find out what \(x\) is! You can look back at the original statement \(x+1=3\) and rephrase it as the question, "what number, if you add one to it, makes three?" Then it is just a puzzle which is easily solved.

That was an example of an algebraic equation. That English language question I translated it into was how algebra was done before the symbols were invented. Could ancient mathematicians have solved this using their lines and circles? Yes. What you do is draw a long line. And then take a compass set to a specific, and completely arbitrary, width. Stick the point anywhere along the long line. Draw a circle. The circle will cross the line at two points, the same distance from the pointy end of the compass. You now have three points, the original bit where you stuck the compass in, and two where the circle crosses the line. Can we answer the question what plus one is three? No, because we only have two identical line segments.

So now stick the compass pointy bit on either of the new points and (without changing the width of the compass) draw another circle. One crossing point will be the very first pointy bit, and the second will be a new point on the line. We now have four points and three equal segments of line. We now have three identical line segments. In this step we added one identical line segments, so the answer to the question must be "how many line segments did we have before we added this third one", and the answer as we saw a second ago was "two".

It should be obvious, but it bears repeating. Whenever you hear of an ancient mathematical proof or theorem, and you think "that's a doddle, I could have done that, and I am an idiot", remember they didn't have symbolic algebra. They had tools to draw circles, and tools to draw straight lines. That was it. And they still managed to come up with good stuff. Geniuses.

Anyway, we started to talk about functions. We have seen a function written out in normal language, so what does a function look like in symbolic algebra? Well, we have already seen one of those as well. A function is basically just the side of the equation that \(x\) is on. So in the above example, the function is:

\[x+1\]

The instructions we follow are really just to add 1 to our variable. We need to give our function a name, and traditionally we give it a name one letter long. We start using the letter \(f\), for the first function we use. If we need to use more than one to solve a problem, we give the next one the name \(g\). And so on. It would be terribly complex if we chose letters nearer \(xyz\) because you would start to get confused between the names of functions and the names of variables. That would be bad.

There is nothing special about the letters chosen, by the way. A variable called \(x\) is not one greater or less than a variable called \(y\). We could just as well be using pictures of clouds, buses, or lawnmowers. It is nothing more than a marker. Same thing applies to the names chosen for functions. I said earlier that we put the variable in brackets after the name of the function. So we end up with:

\[f(x)\]

That means the function is called \(f\) and it involves a variable called \(x\). We know what the function looks like, so we can describe the whole thing as:

\[f(x)=x+1\]

We can then replace the \(x\) with two like this:

\[f(2)=2+1\]
\[f(2)=3\]

We can now talk about \(f(x)\) rather than repeating all the steps every time. This is not too onerous with \(x+1\)) but we will see some more insane functions in due course. Much like the beverage example above you would say that this is "eff of ecks".

The important difference here is that an equation has an equals sign in it, whereas a function does not. OK, pedant, yes the example just about has an equals sign, but that is just telling you what the function is. The actual function (the bit on the right of the equals sign) only has a variable, the sign for addition, and a number. It has no equals sign. An equation, on the other hand, tells you that two different looking thinks are equivalent to each other, whereas a function is list of things that you do to \(x\), say, to get a result.