Adverts for church?
Something thats got me bemused for quite a while are adverts outside churches. You know the ones... "Jesus christ was born 2000 years ago. Worship him here!". They have always mystified me. I am aware that in the US some adverts are even more blatant, with churches actively pushing themselves... but does it work?
I mean, I suppose in a big city where there are thousands of churches, picking where to worship might be a genuine choice, but these signs appear in small towns and villages all the time, where there is a maximum of one church of each denomination. I doubt anyone is going to switch based on a sign telling them Jesus was born 2000 years ago- if they don't know that, they're probably not very good christians. Also it's not like the churches themselves aren't noticable- these little signs are on big stone buildings, generally the biggest buildings around for the most part.
I dunno, maybe I don't get it cause I don't believe. Any thoughts ye faithful?
If I didn't believe in such things, I'd think someone was trying to tell me something. I have decided to apply for an Msc in medical statistics, because I think I wouldn't mind doing a job that involved using those particular skills. Of course, for this, I need a reference. No problem, think I, I shall just ask my lecturer of my module in medical statistics last semester- a perfect match, one would think. Except he's ill, and has been for about a month.
So- back up plan, I decided to go find my lecturer for the previous semester, Ruth Salway. Except she wasn't on the directory and I couldn't find her office- I went to ask at reception... it was closed! ARGH! It seems she has gone on maternity leave........
Oh well. I will get a reference.... from someone......
Maths part 4- functions and maps
So, now we have our numbers. The Natural numbers, the integers, the rational, the real, and the complex. And hopefully you all have a vague idea of what they are. Remember that each collection of numbers contains the one preceding it, so whenever I say the complex numbers, that includes all other kinds of numbers as well.
So what next? Well I'm rather eager to talk about infinity, but along the way I think we had better talk about what a function is. You will have undoubtedly encountered them if you have done any mathematics at all, although they may not have been labeled as such.
So what is a function/ map (the words are essentially interchangable. I am not aware if there is any significant distinction between them)? Well simply put, it's something, that, given an inputted number, give an outputed number. For example f(x)=x^2 is one. You give me a value, x, and f(x) gives you back the square of that number. You will most probably have seen them in the form y=x^2. y and f(x) here are interchangable. Generally we use f(x) when talking just because it is clear that f is dependent on x, where y by itself gives you very little information.
Now, there are certain rules that functions follow. When you define a function, you generally define the "domain" that it works on. This will often be the real numbers, but could be the integers or even the complex numbers. The domain is just the set of numbers you put into the function- all the values your x can take. So for example f:R->R x->x^2 means that our f(x)=x^2 takes all the real numbers and squares them, sending them into R. Notice that becuase we are squaring we will never get negative numbers (unless we are squaring complex numbers), so we could just write f:R->[0,infinity) where [0,infinity) is the set of all real numbers from 0 to infinity. However, we can write R quite happily, as while our map never goes to the negative numbers, it's still mapping to a smaller part of the reals.
The choice of domain is quite important. For example f(x)=1/x is simply undefined at x=0 (it is NOT infinity), so the domain can't include x=0.
Another important rule for a function is that it is well defined. What does this mean? Well if I put the same number in twice, we get the same answer. For example f(x)=2/-2 can give both answers. This is quite sensible, as it means my function will do predictable things.
So, lets recap. We have a function, f:D->S where D and S are some collection of objects. Notice that these don't have to be one dimensional- for example f(x,y)=x+y takes from a two dimensional space to a one dimensional space, and oddest of all do not have to be numbers
. For example the function capital(country) could give capital(france)=paris. This meets all our requirements for functions.
I think we might get to infinity next time. I've just got to talk about sets first......
Maths part 3- the complex numbers
A lot of people don't get complex numbers. It's probably because they think about it too much. The important thing to note about a lot of mathematics is we say "hey, suppose this thing is true" and then see what follows from that. Because mathematicians are generally quite sly, the thing they suppose is true often is true, so their results hold.
So when I say that the root of -1 is i
, (some engineers call it j. This is because they are fools), this is a definition. There is no absolute truth behind this, it's simply that whenever I say i,
I mean the root of -1. So i
^2 is equal to -1. That's all I mean by that. There is absolutely no value that exists that squares to -1, at least not one on our real number line, but that doesn't really matter. We don't care about how useful this thing is just now- we just care that we have defined it thusly. It's a weird concept, I'll admit, but no weirder than the idea that you can't really express the root of 2- as mentioned it's simply impossible to write it in terms of other numbers (other than as the solution to some equation, eg x^2=2).
So, we have our i (I'm going to stop using italics each time I write it, because it's annoying me- you're a smart bunch). So lets do some stuff with it. Well first of all we can write, for example, the root of -2 in terms of i- it will be i multiplied by the root of 2. This way we don't need lots of crazy imaginary numbers to express all of our roots- i will suffice to describe them all.
We can then express all our numbers so far in this form a+ib, where a and b are real numbers. So each number has an imaginary and a real part- a is the real bit, and b is the imaginary bit. In most cases that you'll encounter, b will be zero, and you'll be left with a nice real number, which behaves quite normally.
There are some slightly odd things about the complex numbers- which are numbers of the form a+ib, which are worth noting. An important one is a lack of a sense of order- we know 2<3, but is 2i<3i? Not really- we don't actually know anything about the size of i, so we can't really assign order to the numbers. There is a way of talking about the size of the number, and that's by taking the modulus- when we take the modulus of a real number, we make it positive- so the modulus of 4 is equal to the modulus of -4... it's 4. For the modulus of a complex number, it is defined to be equal to the square root of a^2+b^2.
There is good reason for this- basically you can imagine any number as a line stretching from 0, and the modulus is an attempt to find the length- in the case of -4, it clearly is 4 away from 0. For the complex numbers, we imagine a 2-dimensional graph, with a on the x-axis, and b on the y-axis. To find it's distance from 0 we use pythagoras- c^2=a^2+b^2. This notion of distance, can be, incidentally, extended to a universe with any number of dimensions, even an infinite amount of dimensions!
So we have some kind of frame where our complex numbers live in. But what use are they? Well the thing is is they crop up in this equation-
I have complicated things here by introducing 3 new concepts- cos, sin and exp. Now cos and sin you may well have encountered- they have clearly defined geometric meaning which is why you'll see them when finding angles in a triangle. The exponential (exp) function is quite interesting- basically it's this magical number e^xi. What's e, you might well ask... well thats a story for another day, but for now it's enough to know it's one of the transcendental numbers like pi, and crops up at least as many times as it for various reasons.
Whats neat about this equation is if you plug in x=pi. Cospi=-1, sinpi=0, so we end up with
exp(pi*i)=1. Which is pretty crazy- we put in some imaginary number into our equation, and get a nice ordinary number as a result! It's mad, and the proof is far to involved to get into here, but it's a very important result.
Thanks to this kind of set up being the solution to these things called differential equations, i actually crops up all over mathematics, and in fact in solutions to real life problems. Which is admittedly quite weird, but just goes to show that life does not necessarily work the way you expect it to.
Anyway, that's it for now, next time I will finally be making my way towards infinity, but first describing functions and maps for your enjoyment. I might mention differential equations later on.... if I feel like it.
Maths part 2- a guide to numbers
It is valentines day today, so I thought I'd talk about numbers.
See, there are various types of numbers. The first set we are introduced to, in our infancy are the natural numbers, or the counting numbers. The ones that go 1,2,3,4,..... Basic stuff, you might think, although if you go to the wikipedia article about the natural numbers you will notice sometimes quite heated debates about whether 0 should be a member of the natural numbers. It shouldn't be, and anyone who feels differently is unfortunately a fool. They are denoted with a blackboard bold N, if you care about notation.
Of course, these aren't enough numbers, and the integers are pretty important to- these contain the natural numbers and 0, and all the negative whole numbers. A pretty simple extension, and not a terribly interesting one either. They're denoted Z.
From here we go to the rational numbers, which are basically all the fractions- 1/2,6789/956676 etc, etc. Notice that the integers are contained in this set. Essentially, any number which can be expressed as an integer above a natural number is considered to be a rational number. And, actually, it seems like this should probably be enough- these are the basic numbers you can concieve just thinking about it on a common sense of level. Indeed, many greeks thought that these were the only numbers.
Sadly, it's a bit more complicated than that. Now, as I'm sure you all know, pythagoras has a neat law describing the lengths of right angled triangles- a^2=b^2+c^2, where a is the length of the longest side. Well suppose the length of b and c were both 1 (1 of what is a reasonable question, although it does not really matter- say centimetres if it makes you happier). Then the length of a is the square root of 2.
Well, fair enough, so by our reasoning from before, as it is certainly true that the square root of 2 exists, so it must be expressable as a fraction as above, eh? Well.... no. Euclid proved that the root of 2 was irrational. In fact, you can see the proof here
. So we now have numbers that can be written down as decimal sequences that will never terminate or repeat periodically, and certainly cannot be expressed as a fraction.
This is pretty odd, but if we accept the basic assumptions made by mathematicians, all this follows. Still, at least in the case of these weird irrational numbers they will be the solution of some kind of equation- in this case x^2=2 (incidentally, the solution of an equation is a value for the letter which will make is true- so in this case x equal to root 2 is a solution. As is -root2. Incidentally). Except.... some of them don't.
See, if a number satisfies those nice equations with x to the power of something, we call it algebraic. But there are actually some numbers that don't even do that. Pi is the most famous number- an example of the transcendental numbers.
One might ask how one generates pi. Excellent question. Well if you can find the area of a circle accurately, and divide by it's radius squared, you will get pi. Or if you sum 4-4/3+4/5-4/7+4/9-......... you will also get it. Of course, you can't actually express pi, but it certainly has physical reality, in that it turns up in circles all the time. Actually pi turns up literally everywhere in mathematics, as I may or may not explain to you.
So there we have it. The natural numbers, the integers, the rational numbers, the algebraic numbers and the transcendental numbers. Put them all together and you get the real numbers, a nice smooth number line. Phew. We're surely done now?
Well.... no..... Unfortunately someone got curious about the root of -1.
It occured to me the other day that many super powers are somewhat problematic. It sounds awesome to fly, or stop time..... but what if it stops working. The thing is, they tend to be inexplicable. If they weren't, then they'd just be powers. And somewhat less exciting. So if you don't know how they work, then you don't know if they'll just... vanish. Imagine if you stopped time.... and then couldn't unstop it? Or you flew high into the sky... and crashed to the ground.
I suppose powers like telekinesis are probably fine, although if you use them to thwart villians, it would be somewhat unfortunate if they kicked out mid battle, leaving me as a slightly chubby guy with... well... not many villain thwarting powers to be honest, unless you count crushing sarcasm.
This is vaguely addressed in Spiderman 2 actually, when he loses his powers for no particular reason, and then gets them back for no particular reason. It's kind of odd really, creating a completely inexplicable crisis for our hero.....
Maths (part 1)
I like mathematics. Well, a bit anyway, or I wouldn't have survived these last 4 years of doing pretty much nothing but it. It's kind of interesting in places, and because I feel like it, I'm going to try and talk about why it is.
Mathematics is one of those subjects that no-one particularly understands- you get a tiny sample of it at lower levels, but for the most part you get numeracy skills and a tiny bit of applied mathematics. A lot of subjects start of rather simple and get more complex, but in a way mathematics starts off complicated then gets simple. Although you wouldn't know it to face the subject.
A lot of the time, when a mathematician starts doing stuff in mathematics, they're not sure about whether they can do it or not. For example, when Newton invented calculus (for the second time (that is, someone invented it before him. He didn't invent it twice.), the rigour behind it was created a little bit after him... this is the case for most mathematics.
At degree level we do get a bit more structured, starting with the axioms and working upwards.
The idea of mathematics, for the most part, is pattern spotting. Mathematicians take an idea, like symmetry, or shapes, or the gradient (slope) of a curve, and try and get a set of rules that describe those things. If you can find a basic set of rules that govern something, then you can find other, more odd things, that follow the same things, and know that every single result that follows from those basic rules will apply to this odd thing to.
A simple example is talking about N dimensions. You will be aware of the basic three dimensions, I should imagine (or, if you're a physicist, you can talk about your puny 10 dimensional space). Well mathematicians will talk about a space of an arbitary amount of dimensions, or sometimes even an infinite amount of dimensions. This turns out to be rather useful- in economics you will be dealing with hundreds or thousands of variables, and you can assign a dimension to each of these, and know that the rules you've shown for your arbitary space will work in economics.
With all this, it is somewhat frustrating to tell someone you do mathematics, and they ask you to multiply 79 and 81 in your head (79*81=(80-1)(80+1)=80^2-1=6399), or even worse, ask what practical use this all has. Admittedly the job applications are somewhat slim, but the applications of the most ridiculous mathematical proofs will have real life applications. Admittedly most of it will NOT, because that isn't really why we do it...
Anyway, I hope to to talk about some interesting stuff in mathematics in the next few posts, and perhaps the next time you meet a mathematician you can ask them about the Hilbert hotel, and they will either look impressed or say "what?" because they haven't studied it.....
I meant to
I really did mean to blog. Of course on the weekend Alice was here, and I am always loathe to interrupt a whirlwind of weekend with blogging. But that doesn't excuse monday- the invitation to the pub by Rich, who is moving away next week, did that. Tuesday should have been fine, but I went to see Blood Diamond- an enjoyable film deserving of it's oscar nomination, about a subject that is not as well publicised as it should be. Wednesday? I had the afternoon free, but some friends who I had not seen in a long time appeared, and so the afternoon was gone. And I already HAD plans for the weekend.
And so I've made it to thursday. In fact I almost failed today, as the internet chose this moment to fail. Of course all these obstacles could have been surmounted if I could blog on campus, but apparently Bath uni hates the new google log in to blogger.
So here I am, blogging. Of course, blogging about not blogging is sort of cheating I suppose, although I imagine a significant proportion of the blogosphere is made up of such posts. Never mind. I shall have new posts... on sunday. Probably. Hopefully. Maybe....