Sunday, June 24, 2007

Slimy definition of the week. (Analysis- an introduction)

To have a recurring weekly maths theme to this blog, I think I'm going to have a definition of the week concept. This post should happen on Saturday's (one could say that posting early mornings in the holidays is defined to be a post for the night before!) Then on Sunday, i.e. tomorrow morning at some weird time I may or may not take the definition further. [It really depends on my mood- I could also have more than one in a week!] I've been intending to do this (amongst other things!) for a while, however previously it seemed that the laws of nature where against me. My post on differential equations is sat there, waiting. You see my intentions had been that during the year, I'd post on topics that we're either doing or that I liked. I'd liked differential equations, but then blogging didn't have the 'save now' draft feature. :(

The other thing holding me back was that I didn't know what/how to go about this. So back to square one it is. The definition of the week could either be due to maths that I've encountered during the week, or particular stuff that I liked during the year. As always I hope that you correct any (inevitable) errors that I make.

Now I don't mean that this definition is slimy- no sir (I like saying that!). More of the slimy business will be revealed at the end.

It's weird how the topics in maths which cause me most distress sometimes turn out to be the ones I end up liking the most. This could be partly because they present a bigger challenge for me and I strive to hear that small click in my head. The first ever definition of the week is going to be about analysis. Well one could call this a small introduction into analysis which is necessary.

One of the reasons why this subject seemed really .... I don't know... out of the ordinary (?) was because I had no idea what the heck it was about. One could say that I didn't know the mathematical motivation or reasoning for it. I mean linear algebra on the other hand started of with systems of linear equations. I felt comfortable- in the beginning at least {getting worried about results:/}. It then went onto more wider and abstract things but I felt that I knew where the foundations started. We built up the subject. Even ODEs and mechanics- I knew the score. However, with sequences and series the only thing that can describe what I felt like is:
Why? Because in the big bad scheme of things, I didn't know where this subject fit. My head, as you've probably gathered, isn't the most normal heads in the world. I couldn't place this subject in my minds 'maths map/tree'. Did I give it a new branch? Did it need a new branch? These were some of the questions that harassed me, and for a while I blindly sat in lectures with that face! Anyway, the best way to sum up the question 'What is analysis' is that 'mathematical analysis may be regarded as the study of infinite processes.'1

However, the way it was said for me to be able to find it's branch was something along the lines, that studying this subject allows us to define things like differentiation and integration. By doing this we know when we can or can't differentiate. We're told that to differentiate, a function has to be continuous. What's continuous? 'It has no jumps'. What do we mean by that. There was the motivation I suppose. I mean I was told about other things as well, but knowing where to place this subject on the maths map proved to allow me to change gears. I know that sequence and series is slightly different, but in maths you start of by building the pyramid upside down.2 Then you build layers on top of that, building on from what you started with and know. Sequence and series is the building block since it introduces you to concepts such as convergence.

It's important to have an understanding of numbers (natural, integers, rationals...) and the axiom for real numbers. This is basically what is covered in most first chapters of analysis books. It's also important to know about inequalities etc. However, if I do write about them then I will hopefully try to do so with some explanation. [BTW I won't be including zero in the set of natural numbers]. I think I'm going to 'borrow' the actual definitions etc from my lecturers notes(!) since it's either that or books. (I can't seem to write stuff in my own words. :o )

Definition3
A sequence of real numbers is an infinite ordered list $a_1,a_2,a_3, ...$ , where for each $n \in \mathbb{N}$, $a_n$ is a real number. We call $a_n$ the n-th term of the sequence.

Sometimes we can write formulas to describe a sequence e.g. $a_n=\frac{1}{n}$ is the sequence, 1, 1/2, 1/3 ..... Other notation is $(a_n)_{n\in\mathbb{N}}$ or $(a_n)_{n\ge1}$. [I will edit this post and stick in all the proper LaTeX images, but the time is such that my brain would rather I didn't do that this very instance! Edited!] We write $\{a_n\}_{n \ge 1}$ instead of $(a_n)_{n\ge 1}$ to mean the set containing the sequence. (Not sure whether which you write is such a big deal).

EDIT: I fell asleep doing this!!!

OK, in other words 'a sequence is any string or collection of real numbers, indexed by the natural numbers'4. So the sequence 1, 2, 3, 4.... is different from 1, 3, 2, 4, ....., whereas the sets {1, 2, 3,....} and {1, 3, 2, 4...} are the same.

The first thing that we tend to consider, when presented with a sequence is what happens to it as n gets larger. It's not as important how the sequence behaves for small values of n, since we want to see what's happening when n gets arbitrarily large. Remember that when we say that a sequence is indexed by the natural numbers we mean the following:

eg. the sequence with formula 1/n:

$\text{The sequence:} \quad 1, \quad 1/2, \quad 1/3,........1/n,... \\ \quad\text{It's index:} \qquad\qquad1 \qquad2\qquad3 \qquad.......n.....$
When considering this we mean the n from the bottom line getting larger and larger, and then looking at the behaviour of the sequence. I think I'm writing this out in the above form, because erm.... I had some difficulty grasping it initially! I will probably continue to explain in further detail stuff which I found more difficult etc. (The way I've written the above doesn't seem to sound nice to me at the moment!)

I will be considering these points in another post i.e. the concept of convergence etc. This was just a short introduction I suppose and well I'm going to ponder over which direction to go next. So there you have it- the first definition of the week!

NOTES: This is just something for those who are not aware of such things, but from one of the guides that I linked a while ago, in maths we tend to write using 'we' instead of 'I'. I'm mentioning this because my friend who is doing an English degree, commented on me writing 'we' in my reports as I \sout{forced} gave them to her to read! (She was really taken in by the difference equation report and the bit about stability points and the pendulum).

Also, you probably already know this but when we use variables, say n, we tend to make them italic to differentiate them. I think this applies to formulas and other things maths.

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Before I get some breakfast, I feel that I can't leave the slimy story an open tale! However, because I am quite hungry I'm just going to post this one picture and I would love to hear your conspiracy theories!

More will be revealed tomorrow- but what the heck is that and how did it get there? Like I said, all conspiracy theories welcome! Do I throw a prize in for the closest story.... an infinite amount of money? (muhahaha!) Ahem. Well if you're grossed out by that then stay away from this blog today and tomorrow! Seriously, I won't say that's not 'ew' but you can't deny it being cool. Or is that just me being me once again. If you don't find it 'ew' save it and zoom right into it- it'll help you give it a name. :D

Like I said, I'll reveal the rest till later- but think of that feeling as my initial feeling to sequences and series i.e. slimy! Have a nice day.5 (hehe)

1: As said by Rod Haggarty in Fundamentals of Mathematical Analysis
2: Ian Stewart said something along these lines in Letters to a young mathematician
3: Definition taken from Dr. C's notes (which I didn't know off!!)
4: Taken from Dr. M's notes.
5: Sorry if you're really really grossed out! :o