Saturday, June 30, 2007

Test post

You can skip past this post to the next one... if you want. (two in a day wow!!) This post is going to be where I mess around with LaTeX and how it looks on this blog. I'm just going to be posting random stuff taken from anywhere! I dare not mess about with other posts since it's a pain and takes too long to repair them. Remember bloggers only use &.#.60 for &#60, and &.#.62 for &#62! Without the dots of course, but be careful if you don't want to end up in tears as to where your post has dissapeared to!

Taken from google images.

'Figure 2 shows part of the curve with equation:  y=(2x-1)\tan {2x}, \quad 0 \le x \text{ strictly less than } \frac{\pi}{4} . (-2) Lala ... the question may have been copied wrong- I had to squint!

The curve has a maximum at the point P. The x-cordinate of P is k.

a) Show that k (2) satisfies the equation: 4k + \sin (4k) -2=0 (0pt). (just continuing the line here-ignore).

The iterative formula: x_{n+1}=\frac{1}{4}(2- \sin 4x_n),\; x_0=0.3. (-1pt) is used to find an approximate value for k (-4). random words.


(b) Calculate the values of x_1,x_2,x_3 (-3) and x_4, giving your answers to four decimal places.

(c) Show that k=0.277, correct to 3 sig. figures.

I'll play around with the padding in the morning- just want to make the posts that have Latex user friendly. :) Which pt looks best at the moment? (without padding being changed?)

Friday, June 29, 2007

'Stop being a geek.'

It's weird how certain incidents remind me of what I had intended to blog about some time ago! An anon reader commented:

'stop being a geek and have a rest . . .or ami the only one who does no work over the ols and everybody else is still revising last years material so they are on top form in September'

My reply to that can be found somewhere at the bottom of the comments here. :)

From Google (wiki) I have some rather 'nice' definitions of such word:

' "one who is primarily motivated by passion," indicating somebody whose reasoning and decision making is always first and foremost based on his personal passions rather than things like financial reward or social acceptance. Geeks do not see the typical "geeky" interests as interesting, but as objects of passionate devotion. '

Yes, I made that bold myself. This just made me laugh:

'A person who relates academic subjects to the real world outside of academic studies — for example, using multivariate calculus to calculate the volume of a cake at a party.'

Don't worry I've never done that, but it's always good fun annoying certain people by mentioning words like calculus. :D There are many other sentences which are 'funny' but my left hand has lost some of it's mobility so I'll take it easy. I've been called a nerd a few times as well, (well mystique tends to refer to me as one) and I found this quote interesting:

"My idea is to present an image to children that it is good to be intellectual, and not to care about the peer pressures to be anti-intellectual. I want every child to turn into a nerd - where that means someone who prefers studying and learning to competing for social dominance, which can unfortunately cause the downward spiral into social rejection."

The sad thing is that when I was in college, two guys used to compete with each other to get the least mark in biology tests! That's pretty sad, since many of the people there felt it was 'uncool' to take an interest in studying. I was fascinated by biology and learning about the human body. I let that be known. I have always been a weirdo and that is what ultimately has allowed me filter out all this social acceptance mumbo jumbo. Like I've said previously, you should do what you want without worrying how others will perceive you to be. Everyone is looking at a different angle. (I told you I was lucky!) If the same guys spent less time worrying about their image of being dumb and how they looked then they could lots more. Why is it so hard for other people to understand that there is nothing wrong in learning and acquiring knowledge?

The sayings 'knowledge is power' and 'the pen is mightier than the sword' is what my Geography teacher used to always say. Why is that in todays society that people care more about their social acceptance and how others see them than what really matters? What matters is that you're happy and doing what you love. I take is a blessing in disguise that I'm doing maths- something that I enjoy. Seriously though- there will always be someone who laughs or takes the mick out of you. On my first day in college a group of people took the mick out of me- so what? There are always people out there who share the same interests as you and sometimes we get lucky and meet them. :)

Anyway this is not about me, but I have a question for all you people out there reading this. You don't have to post a comment (sniff) but you can always ask yourself this question. If you have a degree, or are studying for one why did you choose to do your degree? Is it because you wanted a degree, or because you enjoyed the subject you chose to do? Maybe it's because of career prospects?Also if you're doing post graduate studies, then why? If it is maths that you're studying or have studied, why did you choose to do a maths degree? It is maths that I'm really interested in, and really about why someone would choose to do a maths degree.

Going on them 'definitions' some may call me a geek-nerd. I don't take that to be derogatory. I am 'me', and as I always say to my brother, it's a complement. :D However, I chose to do a maths degree because I have always loved and enjoyed maths. It was something that I had a passion for from an early age and, thanks to some positive influences (teachers and the folks) I was able to realise that it is maths that I want to study. I had a wide choice of what I could apply for, but in maths is where my heart really lies. To be honest I was ignorant of what maths really was when I was applying to university. I just thought- three years of maths, maths and more maths, sounds great. (I was looking to apply to university for Maths and Chemistry at the end of my AS exams since I'd messed up further maths, however A2 further maths was much better and I then started liking chemistry less!) It's a good job that now at university - having decreased my ignorance somewhat- my love for maths has remained.

I have been researching this post for some time by asking various people why they chose to do maths. One person said that they chose it because they liked maths and another said because it was a degree. A few have said because they didn't know what else to do and they were good at maths so they chose to do a maths degree. Earning lots of money after graduating seems to be a common theme as well. I've realised that I'd probably never want to leave the whole academic path. I can't envisage myself doing any other job, although some jobs do sound cool (sales person...). If it was up to me I would probably continue doing maths forever and forever- just learning it and appreciating various forms of it that is. When asked by Milo once upon a time, I replied that learning and understanding maths was a challenge. I always like pushing myself (unfortunately at times!) and I always enjoy challenges. Winning or losing isn't important, but it's what you gain from them that is. Challenging yourself allows you to improve and always sometimes question things. It's about setting yourself goals and trying to accomplish them. I wanted to understand maths- that was the challenge. I wanted to do more of it and be able to do. In college I liked what I saw- I wanted more. University maths it was for me then!

Just now I've eaten (half way through writing this), and the 'gang' classed me as a 'Geeky Nerd' or 'Nerdy Geek', however they deduced that it should be taken as a compliment as well. :D Unfortunately many of you might not find this funny, but to enhance my freaky nature I did a 'freaky' thing. Po always starts singing a line of one song and for the rest of the day never shuts up. In the end most people have that song in their head and so Po is subject to many evil looks! Today the line Po was singing was 'it's Real Love that you don't know about'. If you know what I'm going to say next and want me to not write it then I'm sorry I have to. If you don't know and are already worried about my freakiness then it's bye bye for now! I couldn't help it and so told Po that in my head I keep on hearing, 'It's Real Analysis that you don't know about'. :o

(Have you all left the building?) It's in that silly tune as well, but I said that followed by my \sout{evil} normal laugh! Hahaha. That shut Po up though- although I was subject to that 'what the heck face'. You can come back now!

Anyway, I guess I want to hear about your motivation for studying and doing maths. I have never thought that by studying maths I'm 'studying'. When I'm doing maths it's like playing video games. It's like playing football (although I'd choose to play football any day!) It's something that I enjoy doing. Obviously when it's time to revise I become bogged down by things but on the whole, it's no longer a subject that I have to attend school and study. I don't have to do it. I chose to do it because I enjoy doing so! The 'top' thing about university maths is that it's very diverse. If you don't like one topic move on. I've not started 'learning' or doing maths yet, but I intend on doing so. Not just because of the PASS thing, but because I want to explore my subject further. Maths is cool. *draws black square* However, my 'thirst' for knowledge is not restricted to maths. :) (I also enjoy biology, chemistry, a little physics some history and.... do I have to continue? Why not is what I say.)

----

To my random doings now! I've been helping someone fix and repair their house. Naturally I volunteered, doing random jobs like getting rid of the carpet and helping to knock down walls! (It was a partition in the room made out of wood). Yes, I stuck my bob the builder hat on today but alas I did some stupid things. I was using the mallet to hit the wood and my foot was at a weird angle. It slipped and my hand scrapped against the wall. It started bleeding but I didn't realise it (it wasn't too bad), but I have lots of other bruises and pains to be proud off. I think I did something to my left arm (I got carried away I guess) and typing is indeed painful. Using one hand is to slow, but it'll get better. What else- well I picked pieces of wood up and a nail went into my thumb. :/ That hurt! It was good fun though and I look forward to tomorrow. (Let me introduce you to the gang: there's Bob, Wendy, Lofty, me, Rolly and Spud. :D)

Also I got the work shadowing/work experience placement at the secondary school! That's a relief but for the next two weeks I have to be up at 8am. Noooooooooooooooooooo. (That means I can't sleep at 3am!). I read something today and realised that blogging might not be a good thing after all. Actually I realised that I should be careful about what I blog about (i.e. careful about all things\{maths}). There's a huge chance that someone who I might write about may come across this blog. This is just reminding myself. :) [The post feed will go off tonight- my guilty conscious means I have to change someones name in previous posts. :o ] Initially I didn't expect my blog to be read and so probably posted a lot of silly things. If you create a blog and don't want anyone to read it, my advice is to not link anywhere! (I don't mind this being read :p but I think that I shouldn't write nasty or bad things about other people. Or things which I know if a certain person read would find offensive.) Another reason why I should be careful, is that I'd rather that someone not figure out who I am! Well not yet anyway...

To quote something I read somewhere, 'I shouldn't write anything that I wouldn't write on a postcard which could be found by any random person'. I hope that I haven't written anything as such, but one may never know! (I don't intend on reading my posts just yet, but 'control f' should help.') However, that being said, I suppose from now on I'll take care not to write anything which I wouldn't normally say to anyone. This blog has definitely helped me a lot with my maths, and as have many of the readers and commenters. :) It's inevitable that I whinge, but I'm trying to steer this ship into mathematical waters where it belongs! Don't worry I'm not going to stop blogging (if that had you worried, and if not then haha!). Just hoping to blog about the right things I hope. :)

Madness - a Maths Christmas Quiz

I had actually typed a rather long post and finished it a few minutes ago. However, upon finishing it I decided not to post it. It's a bit depressing, and well it had too much history in it. (My history to be precise!) On another day I probably would have posted it, but having written it and gotten the stuff out of my system I realised that it wasn't exactly postable. (Like I said too much beans history :)).

Hopefully during the next two weeks I'll be doing work shadowing. I say hopefully because my lazy self has yet to phone and enquire about it! No topic today, but during the day I realised that I want a blackboard in my room. :D Not a massive one, but a while back I had a white board and it was cool! (Actually nuno had a small a4 blackboard and then this idea came to me! The question is where can I get one from in the size I want it!)

This bean misses university it seems. I'm tempted to use exam results as an excuse for my 'mood', but that's what it'll sound like- an excuse. I've gone mental. I can't stop checking my emails! Does it matter that they're provisional? Why can't they at least update self-service? (Yes, I've been checking that all the time as well and it always says, 'exam marks will be available after ratification of the exam boards.')

That all probably sounds disjoint, however I do apologise. The song going through my head to\sout{day}night has been 'people just ain't no good' from Shrek 2. That's the only line I know of it. :D I met someone tonight and this magneto had tried to 'scam me'. Well he'd asked me to do something and I trusted what he said to me about his own role in everything. It wasn't a 'big thing' but he'd 'disappeared' and so I had basically messed up, him acting like it was no big deal. The sad thing is that he had planned this from the start. He had planned getting me into a tight corner. Not because he had something against me, but that was all he could do to reach his 'goal'. Ah well, he'll soon learn that taking 'shortcuts' in life may seem easy and a thrill now, but one day it'll catch up to him. :) He laughs about it and doesn't understand that what he did was wrong! I had trusted him and considered him genuine and so had wanted to help him out. Yes, I'm too darn gullible (trusting?) at times, but I won't be trusting him again (well at least now I know what he's capable of. :)) That was basically a summary of the other post!

So there you have the reason for my disjointness, and so with that in mind I present you with section A of a maths Christmas quiz that we did in college. :D Is it too much for me to ask that you don't Google the answers? (How many can you answer without googling- do post your solutions :D) 'An infinite amount of money for the person who either has the funniest answers or the most correct. (haha)' I guess I can't say most correct, since the first person to comment might answer all questions! (You're an honest bunch right?)

Section A

  1. When do elephants have eight feet?
  2. What has four legs and only one foot?
  3. What has four legs and flies?
  4. How many times can you subtract 6 from 30?
  5. A woman has 5 children and half of them are male. How can this be?
  6. It happens once in a minute, twice in a week and once in a year. What is it?
  7. What goes up and never comes down? *
  8. When things go wrong, what can you always count on?
  9. Which is correct: 9 and 5 is 13, OR 9 and 5 are 13?
  10. What is a kitten after it is 7 months old?
  11. What has a neck, but no head?
  12. What kind of table has no legs?
If you're eager for the answers and that might cause you to Google, then don't worry I'll be posting the answers in the comments section soon! Have fun, and if it's any encouragement I didn't do too well. :) Enjoy. (and if anyone watches Wimbledon, Henman is once again out. :( But isn't Federer awesome to watch- he's great and you can feel when he's upping the tempo and he always plays an attacking game!)

NOTE TO BEANS: sort work shadowing out, and phone the darn opticians. :o

Wednesday, June 27, 2007

The first chapter- truth tables, and the 'language of maths'.

I think I've been jumping from one topic to another, but I can't help that. We've previously discussed the importance of proofs and also proof by induction. (Analysis can wait for a while). From the comments in that post I've realised that knowing the language of maths is crucial. If you're going to stay in some other country then it's necessary that you can communicate with the residents of that place. In certain languages pronunciation is crucial. If you say a word differently it can be taken in some other context. The same applies to the language of maths. It's important that we know when we can use certain things and basically understand what them things are.

Definition1
A declarative sentence is a sentence that ‘declares’ a fact or facts.

A statement is an assertion which is typically expressed by a declarative sentence. Examples of declarative sentences are:

'The grass is green.
1=0.
\pi=100'

Then we move onto propositions.

Definition1
A proposition is a declarative sentence to which we can assign a truth value of either true or false, but not both.

All of the examples above are propositions. 'The grass is green' is true; 1=0 and \pi=100 are false. Declarative sentences can also be predicates. A predicate is of the type: m < n. They involve free variables, and once these are given values they become propositions e.g. if m=10 and n=9 then 10 < 9 is a proposition (a false one). From now on I will use the word statement to mean either a proposition or predicate. (Simply because it's smaller than 'declarative sentence'!)

Generally propositions or predicates are denoted by P, Q, R, S or p, q, r, s. Simple propositions are combined by logical connectives to form composite statements (arguments), which can be quite complex at times. So to deduce the truth value of a complicated statement we have to determine the truth value of the simple statements which it is made up off. In this post we'll be looking at the logical connectives 'and', 'or' and 'not' (implications will be done next). Initially my all famous 'shrek face' may surface however it is vital that these are understood since they may(do?) aid your understanding of other things like proofs.

Now in different text books and notes you'll find various notation. For propositions p, q:

  • p \wedge q is the proposition 'p and q i.e. the conjunction of p and q.
  • p \vee q is the proposition 'p or q or both' i.e. the disjunction of p and q or the inclusive or.
  • p' or '~p' is the proposition 'not p'. i.e the negation.
I won't be using that notation, but you may come across it. 'P and Q', 'P or Q' and 'not P' will be what I'll be using. Moving on:

Definition1
A propositional form is an expression involving logical variables (i.e. the letters denoting propositions) and connectives such that, if all the variables are replaced by propositions then the form becomes a proposition.

An example of propositional form is '(P and Q) or R' (for propositions P, Q and R). {(P \wedge Q) \vee R} and(?) 'P and Q'.

Truth tables are used to give propositional forms truth-values (the propositions which are used naturally have truth values). I believe that there is a standard way of writing the truth tables, and if you have an exam on such material it's important to know how your lecturer likes them to be, in fear of being marked incorrectly.

The truth tables for 'P or Q', 'P and Q' and 'not P' follow respectively:

Table 1
Table 2
Table 3
So if we were to look at the second line in table one, it indicates that if P was a true statement and Q was a false statement, then the statement 'P or Q 'is true. As I've said in a previous post, in everyday speech we tend to 'or' in the exclusive sense, eg do you want the paper or the pen? However, in maths we use it inclusively, and you probably will have to mention otherwise. 'Or' occurs in many mathematical statements and we never realise this. We know that a \le b, means that a is less than or equal to b, but notice the hidden or there. The inequality a \le b is not telling us that a single inequality is true, but it's saying that one or both of the following are true: a < b, or a=b. Which is why both the statements 9 \le 10 \text { and } 10 \le 10 are true. (from the 'or' truth table').

From table two, for an 'and' statement to be true, both the statements P and Q must be true. Once again, we probably have used this connective without being aware of it. . The following: 2 <>2 and e<3, hence the 'hidden' and is there. If one of the statements in false then 'P and Q' is false.

The negation of a statement is true when the statement itself is false, as can be seen from table 3. To negate a statement P, we can simply write 'not P' or we can usually obtain a negation by sticking 'not' at the start of the sentence. However, sometimes we have to be careful as to what the actual sentence is saying. I believe to fully be able to negate sentences like 'all my pens are in the pencil case' you have to have some understanding of quantifiers (or be good with words!). I think the negation to that is, 'One of my pen is not in the pencil case' since for the statement 'not' to be true we only need one counterexample you could say. (Sorry if I'm not making sense- I'm still a little iffy with this whole business!). So a question: negate 'If no one else cares, then I don't either.' (Yes you guessed it- I couldn't). A strategy is required for sentences with words like 'all' and 'every', and if I do ever post on quantifiers this will surely be resurrected.

It's important to understand negation and this whole 'not' business, since I think it helps explain proof by contradiction. I wouldn't worry so much about quantifiers at the moment, but they did help me to some extent. Negations will be looked at in more detail in the 'chapter' on proofs. [Am I not shutting up about negations because I couldn't get my head around them? All I'll say is: negations- to be continued... and leave it at that!]

Before I do an example, I forgot to mention that two propositional forms are equivalent if they have the same truth table. I'll end by doing two examples. One example1 with three statements and the second a truth table for P and (not Q).

Example 11
Write the truth table for ((not P) or Q) and (not R). (you might want to try it before scrolling down! ;) )

First we draw a table with columns P, Q and R as follows:
I suppose that's the 'generic' way of assigning truth values with three statements involved. Now we look to the question and add the columns which we require i.e. first '(not P)', then '(not P) or Q' and then '(not R)'. Then make a final column which has the question in it. E.g. the first row would be found by the following process.

Since P has the truth value T, then (not P) will be F. (same applies for (not R). So looking at '(not P) or Q', with truth values 'F' and 'T' we see from table 1 that this corresponds to row three, and hence the statement '(not P) or Q' is true. Therefore, looking at the question we see that we effectively have a conjunction with truth values 'T' and 'F' (i.e. ((not P) or Q) is a true statement and (not R) is false). From table 2, row two, the truth value for this statement is F, giving:

And so the rest of the table is completed in the same way:


Posting at 3:30am is never a good idea, and so I will do example 2 tomorrow (I mean later on in the day)! You may notice that I started this post 'yesterday', but that was a busy day indeed. Obviously, due to the time my excuse for errors comes into place so once again please do let me know of any errors.(Reading over it at this time doesn't make sense!) BTW, I only intend to post the content of our first weeks lectures and after that I will go back to analysis. I'm just hopefully trying to show what is is necessary for us to understand before we go further. Hope I haven't made too stupid a mistake, or missed something out!

EDIT 1: time- too early. The darn post wouldn't post because of conflict of html tags. :( Using the strict inequalities got it in a jumble! So for any bloggers out there never use <

EDIT 2: editing this post has messed up the html tags again. :( I will just post the tex code for the second example in the comments. :( (I've lost bits and bobs of the post!)

1. Dr C's notes.
2. Bits and bobs from my lecture notes and Dr E's book.

Tuesday, June 26, 2007

Enough of this slimy business.

So you got me to confess after all then! You guys are pretty good... too good. (Note: I've only had 5 hours of sleep- I'm not making excuses, seriously!) However, to gross you out (muhahaha) I might as well finish the picture story with the last two picture. How about a caption competition?

'Creature x's break for freedom!' (Rubbish I know- that's why I thought I'd offer it to the house!)
Consider that my revenge. Ahem. Well some may call that smile ball a slug- I call it a yeerk. I can't help but be fascinated by it! How it got into my dad's car remains a mystery- I'm sure my Uncle had something to do with it since he was the first to spot it! It's trail that is. After some investigation by Major Beans, we have good reason to believe that the yeerk was planted in the car overnight. Who the target body was, we don't know. Last night the yeerk escaped from it's prison compound and we were unable to interrogate it. Investigations are still pending, and our search parties are still out there. Our suspicions suggest that the target may have been Bean's dad. Thankfully, we managed to sabotage the yeerk's plan before that happened. Or did it happen?! Dad has been missing for a day now..... mysterious indeed.

However, there is no reason to panic! We believe that only one yeerk remains on this planet and soon it'll be in our hands. The little so and so somehow crawled into that little space! We all thought it was dead, and so my dad was going to get it removed. Yes, my Dad drove the car whilst that evil Yeerk relaxed. Gives us more reasons to suspect some foul play. Why did my dad not insist on getting rid of it straight away? Has this creature evolved? Does it have more powers? Now you best start panicking!!! (let me know when you calm down).

Eventually, my dad did get sick of the yeerk getting a free ride all over town and not paying a penny. Attempts using the screwdriver were cunningly dodged by the evil creature, and so it was a small stick that eventually drove it out! Interesting indeed. Maybe we should switch tactics! My dad poked it a few times with the stick and it started crawling out. This was when everyone went mad. Well actually it was just me and my brother there since we'd just got back home! My brother was told to get some paper. I was told something, but instead I ran upstairs to get the camera. (:o). In my absence my brother had stupidly got a small paper and so my dad dashed inside and grabbed the card. Operation Yeerk liberation was a success. Rejoice men, for tomorrow we're going to get that little piece of slime back. It was actually quite weird! And yes I've been grossing all the teletubbies out by showing them the pictures as well. :( [Sad face because I think by seeing the pictures the yeerk might have taken control of their brains!]

I had to get that slimy tale of my chest, and consider this the end of it. Well unless the yeerk resurfaces that is! I woke up early today because I had to return some library books back to the university library. Drat- my plan of no one wanting any books in the holidays has failed miserably. That didn't stop me from getting two more! (Well I've got to keep low on the buying front so borrowing it is.) I resisted the temptation to visit my office. :( I believe we said our goodbyes last time. :D Didn't want to put myself through that emotional turmoil again, and of course I thought of you guys as well! (You obviously don't want to be reading another post about the mss building!) I seriously do miss lectures and the hustle and bustle of going from one to the other. Call me a freak, but I'm itching to get started again? Maybe if we started earlier we'd have more time to cover stuff ... Yes, I jest. That's unlikely! Nothing beats active learning, books are good but getting myself to do things is indeed a difficult task.

Another reason why I didn't go to that building was because of ER. (three points for the correct guess. ;) ) The provisional ones have come through I believe, but we can't be told of them. This is frustrating since I previously had reason to believe that on Monday, after the exam board meeting we'd be able to get our results. Same procedure again I must confess- I keep on checking my emails and well.... I'm anxious to know how I did in Linear Algebra truth be told. (I know that I messed up and want to know how badly!) It was the first exam after all (I try to console myself.)

Oh and I've just remembered to mention something (thanks to Jake). Comment moderation will probably, annoyingly, be on for a while as a precaution. :) Sorry to make you suffer like this, but it's only for a week or so. It's not like I'm going to be anything but checking my emails in the next few days, so hopefully it shouldn't be too much of a hassle.

I have a question: My brother is in year seven and he's pretty ok at maths. He claims that he doesn't like it (due to a teacher) but he's ok in it. How do I go about encouraging him? I mean I want to help him but what do I effectively 'teach him'. What's the limit? (the sky I hear!). I do want to push him from an early age since that does make a lot of difference later on. Do you think doing Sudoko's or Kakuro puzzles is good for ones brain?

Anyway, this randomness will end on a pleasant note. (I hope). I at least feel that I can post random mumbo jumbo like this after posting about induction this morning! Tomorrow hopefully I'm going to go back to square one. Watch this screen. (That sounds so .... *shudders*.)

In my defence I've yet to have my cup of tea. Here are some pictures that I have just taken a while ago:
I don't know why but I like this one. (I took the right hand side bit with the camera upside down!)
The moon playing hide and seek.

Anyway, to summarize: Yeerk wars are over, I haven't got ER yet, started reading Fermat's Last Theorem (interesting) and Dad's not really gone missing! Oh and comment moderation will be on for a while without me doing any moderation that is. Why I'm feeling particularly jolly I have no clue! Honestly- it's 11pm and I feel electric. (BTW that is my favorite metaphor ever. Surprisingly my first time using it in this blog! Whenever I do write stories, I always have the sentence, 'the sky was electric' or something along them lines! Why- because in year 5 when we were doing Willy Wonka and the chocolate factory, I got 3 house points for my work which had that in it. Yes I'm easily pleased!) Oops I just gave the game away. (Is it that obvious that I've forgotten what I was going to type? :( )

Yet another slimy Proof (by induction?)

If you've just fallen of your chair to find me awake at this time then I do apologise. (What you don't think I haven't slept yet do you?!) Due to unfortunate (I claim!) circumstances I am awake at this weird time! I'm not too happy, since five minutes ago I learnt that I could have slept another hour. Not exactly five minutes -that's just me being vague as you do. Unsurprisingly lack of sleep initially means that I'm fully wake, and as this post will demonstrate slightly .... ? (Can't think of that word). BTW I haven't forgotten about my slimy story, which it seems is only fascinating to myself. *sniff* More on that at the end.

So to business. I didn't wake up and think 'bham I've got to post about this,' so we can all relax. (Un)fortunately my posts about maths are going to be quite random maybe. I'll try to avoid this, however upon reading this post at Modulo Errors I thought it necessary to discuss Proofs (note the capital P!) amongst other essentials in Maths. Analysis really refines the art of proofs and I believe it's important that I lay out the path nicely rather than trying to jump over the wall.

Initially it is easy to be overwhelmed by Proofs. We don't realise what is happening, but know so much to realise that we don't like the module Sets, Numbers and Functions (SNF). We prefer calculus. Proofs do crop up there, but we're more familiar with them (eg trig proofs). It's like being injected with a drug and your body has a bad reaction to it. That was what happened to me. We all react in different way (unless of course you're twins and have the same DNA... Biologists don't shoot me if that's wrong!). Thus our immune systems help us to overcome this 'illness' at different rates and different ways. Some people might need medication (eg me!) whereas others are able to battle it out on their own. Basically, for a majority of students starting on a maths degree either you 'don't like' proofs or you have no preference towards them. However I think a majority of students don't really understand how vital proofs actually are.

There are many reasons as to why students could possibly not like proofs. I have probably said this a million and one times, and so will say it again! I did further maths and if memory serves me well in FP1 we learnt about induction. I'm not sure whether or not to think that my teacher did a brilliant thing, since effectively I'd learnt it 'algorithmically'. I was clueless really as to what I used to be doing. All I knew was that if I didn't mention 'since this is true n+1 times, it's true for all n' or something along them lines I'd lose a mark! [Having now come across quantifiers, I do recall Mrs. B telling us about this symbol,\<span class=forall"> (for all), once upon a time.] The fact remains that when I was taught induction at university I was actually worse off than students who probably hadn't done it before. (maybe). Whilst we were taught it I used to get muddled up a lot since all I wanted to do was chop something from here and move it down there, and rearrange to get the right hand side. I never understood it and always used to assume the result and then say it's true!! Like I said credit has to be given to Mrs. B, since I don't think I ever really understood understood FP1. I say this because I find it extremely difficult to remember anything else that we did.

As it has been mentioned in Blogistan it is fundamental that you understand and get your head around proof by induction. Another confession I have is that the only reason I didn't answer the question on Induction in our January SNF exam, was because even then I had sucked majorly at proof by induction. I had tried to revise it but the cogs just wouldn't click. The Tweenies were in shock at this, since they felt that was one of the easiest questions on the paper. Even now I'd probably disagree since I do tend to find induction a challenge.

Before I give an example of induction, I'd like to continue discussion as to why students don't like proofs. In college C3 had an element of proofs to it but we weren't taught this chapter since the teacher said it should only be a few marks, and it doesn't matter if we drop them marks. At the end of the day a question did come on the exam paper- it wasn't too taxing, but that being said I still missed one part of it out. (Something about Pythagorean triplets). I do also recall the C3 text book having a chapter about proofs and the word 'contradiction' cropped up as well.

In my case the problem with proofs probably began in college. In FP2 I can't recall ever consciously thinking that 'yes I'm proving this'. It's about notation(?). The nature of proof should be made clear in college. You see I say all these things, but I'm not complaining about my experience of such things. I don't know how different things would have been if I'd understood said concepts, but my experience has got me to where I am today. I've learnt from it (hopefully) and glad of it.

So we start university with a vague idea of proofs. The first week of lectures we are introduced to most types of proofs- contradiction, example, direct etc. We don't understand these but it's our first week, things should fall into place soon. It's time to do the problem sheet. Things haven't fallen into place!! We look towards the example in lectures and try to follow the book, but still we fail miserably. 'Contradiction- it's stupid' and a few other well chosen expletives we conclude. Why- because we have no idea how we're meant to go about our business.

The weeks go on and all we're seeing is different colourful proofs in all shapes and sizes. You copy them out blindly, wishing for them to make sense. Slowly we start hating them. Not hating hating them, but we become sick of them. We don't want anymore- they don't taste nice. We have yet to understand how important they are. Then one day, after numerous complaints on the feedback forms, the new lecturer (who gave lots of proofs!) explains that why do a maths degree if you don't want to do proofs? We know somewhere in our heads that he's got a point, but still we don't like proofs. Not for long though.

Now in terms of university proofs and being able to follow them I think it is absolutely crucial that you understand the DEFINITIONS of things. Yes, that is one of the main things when it comes to formal proofs. If you don't get the bigger picture in lectures- don't worry. All you should initially worry about is getting to grips with the definitions. Be able to say them from the top of your head in your own personal style. Trust me on this- definitions are really important. After that focus on trying to understand what a theorem is actually telling you. I will write more about this in due course but it starts with the definitions in my opinion.

When does it finally fit into place, I hear you ask? (ahem!) Like I said- how we react and get over this illness is different. My initial motivation kicked in when Dr C started lecturing us (his proofs were plain cool- I could follow one step to another most times even if I didn't get the big picture). This motivation caused me to work. Working allowed the rusty cogs in my head to start moving. Working slowly started curing me! That's what did it for me, however I needed motivation to work, whereas the next person may not. I'm not going to pretend otherwise but some proofs still go whoosh over my head. However the difference is in my change of attitude towards them. No longer do I internally suffer when I hear the word proof. Now when presented with a theorem I want the proof!

A proof is a logical argument which establishes the truth of the statement. Proofs are necessary because they allow us to make statements which are true! We could claim a certain thing (conjecture) however until we have a proof we are not to know whether this claim is actually true. We cannot assume it to be true since, even if the counter example isn't eg 900000, it could be a bigger number-we don't know. The point being that one may exist. 'The concept of proof is absolutely fundamental to mathematics. In fact one may fairly claim that without proof (pure) mathematics does not exist.'1 Each step in a proof must be true, either deduced from previous statements or the assumptions made. I will be writing more on different types of proofs and what exactly a proof consists on another time, however they really are the building blocks of maths and it is important that we understand that. Only when we properly understand the concepts of proofs and the different types of proofs will we be able to construct them ourselves. (I live in hope- that's one thing I'm rubbish at!)

So back to induction now and my example. Before I go further, I would once again like to say a big thanks to Dr Coleman. Yes- you've got it, he helped me understand proof by induction after one popped up in sequences and series! (They're everywhere!) Honestly, I was slightly embarrassed, since the Tweenies were all trying to explain to me what's going on but it wasn't making sense. Why, because I didn't understand what I should be doing. I had been assuming what we're supposed to proof and then messing around with it. It's not necessary to understand the 'tricks' etc of proofs but the concept should be understood. The concept of induction should be understood since it also pops up in linear algebra and everywhere else! Thanks to Dr C. I feel I know how to tackle proofs by induction and how to go about doing them- trust me it was a a struggle.

From my lecture notes, the induction principle is:
Suppose that P(n) is a statement involving a general positive integer n. Then P(n) is true for all positive integers 1, 2, 3... if,
i) P(1) is true (i.e. the base case)
ii) P(k) \<span class= for all positive integers k.

That is the general case, however we can use the same idea and start with any other integer as the base case. 'Suppose that n_0 is an integer- positive, negative or zero. Induction can be used to prove that a statement P(n) is true for all integers n such that n\<span class=ge n_0">. The base case is now P(n_0) and the induction step is P(k) \<span class= for k\<span class=.'2

I'm delaying the example, but if I claim A \<span class=, then you'd either start with A and deduce B, or start with B and show A. You only consider one side of the argument, and then use that to show the other side. You can't start with one side and then all of a sudden jump to the other! In this case you have a choice of which side to consider first.


The example3:

Claim \<span class=

You may find it useful to identify P(n), which in this case is 2n+1 \<span class=.

So we first check the base case, and in our case that is when n=3. We are trying to show that P(3) is true.

\\2 \times 3+1 =7\\ 2^3=8\\ \text{and } 7 \<span class=

Then we proceed by assuming that the result is true when n=k, i.e. that 2k+1 \<span class= is true.

The next step is one of the most important ones (in my opinion). We want to show P(k+1) is true. That is what we want. We don't know anything about P(k+1). It might be true but we don't know that -we WANT to show it's true. That is why I mentioned the bit about trying to show that A \<span class=. We start we one side on the statement to show the other, and in our example we apply the same argument. We start with one side of P(k+1) and try to deduce that the other side is true.

So*:

\\\text{The LHS of }P(k+1)\text{ is }2(k+1)+1\\ =2k+3 = (2k+1)+2\\ \le 2^k+2^1\text{ by the induction hypothesis}\\ \le 2^k+2^k \text{ since } k \ge 3 \ge 1 \\ =2 \times 2^k=2^{k+1},\\ \text{which is the RHS of }P(k+1).

\text{So <span class=

Hence, by induction 2n+1 \<span class= is true \<span class=.

\<span class=

What to do: make it clear what P(n), the base case and P(k+1) are. I think it's good to let the reader now what you're doing and so have written things like 'by the induction hypothesis' when they were used. We were able to do that step since we have assumed that P(k) is true.

What not to do: Say 'assume P(k+1) is true'. Yes, I used to do that all the time- it's wrong. We don't know anything about the truth of P(k+1), we just showed that. The aim of the game is to show the truth of P(k+1). We cannot start by saying it's true!

I'm sure there are many other dos and don't so feel free to add to them. If I was being perfectly honest, I don't really fully get the 'trick' when we use k \<span class=. I slightly understand why we do it, but it's not properly 'clicked' yet in the sense that if I was to help someone doing this proof I wouldn't know how to explain it to them. It's used a lot in such proofs and so if you understand it that's great.

There you have it- induction in a nut shell. If I've missed anything out, or written something absurd please do let me know. I'm not going to get the chance to check through this straight away. I also can't remember what else I was supposed to write. :/ The slimy tale I'm afraid will have to continue in a post later today- I am running late, but if you haven't been following the story here's your chance to catch up!


*(meh- nothing apart from TeXnic centre understood my formula. :( ) EDIT: fixed thanks to Steve. :)
1 Allenby: Numbers and Proofs
2 Dr Eccles: An introduction to Mathematical Reasoning

Monday, June 25, 2007

The Slimy tale continues...

Due to my brain deadness I have no intentions of posting intelligent(?) stuff. :( Exam results are looming and I've been getting slightly stressed out. OK maybe not slightly, but majorly! Naturally when pondering over such cursed events one always assumes the worst. Also I seem to keep on remembering not reading the question as one-to-one but as onto. Humbug.

\\\text{However, now I can do this without too much work,}\\ \text{which makes beans very happy!}  And this too:

\displaystyle\int^1_0 \frac{x^4(1-x)^4}{1+x^4} dx = \frac{22}{7}-\pi

More will be revealed during the week. ;) But if you didn't catch the comments to the previous post, thanks to Steve I can post LaTeX stuff without too much hassle. *cue evil laugh!*

At the moment I've got my trainers on (yes at this time!) because of this thing crawling around my room.
Well actually I believe I've seen mummy and daddy spider about which is the reason for the trainers. I don't mind spiders but since I picked up a particularly hairy one when I was slightly younger I hate the idea of them crawling on my hands or feet! Hence the trainers. I still do pick them up if I have to, but ONLY with gloves on. (hehe) I hate the big ones- they're too fast and ... hairy! I'm slightly unsure of going to sleep on the floor since images of them crawling into my mouth are uncontrollable. :(

Anyway, I just happened to take that picture now which has distracted us from our slimy story. I've yet to hear your comments on the guest in my dads car yesterday- you're a boring lot ain't you. ;) (JOKE- yes I'm emphasising that!!!) However I cannot let that demotivate me and so must continue this tale with another picture. So I guess if you were 'upset' yesterday then you might have t give tomorrows post a miss as well.

If you were unsure on the truth of the picture then don't be. I took that picture whilst everyone was either screaming and running for their lives, looking for large paper, throwing up, going mental etc etc. I did get told of for being in the way, but I survived the deadly, powerful mental powers of that creature which seemed to have affected everyone-even my Dad! :D Blah- you want me to give away the story don't you. Not going to happen. I'm still interested on what you'd call that thing and how you think it got there. Don't look at me- I didn't put it there....


I need to get a life. :D :( {Not sure whether or not that's a good thing!} Good luck with your conspiracy theories.... no one. (When I'm tired and not sleeping I tend to write more rubbish- please do excuse it!) I should just shut up but.... fine night night. :)

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.

----

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