Something to Cheer Me Up

Cricket related blues aren't fun...but here is something which never fails to cheer me up.

Feeling Even More Special

So what did I do with my birthday?

Well I went to work...and it was AWESOME!

Here are some pictures of what happened, all at school, and most from students.

 

 Lots of messages from various students and classes...

 A card from my grade 9 class

A little party in my classroom with my family after school...

And it's not even over yet...

Feeling Special

This post has nothing to do with English or Maths...but everything (I suppose) to do with teaching.
Teaching is the best job in the whole world, despite occasionally and temporarily being the worst job in the whole world.

So tomorrow is my birthday. And today my grade 12s threw me a surprise birthday party.

Don't get me wrong, we still worked. Hard. But at the beginnning of the lesson I was presented with some cooldrink, and biscuits and chips and sweets and best of all, a birthday balloon covered with messages from the class:

So guys...this one goes out to you: Zintle, Akeelah, Aqeela, Kristi, Brent, Saphron, Tariq, Benzly, Kyle, Greg, Dhanyal, Bevanee, Songezo, Gcina, Brian and Boeba.

You rock!

Generating Fractals in Excel

Being interested in fractals recently, I found this web-page...Haven't completely finished working through the process, so I can't promise it works easily, but the Maths behind it is extremely interesting...It does involve complex numbers, so when I have the energy I will post a little something about that too! In the meantime, if you have plenty of bandwidth then try it out...

http://pcplus.techradar.com/node/3095

Virtual Maths Worlds...Another discovery!

Go look, go look.

http://www.cafepress.com/cattail_nu/1988004

I think someone else may have a similar sense of humour to me...at last!!!

P.S. The link is there, just hover over the blank space and it will appear. Magic!
P.P.S. Eventually I will get round to changing the colour of my links. But in the meantime you'll just have to survive.

The Weird World of Exponents

Exponents are weird. Let's face it. We have these tiny little numbers floating around in the air like gnats or magicians' familiars, and they do all these magical, weird things to whatever they touch...

When exponents are involved, numbers mysteriously get massively bigger, or massively smaller. Sometimes they just disappear into thin air.

Ah Maths...

You see, having come up with a new notation (i.e. xⁿ meaning x multiplied by itself n times) the mathematical community has to decide on certain logical and notational conventions which allow the notation to be used consistently. Here they are:

Or, more "mathematically" (i.e. using a whole lot of funny letters instead of colourful blocks):

When confronted with this list, most people justifiably panic. I mean, that's just weird, right? 

Well actually, it's not as weird as it may seem at first...Allow me to demonstrate.

Okay, so if  xⁿ = x.x.x.x.x.x... (n times), then it makes sense that 

xⁿ.x^m = x.x.x.x.x...(n times).x.x.x.x.x.x.....(m times) = x.x.x.x.x.x...(n+m times)

 (thank you http://www.loisterms.com/powers.htm for this and a couple of other images in this post!)

In which case, it makes sense that 

xⁿ÷x^m = x.x.x.x.x...(n times) ÷ x.x.x.x.x.x.....(m times) = x.x.x.x.x.x...(n - m times)

Bear with me. We're getting to the exciting bits now.

At this point we have to deal with the fact that we could have the same number of xes on the top and bottom. In order for our new system to be consistent with other mathmatical systems involving the multiplicative identity (i.e. 1), we also have to state that

x⁰=1

Okay, great, but what if we have more xes on the bottom of the fraction than on the top? Looking at this problem through the lens of the previous "laws" (or logical corollaries of the original notational statement), we find ourselves with a new statement

x^-n=1/xⁿ

The last few exponent laws follow in a similar manner...why not try justifying them yourself? 

Call me a nerd (yes, fine I'm a nerd) but I find the system extremely satisfying. Each law demands the next law, and if a single law was missing the whole system would come crashing down as fast as the Wall Street in 1929.

Ah Maths...

 
And that's just for fun; a fractal based on exponents :-)

This guy called Will

A few years back I came across this guy called Will. A pretty average guy really, from a pretty average small town family. His dad sold gloves and hats, if I remember correctly. Plenty of money, but not all that much education. Not posh people anyway.

Well in any case, by the time I met Will he'd had a fairly interesting life, with a fair number of divorces, affairs and general angst behind him. Rumours about his sexual orientation abounded, and most people had a strong opinion about him, whether it be positive or negative.

You see, the thing about Will was that he had a knack for writing (and selling) plays. He was prolific, and popular, and totally low brow. His plays were full of dirty jokes and lewd suggestions and blood and guts and drama...

A lot of them are still among the most popular and frequently produced plays in the English language!

Proof vs Truth

The famous saying is that you can prove anything with statistics...a lesser known and probably less, um, catchy saying is this:

Given anything, you can prove anything.

But the truth is that given enough hand-wavy argumentation, impressive numbers, big words and confidence, most people will believe anything. Speaking as a person with a strong tendency towards gullibility, I can tell you that this can be a real problem...

You see, the fact that I am convinced of something does not make it true. And not all true things are necessarily all that convincing...

My New Mission

oh wow

long time no post!

The last few weeks have left little time for daydreaming and even less for writing about daydreaming. Marking, entering marks, chasing down missed tests, marking again, entering marks again, altering marks, marking again... lots of mental arithmetic anyway!

It made me wonder (again) about why exactly we all end up doing a whole pile of stuff we don't enjoy just in order to do whatever it is we actually wanted to do in the first place. Studying is often (sadly) the prime example. Others include washing up, marking, exercising, driving in traffic etc...

It seems unavoidable - behind, before and in front of any pleasurable or fulfilling task lies a sea of unpleasurable and unfulfilling jobs.

Um, silly question but WHY?

Surely they can't all be necessary???

New mission for me: get rid of all unnecessary, time consuming and soul-draining tasks (without getting fired). Anyone keen to join me?

Peano's Axioms for Non-Mathematicians (not dummies)

So when I looked at my previous post on Peano's axioms again, trying to see it as if I'd never seen the axioms before, I realised that maybe they aren't too accessible as they are...

Here is my attempt to explain in plain(er) English. I'll leave out some of the definitional fluff at the beginning and cut straight to the axioms themselves

1. We assume that there is at least one natural number: 1 (some people prefer to start with 0, I'll just use the same convention as the one in the previous quotation). The set of natural numbers is not empty.
2. For each number that is a member of our set (i.e. for each natural number) there exists a successor to that number. In other words, every natural number has a "next" number. The successor of 1 is 2, the successor of 2 is 3 and so forth. We'll label the successor of an unknown natural number x as x'.
3. There is no natural number whose successor is 1. In other words, x'≠ 1, no matter what x is. In still other words, 1 is the first natural number. (It also works if you let 0 be this "first" natural number).
4. Every natural number has one successor or no successor, i.e. the successor of a natural number is unique (if it has one). We write this symbolically as: x'=y' means that x=y.
5. Let's make a special set of natural number that obeys the following rules. A: 1 is a member of this set. B: if a certain natural number is a member of the set, then its successor must also be a member of the set. THEN all the natural numbers have to be members of that set. This is the principle of induction.

I think perhaps I'll post about the principle of induction in more detail another day...it certainly merits a post of its own!

In the meantime, I hope that helps you to understand Peano's axioms a bit better.


Oh, and by the way:

The coolest thing about the natural numbers is this: any set that obeys those five axioms (considered as rules) is, for all extent and purposes, essentially identical to the set of natural numbers.

This is deep. Think about it.

Contingency and Cricket

Cricket is a game of contingency. Every decision made in the game relies on a series of apparently insignificant pieces of information.

If we win the toss...
If we bat first...
If the sun comes out...
If so-and-so gets into his innings...
If so-and-so is unable to complete his bowling spell...
If the Duckworth-Lewis system kicks in...
If he bowls a no-ball...
If he scores a boundary...

This form of decision making is actually very mathematical. Logic (which is largely based on contingency) forms an enormous part of modern mathematical research.

What are the real, logical implications of a certain piece of data?
I guess maybe we'll have to watch the rest of the match!

Peano's Axioms

Have you ever wondered about how the Natural Numbers (1, 2, 3, ...) are defined mathematically?

Well, no you probably haven't. Most people don't.

After all, the Natural Numbers are...um...natural. They're a fundamental part of how we think about numbers. But believe it or not their official definition (originally stated by Peano as a set of axioms) is fairly complex!

The following is quoted from Edmund Landau, Foundations of Analysis, Chelsea, 1951, pp. 1-18 (via http://www.ms.uky.edu/~lee/ma502/notes2/node7.html). 

We assume the following to be given:

A set (i.e. totality) of objects called natural numbers, possessing the properties--called axioms--to be listed below.

Before formulating the axioms we make some remarks about the symbols = and ≠ which will be used.

Unless otherwise specified, small italic letters will stand for natural numbers...

If x is given and y is given, then:
either x and y are the same number; this may be written x=y (to be read ``equals'');
or x and y are not the same number; this may be written x≠y ( to be read ``is not equal to'').
Accordingly, the following are true on purely logical grounds:

1. x=x for every x.
2. If x=y then y=x.
3. If x=y, y=z then x=z.

Thus a statement such as a=b=c=d, which on the face of it means merely that a=b, b=c, c=d, contains the additional information that, say, a=c, a=d, b=d.... Now, we assume that the set of all natural numbers has the following properties:

Axiom 1:

1 is a natural number. That is, our set is not empty; it contains an object called 1 (read ``one'').

Axiom 2:

For each x there exists exactly one natural number, called the successor of x, which will be denoted by x'.

Axiom 3:

We always have x' ≠ 1 . That is, there exists no number whose successor is 1. That is, there exists no number whose successor is 1.

Axiom 4:

If x'=y' then x=y. That is, for any given number there exists either no number or exactly one number whose successor is the given number.

Axiom 5 (Axiom of Induction):

Let there be given a set M of natural numbers, with the following properties:

I.

1 belongs to M.

II.

If x belongs to M then so does x'.

Then M contains all the natural numbers.