Brilliant Mathematical Thinking (or is that brilliant daydreaming??)

Transcript:
In algebric equation if 1+y is the sum you cannot add 1 and y together for example 1 would be a man and y would be a crocodile. But 1 [times] y can be possible because for example 1 would be a man and y would be a crocodile but [times] would be the fairy that change 1 and y into a frog so they can stay together then the answer is 1y.

Smiley face but no mark...

To Infinity and (not) Beyond: Part I

 

...awesome disney blog where I found this...

This mini-series of posts is dedicated to those of you who never got the full impact of this wicked Disney joke (much like me, until Maths III).

Newsflash: the whole point of infinity is that there is no such thing as "beyond" infinity. Infinity is it. The biggest. The most. The furthest.

To be honest, I'd be lying if I said I understood infinity. I don't even understand Mobius strips, and those are about the best visual representation of infinity we have... Escher drew the best one (of course). Follow the marching ants and you'll see what I mean by the Mobius strip being a visual representation of infinity. Sort of, anyway.

Much like me, the ancient Greeks didn't really like infinity much. In fact some of them refused to believe in infinity at all (you've got to admire these guys for their stubbornness if nothing else). They were forced to acknowledge the idea of unboundedness (time appearing to have no beginning and end, for example) but all applications or references to infinity were... tricky. Irrational numbers like

π

(which has an *infinite* number of digits) were regarded with deep suspicion and generally shunned.

The Arabs used the notion of infinity, because they started solving equations like

x² = 2

which (as you will know) only has irrational roots. They still weren't too keen on infinity though, and didn't examine the alarming endlessness of irrational numbers too closely.

Meanwhile in Europe quite a lot of people used the notion of unboundedness or infinity as a handy notion to explain divinity. Theologians like St Augustine and St Thomas Aquinas used the idea fairly liberally to refer to God, the unlimited being.

In fact most great minds have flirted with the idea of infinity here and there. The famous Galileo said that “It is wrong to speak of infinite quantities as being the one greater or less than or equal to the other.” 

...super-awesome collection of fractal art...

But still no-one had really studied infinity for its own sake. Until one day along came...Georg Cantor.

I think he deserves a post of his own...watch this space!

(oh, and go here to download an article about the history of infinity which goes into a lot more detail. 27 pages of detail, but fairly accessible. strongly recommended if you found this interesting)

The History of Zero (Part II)

'Right,' said Colon....'I heard this wizard down the University say that the Klatchians invented nothing. That was their great contribution to maffs, he said. I said, "What?" an' he said, they come up with zero.'

'Dun't sound that clever to me,' said Nobby. 'Anyone could invent nothing. I ain't invented anything.'

-Terry Pratchett JINGO p.29

(via a comment from samart, thanks!!!)

So Brahmagupta has come up with his fascinating theories about zero in India, and various tradespeople and explorers have transported these ideas (along with their spices and so forth) back to Baghdad and the rest of the Middle East. 

The most famous of the Muslim mathematicians to use the new zero in his work was Mohammed ibn-Musa al-Khowarizmi (the gentleman on the stamp). He "invented" or at least coined the term for algebra ("al-jabr" means "completion").

In any case, al-Khowarizmi investigated linear and quadratic equations equal to zero. He made enormous strides in demonstrating the great practical use of al-jabr and ultimately of zero, not simply as a placeholder but as an extremely important number in its own right (read more about al-Khwarizmi). 

As a brief aside, in algebra one of the most exciting things about zero is its remarkable multiplicative property. Anything multiplied by zero is equal to zero, and (crucially for quadratic equations) if the product of any two numbers is zero then one of them has to be zero.

*Ahem*

Interesting though it is, this doesn't get us much further in the story. The next player to enter the scene was a little boy called Leonardo. His father was a customs officer and merchant from Pisa, and young Leonardo would accompany him on his trips in North Africa and the Middle East. Leonardo grew up to write a book called Liber abaci, in which he explains the Hindu/Arabic numerals, including zero. You've probably heard of Leonardo. He's better known as...Fibonacci! (read more about Fibonacci).

Since Fibonacci was Italian, this brought zero firmly into the European sphere of thought. As is usual with revolutionary ideas, the private sector cottoned on to the wonderful properties of zero quickly, while governments and officaldom in general was more suspicious.

However, the work of several influential mathematicians (including Descartes) gradually brought zero into common usage. Eventually Newton and Leibniz were to find it indispensible to the furtherance of calculus...  

And now...well, can you imagine maths, or life for that matter without zero?

I certainly can't!

By the way, go here for a wonderful discussion of the history of zero in general...This is where I got a lot of my inspiration for this mini-series.

Identities and Inverses

Before I go any further in the History of Zero, perhaps I should spend a bit of time elucidating... 

Why is zero so special? And why are Brahmagupta's insights so profound anyway?

In order to explain, I will need to state a few definitions. Bear with me, they are a teensy bit abstract, but lots of fun. News flash: by Identity I do NOT mean "who you think you are"...

***

 Identity:

Assume that we have any set "S" and an operation "%" that relates any two members of S (we'll call them "elements of S") to a third element of S. Then the identity of S under % is an element x in S such that x%s = s for any s in S.

Note: It is easy to prove that such an element x is unique for a particular set and operation (I'll show you another time; let's not get sidetracked).

***

To be a bit more specific, in the set of all real numbers we can consider either addition or multiplication as our operation.

The multiplicative identity for the real numbers is 1 (because anything times 1 is itself).

The additive identity for the real numbers is 0 (because anything plus 0 is itself).

 

***

 Inverse:

Assume that we have any set "S" with identity i, an operation "%" that relates any two members of S (we'll call them "elements of S") to a third element of S and any particular element of S called s. Then the inverse of s in S under % is an element x in S such that x%s = i.

Note: It is easy to prove that such an element x is unique for a particular set, operation and element. We'll label the inverse of s as s^-1.

***

To descend to a more concrete level once again...(i.e. the real numbers for our purposes)

The additive inverse of 2 is -2 (because 2+(-2)=0)

The multiplicative inverse of 2 is 1/2 (because 2*1/2=1)

Now I suggest that you go back and re-read Brahmagupta's claims about negative numbers, positive numbers and zero. Hopefully you'll find them a bit more exciting in the light of this (wonderful) information...

The History of Zero (Part I)

0

When you come to think of it, there isn't much we can do without zero. Zero is just zero...the additive identity to some, just plain nothing to others, a placeholder in the units, tens, hundreds (and hopefullly thousands) columns to still others.

But the number zero has an amazingly chequered history, ranging from the ancient Sumerians to the classical Indian and Arabic mathematicians...so here goes:

Sumerians were farmers, so they needed to keep track of numbers of livestock etc... They had a complicated record keeping system of base 60 (a decent explanation can be found here: http://www.crystalinks.com/sumermath.html).

The Sumerian system was positional. In other words, where a particular symbol is in relation to all the other symbols tells you that symbol's value. An example in modern numerical notation is this:

431: 1 has the value 1, because it is in the units place. 3 has the value 30, because it is in the tens column. 4 has the value 400 because it is in the hundreds column and so on...

Any kind of positional system ends up requiring something to mean "empty" - for example, there is nothing in the 10s column in the number 604, so we write "0". So eventually the Babylonians, who had inherited the Sumerians' system, came up with a special symbol to mean exactly that. At first they just left the space empty, but since spaces tend to get squashed and disappear, at length they actually used a symbol as placeholder:

Well, so far so good. As a human race we have "invented" zero, at least as a "punctuation mark between numbers" (http://www.scientificamerican.com/article.cfm?id=history-of-zero). There's still a long way to go.

The next distinct step (that we know about) took place in 6th century India during the Gupta dynasty. The mathematician Brahmagupta began to work with negative and positive numbers ("debts" and "fortunes") and realised that a sum like 3 - 4 was far from meaningless, despite the fact that it is difficult to imagine it concretely.  He started to see numbers as abstract entities, not just representations of quantity (more discussion at http://www.storyofmathematics.com/indian_brahmagupta.html). 

But of course if you're dealing with negative numbers then you have to face the problems of sums like

(-1) + (+1)

 So he came up with the idea of zero, as a special number rather than just a placeholder. And he developed a whole lot of rules to go with it...

Though a lot of these statements may seem obvious, at the time these ideas were revolutionary! They were so exciting that they quickly (relatively quickly anyway) made their way over to Baghdad and the rest of the Middle East.

Bertie Russell and the Incompleteness Theorem (Part II)

Right, where was I? Bertie has just discovered his crazy uncle and a new fear - of going insane.

Ok, so the next big event in our young hero's life is that his grandmother (finally) hires a Maths tutor for him. Mathematics opened a skylight in Bertie's life. For the first time, Bertie came across the idea of proof. In a world that was full of terrors and things that go bang in the night, he realised that if something is true, you should be able to demonstrate that it is true. And thereafter you need not agonise over it any more - you've shown it to be true. Equally, if something is untrue, you should be able to disprove it and move on.

Of course the other way in which Mathematics changed Bertie's life was that it soon became clear that he was very good at it...and he was only to get better!

At that point we will leave Bertie for a few years. He grew up in more or less the ordinary way, eventually went off to university at Trinity College, Cambridge where he steadily demonstrated himself to be more and more brilliant.


But as Bertie found out more about Mathematics, something began to worry him. Everything was so logical, up to a certain point. Theorems built on theorems. Propositions were proved or disproved. Yet something was still missing. The whole edifice of Mathematics was built on axioms. In one way or another, something foundational had to be assumed.

Given that in some ways Bertie's happiness and sanity depended on the certainties of Mathematics, he did not take kindly to this, and quickly set about trying to remedy this foundational problem in the field. Between a marriage (begun and ended), several affairs and multiple academic papers; conferences and international unrest; Bertie attacked the problem with vigour. Together with his great friend Arthur Whitehead he wrote a massive tome on the fundamental principles of Mathematics. He became extremely famous. But he still couldn't settle the foundations of Mathematics to his own satisfaction.

And then something really remarkable happened. A young Mathematician called Kurt Gödel proved what became known as the Incompleteness Theorems. One (and only one) of the shattering implications of these theorems is this: there will always exist some things that are true, but not provable. And there will always exist some things that are untrue, but not disprovable. Another way of saying this is that is impossible to have an entirely internally consistent set of axioms for Mathematics.

Wait...just think about that for a second...


Can you imagine how devastating this must have been? Bertie's life goal was crushed in one (or two) little theorems...

We will leave Bertie at this point. As a brief postscript, I am happy to report that Bertie survived this setback and went on to make many more contributions to Mathematics, Philosophy and even politics (particularly as a pacifist during both World Wars).

I leave you with the thought of the Incompleteness Theorems.

May they keep you awake at night.

They truly are remarkable.

As was Bertrand Russell.

Inaugural EOTSDS: Bertie Russell and the Incompleteness Theorem (Part I)

I have inducted a new tradition...the last-day-of-term shaggy dog story (otherwise known as the EOTSDS)! After all, no-one wants to be at school, but we all have to sit in class and try to be reasonably productive. So I've decided that the thing to do is tell a very long, moderately entertaining and somewhat educational shaggy dog story.

This is the first EOTSDS of 2011. I took it (loosely) from a remarkable book called Logicomix http://www.logicomix.com/en/, which you should all go out and beg, borrow, steal or even (gasp) buy immediately. You'll have to forgive inaccuracies in my version and kindly take into consideration the conditions underwhich this story was first told. You'll also have to fill in a lot of shaggy dog details which just don't work in text...

Are you all sitting comfortably?

Once upon a time, a very long time ago, there lived a small boy called Bertrand Russell, but everyone called him Bertie.

Now Bertie got off to a very unfortunate start in life, due to both his parents dying. So he had to go live with his grandparents, who were very important people who lived in a stately home. Think gloomy, old, important, and really not very nice to live in, especially as a very small boy.

Poor Bertie was given a large, dark drafty bedroom all on his own in this ominous dwelling, and on his first night he heard this terrible groaning noise from somewhere in the attics. Being very young, and since the night was very dark and there was no one to call, he lay in bed and trembled with terror. Was it a ghost? Was it a demon?

This nightly groaning continued to terrify Bertie for many years. All in all Bertie was growing into a very frightened little chap. His grandmother's approach to child-rearing didn't help. She was extrememly strict and extremely religious. She didn't believe that children should ask questions, and she told Bertie lots of stories about the awful things (such as burning in hell) that would happen to him if he was naughty. So of course Bertie was convinced that the groaning was a devil sent to punish him for his misdeeds...


Well, the years went by, as they do; and Bertie grew up, as one does. And soon enough, his grandmother started to hire a series of tutors for him (in those days the children of the very rich didn't go to school, they had private tutors). He fell in love with his beautiful German teacher of course, but much more important was the science teacher. He didn't exactly learn brilliant scientific method, but even the simple observations that they made together began to make Bertie think. And one of the first things he thought was about ghosts and devils. In fact not to put too fine a point on it, Bertie began to have serious doubts about the "ghost/devil theory" of the nightly groaning.

So, one night Bertie snuck off and followed the sound of groaning. He made a momentous discovery, which was to cure his fear of ghosts forever. However that fear was replaced with a far more serious one... What he discovered that the groaning came from a mad uncle whom his grandmother had been keeping hidden in the attics. True story. In grandmother's defence, attitudes to madness or mental illness in those days were such that hiding a crazy relation may have seemed like a very good idea. Mental illness was seen as hereditary, and no-one would want to marry into a family in which there was a known "madman". And the official mental asylums were so horrible that you would not want to send any family member to one of them, no matter how serious his condition (see the opening scene of the movie Amadeus...)

So Bertie's fear of ghosts was replaced by the very real fear of going crazy; a fear that would stay with him throughout his life. But at first it wasn't just a vague fear. It crippled him.


Do not fear, good readers. All is not lost. Stay tuned for the next exciting episode (my fingers are getting tired of all this typing).

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 :-)

Field Trips

After school today I marked 87 scripts. Some scripts were almost flawless, other struggled to make their meaning even vaguely accessible. Some covered fairly standard grade 9 work, others dealt with fairly advanced grade 12 material.

But they all had one thing in common. Each of them took me down a unique (and often alarming) journey through Mathematics.

The geography of Mathematics is far from dull.

There are the mountains when a wonderful insight comes from an unexpected source. The light of clarity shines on beautiful, competent (and legible) work.

There are also valleys, where it is dark and murky. Here the intrepid traveller peers through the mist, trying to find the path, or at least enough of the path to credit with some marks.

Sometimes the traveller happens upon a sparkling stream of new ideas. Sometimes she trips over a rock of a silly mistake. Occasionally she sinks into a bog of general misunderstanding and confusion...

Amazingly, each of these features (and doubtless many others that my metaphor has not yet uncovered) can be found in almost every single script at one point or another.

Not an afternoon wasted then?

Maths Is Everywhere

This is my most favourite TV show at the moment, and thanks to You-Tube I can steal a little tiny bit of it to show to my classes and you, oh reader...

Mathecaticians

This is not a misspelling (though my spelling has always been a little bit suspect). It's a profound thought.

Gauss and Pascal are our kittens. They behave much as I imagine the original famous old mathematicians would have done. In other words:

They poke their noses into everything!

My favourite story about Herr Gauss (original) is probably entirely urban legend. I have not verified it, and if you know it to be false please keep that to yourself. It's a wonderful story.

So when he's about 8 or 9 (maybe 10, let's stretch a point) Gauss is sitting in his Maths class. As usual, he's completed all his work. And his homework. And the next day's homework. And his poor teacher is going crazy, probably partially because Gauss is pulling Heidi's hair and throwing thumbtacks at Kurt.

Finally, the teacher tells Gauss that his next assignment (I expect he was told to complete it before he went out to break) was calculate the sum of the first 100 natural numbers.

So, being eager to get to his bratwurst sandwich before Kurt stole it, Gauss does just that. Except, as he's a lazy little brat who doesn't feel like taking the long way round, he invents and proves a formula for calculating the sum of any arithmetic series.

No, I don't think you were paying sufficient attention: he proved the formula for calculating the sum of any arithmetic series.


The same elegant, easy-peasy-once-you-see-how-to-do-it proof that we use today...

Long live curiosity!

Daydreaming in Maths

I feel like a fraud.

Actually, I just feel like the same grade 9 girl who swore never EVER to study Maths for a single second longer than she absolutely had to.

Or maybe like the grade 12 girl who wrote sonnets, sestinas and villanelles in Maths lessons. In between absentmindedly creating calculus doodles.

Or like the university student who liked to sit next to the window during Maths lectures, watching the human traffic go by on Jammie plaza and mulling over one of John Donne's more obscure puns.

So this isn't a blog about Maths (despite the fact that I'm a Maths teacher, *gasp*). Nor is it a blog about English.

It's a blog about daydreaming.