Recent Reading and Rants

Big news guys: I've been cheating on my blog! With an old hobby - reading! At least two or three times in the past two weeks of silence I have stayed up late finishing a particularly delicious novel. Ah...

These are some of the things I've been reading. I may have missed a few in this list. As you can see, my bedside reading pile is getting a bit out of control

The Hunger Games (Suzanne Collins): dystopia fantasy. Thrilling, unexpected, sensitive. I will definitely be reading the sequels as soon as I can get my hands on them.

Sabriel (Garth Nix): classic fantasy. In a word - brilliant. Scary, but brilliant. If you're going to read one fantasy novel for teenagers, this is it.

Postcards from the Edge (Carrie Fisher): the story of an ex-druggie Hollywood actress, told in fragments. Funny, tragic, absurd and random. I reserved judgment until the last chapter, but the ending pulled the whole thing together beautifully.

News from Thrush Green (Miss Read): old, kitsch, adorable. Restful reading.

The Avenging Saint (Leslie Charteris): old, predictable thriller/teccie. Also restful if you like that kind of thing, which I do. Involves car chases, fights on yachts, rescuing millionaires' daughters from certain death etc..

The Physics of the Impossible, Introducing Stephen Hawking, Newton's Notebooks: I know these last three seem a bit highbrow for my taste, but don't worry. I've only been dipping into them because they're living on my husband's bedside table. So no review of these ones I'm afraid. You'll have to wait till I get a new brain.

You will also observe that there is very little official "literature" on my list. This is because (let's face it) most literature is difficult and/or unpleasant to read. So I don't, unless I have to. [disclaimer: there are many exceptions to this rule. wonderful ones. but they still aren't exactly leisure reading. and who the heck decides what is literature and what isn't anyway.]


Rant Over.


I hope to contribute a little more to the stream of internet *stuff* this week, thus in my own small way contributing to the creation of a species entirely dependent on machines (this is my way of saying "I'll try to post more")


Goodnight to you all.

Contingency in Hockey

It seems that a lot of sports are actually exercises in contingency. Today, for example, I had a long conversation on the topic of contingency with the poor harried coach of the U14 hockey team that I manage (those of you who are paying attention will observe that I seem to manage only U14s. this is in fact the case. I think its the penalty of being new.)

Back to the point. The coach has desperately been trying to teach the girls the art of "two vee one" (and I quote). The idea behind "two vee one" is simple: try to take the ball to a part of the field where your players temporarily outnumber the other team's players.

Though it makes perfect sense in theory, in practice "two vee one" requires some very advanced contingency thinking under pressure. For example

IF she gets the ball AND I am behind her AND there are three opposition players coming from the left AND there is one opposition player coming from the right AND our strongest striker is running towards us from the centre THEN...

All of this in the split second it takes to run down the line on the right hand side of the field before the aforementioned three players reach you.

I am filled with admiration for those who manage to achieve this even once in a lifetime.

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.

Daydream...

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.