By Diogenes
NB: Euler is pronounced "oiler".
We found this wonderful piece of literature a few years ago via Prof John Webb - thanks Prof!!
Showing posts with label hectic maths. Show all posts
Showing posts with label hectic maths. Show all posts
Saturday, January 12, 2013
Wednesday, May 23, 2012
Mathematical Constants III
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| from gogeometry |
I will never forget my first encounter with the constant e. It was my first week of university. I was trying to complete the first of many Maths tutorials. I came accross this funny symbol: ln(...)
Since I couldn't find an explanation of what "ln" meant in my 700 page textbook (which was otherwise excellent, by the way), I went to my tutor. Our dialogue went something like this:
Me: What does ln (pronounced lin) mean?
Tutor: You don't know what lin means?
Me: That's right, I'm asking what ln means.
Tutor: Ln is the natural logarithm.
Me: Okay...
Tutor: It's just a logarithm with base e.
Me: What's e?
Tutor: It's a constant that forms the base of the natural logarithm.
Me: But you said that the natural logarithm was a log with base e.
Tutor: Yes.
Me: And you defined e as the base of the natural logarithm.
Tutor: Yes.
Me:...
Needless to say, this dialogue continued around in circles for about ten minutes before we both gave up in disgust. Even though I learned how to use ln and e, they have remained largely mysterious to me.
It's gonna take a few posts...but I plan to correct that! Join me on my mission...
And just to lighten up your day (and its ALMOST on topic):
Wednesday, May 9, 2012
Famous Mathematical Constants II
3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679 82148086513282306647093844609550582231725359408128 48111745028410270193852110555964462294895493038196 44288109756659334461284756482337867831652712019091 45648566923460348610454326648213393607260249141273 72458700660631558817488152092096282925409171536436 78925903600113305305488204665213841469519415116094 33057270365759591953092186117381932611793105118548 07446237996274956735188575272489122793818301194912 98336733624406566430860213949463952247371907021798 60943702770539217176293176752384674818467669405132 00056812714526356082778577134275778960917363717872 14684409012249534301465495853710507922796892589235 42019956112129021960864034418159813629774771309960 51870721134999999837297804995105973173281609631859 50244594553469083026425223082533446850352619311881 71010003137838752886587533208381420617177669147303 59825349042875546873115956286388235378759375195778 18577805321712268066130019278766111959092164201989 38095257201065485863278865936153381827968230301952 03530185296899577362259941389124972177528347913151 55748572424541506959508295331168617278558890750983 81754637464939319255060400927701671139009848824012 85836160356370766010471018194295559619894676783744 94482553797747268471040475346462080466842590694912 93313677028989152104752162056966024058038150193511 25338243003558764024749647326391419927260426992279 67823547816360093417216412199245863150302861829745 55706749838505494588586926995690927210797509302955 32116534498720275596023648066549911988183479775356 63698074265425278625518184175746728909777727938000 81647060016145249192173217214772350141441973568548 16136115735255213347574184946843852332390739414333 45477624168625189835694855620992192221842725502542 56887671790494601653466804988627232791786085784383 82796797668145410095388378636095068006422512520511 73929848960841284886269456042419652850222106611863 06744278622039194945047123713786960956364371917287 46776465757396241389086583264599581339047802759010
Tada! The first two thousand digits of Pi! And its growing everyday...*sniff*
Disclaimer - I did NOT calculate this myself. I got it from here.
Friday, May 4, 2012
Famous Mathematical Constants I
Pi (not pie) is usually attributed to Archimedes. In any case, it's been around for a long time. And we use it in a lot of places (more on that later).
But first - what does pi represent?
In order to answer this question, I need to remind you about a tiny piece of geometry - about circles.
The measurement around the rim of a circle is called the circumference. The measurement across the centre of the circle (from edge to edge) is called the diameter.
Now at some point someone started investigating the relationship between the circumference and the diameter of the circle. Something like I did (and you can do it too if you don't believe me), except they did it a lot more times and probably a lot more accurately.
Draw a circle using a compass or other circle drawer. Measure the circumference of the circle with a piece of string.
Measure the piece of string.
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Calculate circumference divided by diameter (called the ratio between circumference and diameter)
Repeat...
You see, if you do this accurately enough (which I definitely didn't) then you find that ratio is always the same. In fact it's always roughly equal to 3,14159... an irrational number which we call pi!
Now the first (and I think the most important use after all this fiddling around with string) of pi is to find the circumference of a circle more efficiently than measuring it with a piece of string!
Knowing just the diameter (or radius - half the diameter), we can use pi to calculate the circumference!
C = dπ = 2πr
And then we can also use pi to calculate the area of a circle (you might notice that the circumference is the derivative of the area :-) )
A=πr2
Here are some alternative ways of writing pi. As you can imagine, each one is linked to a very special and mostly very advanced branch of Mathematics. Here are some super awesome everyday applications of pi.
That's all from me for today...we'll tackle e next time!
Tuesday, August 2, 2011
Limits of the Imagination
This evening I'm doing an "academic lecture" as part of a series at school. My lecture is going to be on "The Limit". Here's a preview:
So let's create an accurate definition:
Are we getting somewhere?
Maybe not...
When it comes to limits, expect the unexpected!
So let's create an accurate definition:Any Questions?
Monday, June 20, 2011
To Infinity and (not) Beyond: Part II
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| photo from the nytimes archives |
I'm sorry that this post is so belated, but the creative side of my brain has been dead, and understanding Maths takes a lot of creativity.
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| everything maths |
I promised you a discussion of Mr Cantor and his contribution to infinity... Cantor realised that there is more to understand about infinity than previously thought.
He started by thinking about different sets that we use everyday and think about as "infinite", such as the natural numbers (0,1, 2, 3, 4...). And the real numbers (0, 0.1, 0.0000000002, -0.0547829 etc...). In fact he thought about a lot of sets, and these are just two simple ones. But they will suffice for my highly simplified explanation.
Most of us would agree that there are infinitely many natural numbers. You could keep counting forever (though why you'd want to I'm not sure) , and you would never "run out" of natural numbers - even if what you were counting was the number of stars in the sky!
To briefly introduce some terminology, Cantor called the number of elements in a set the set's cardinality. He called the cardinality of the natural numbers "aleph null" (HINT: the squiggly N is the Hebrew letter aleph).
Well so far so good. All we've really done is make another label for infinity. Who cares?
The really exciting part comes now! Cantor went on to observe that the real numbers behave...differently...to the natural numbers. In fact, he demonstrated that there are infinitely many real numbers between every two natural numbers.
This is the core of the Cantor's insight, and it is demonstrated visually by Cantor sets, which you can learn how to draw here.
The idea is that if you keep dividing the interval between 0 and 1 into thirds, deleting the middle third each time, you could keep dividing forever. Since the real numbers between 0 and 1 essentially represent every possible fractional value between 0 and 1, they can in some sense be represented by the infinitely many "segments" you would have at the end of this process. If there was an end, which there obviously wouldn't be!
And yes, the Cantor set is a kind of fractal - go here for another (probably more rigorous) discussion!
Cantor went on to say (very reasonably in my view) that you can't say that the real numbers have the same cardinality (number of elements) as the natural numbers if there are infinitely many real numbers between 0 and 1 and further between any two natural numbers. It just doesn't make sense! So he labelled the cardinality of the real numbers aleph, with the rider that aleph is bigger than aleph null.
Um, aren't we forgetting something here??? We agreed that both the natural numbers and the real numbers had an infinite number of members. And do you remember Galileo: ...“it is wrong to speak of infinite quantities as being the one greater or less than or equal to the other”?
Cantor disagrees. According to him, there are at least two types of infinity. The first has the same cardinality as the natural numbers. This he called a "countable" infinity. The second has the same cardinality as the real numbers. This he called an "uncountable" infinity.
As you can imagine, this has some interesting consequences...but that's a story for another time.
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| from here |
Monday, June 6, 2011
To Infinity and (not) Beyond: Part I
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| ...awesome disney blog where I found this... |
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
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.”
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| ...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)
Tuesday, May 10, 2011
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).
***
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...
Sunday, April 10, 2011
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.
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.
Saturday, April 9, 2011
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).
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).
Sunday, March 20, 2011
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
http://pcplus.techradar.com/node/3095
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.
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
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.xm = 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÷xm = 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...
Tuesday, March 15, 2011
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...
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...
Friday, March 4, 2011
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
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.
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
- 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.
- 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'.
- 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).
- 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.
- 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.
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.
Wednesday, March 2, 2011
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!
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:
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:
- x=x for every x.
- If x=y then y=x.
- If x=y, y=z then x=z.
- 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'.
Wednesday, February 23, 2011
Definitions and Theorems
So no-one was brave enough to take a guess?
Well, the problem lies in the fact that I failed to differentiate between definitions and theorems. And the difference is crucial. Absolutely crucial.
A definition is basically the naming of a certain property or group of properties. So for example, say we observe that pretty often the order of multiplication in the natural numbers does not change the value of the product (i.e. 3x4=4x3). So instead of saying that whole shpiel each time, we decide to define the term "commutative" (which in "real maths" loosely means a x b = b x a or a + b = b + a). You can't prove what commutative means. That's just what you've decided to call that particular property.
If you are alert, you will observe that this is pretty similar to how I explained what an axiom was. I promise that there is an important distinction. An axiom is something you assume to always be true. A definition is simply the name you give to something which is sometimes true in some situations.
So, back to theorems and definitions.
So we've defined commutativity. Now the more we look at the natural numbers, the more we get the feeling that all natural numbers will be commutative under multiplication. Now we're getting into theorem territory. The minute we make the claim "natural numbers are commutative under multiplication (i.e. it doesn't matter which order you multiply natural numbers in)", we are saying that our definition of commutativity applies to a very large group of numbers.
This has to be proved. And I'm not going to tackle that enormous topic now. Suffice it to say that it is important to distinguish clearly (in Maths notes or elsewhere) between theorems and definitions.
Briefly (and slightly poetically):
Well, the problem lies in the fact that I failed to differentiate between definitions and theorems. And the difference is crucial. Absolutely crucial.
A definition is basically the naming of a certain property or group of properties. So for example, say we observe that pretty often the order of multiplication in the natural numbers does not change the value of the product (i.e. 3x4=4x3). So instead of saying that whole shpiel each time, we decide to define the term "commutative" (which in "real maths" loosely means a x b = b x a or a + b = b + a). You can't prove what commutative means. That's just what you've decided to call that particular property.
If you are alert, you will observe that this is pretty similar to how I explained what an axiom was. I promise that there is an important distinction. An axiom is something you assume to always be true. A definition is simply the name you give to something which is sometimes true in some situations.
So, back to theorems and definitions.
So we've defined commutativity. Now the more we look at the natural numbers, the more we get the feeling that all natural numbers will be commutative under multiplication. Now we're getting into theorem territory. The minute we make the claim "natural numbers are commutative under multiplication (i.e. it doesn't matter which order you multiply natural numbers in)", we are saying that our definition of commutativity applies to a very large group of numbers.
This has to be proved. And I'm not going to tackle that enormous topic now. Suffice it to say that it is important to distinguish clearly (in Maths notes or elsewhere) between theorems and definitions.
Briefly (and slightly poetically):
Theorem = Claim
Definition = Name
Sunday, February 20, 2011
By Definition
When I was young and foolish I used to make the most beautiful Maths notes you've EVER seen. I had different coloured pens, and each chapter had a colour with matching headings. Theorems and definitions were done in a certain colour so that I knew they were important and there was yet another colour for examples and helpful hints...
What was I doing wrong?
Give it a guess, oh reader! I'll post the answer next time...
What was I doing wrong?
Give it a guess, oh reader! I'll post the answer next time...
Friday, February 11, 2011
Language, Axioms and Bananas
Talking about Algebra with my grade 8 class today, I was struck again by how confusing and apparently random Maths can be. I remember being totally mystified by Maths at various points in my relationship with the subject. Occasionally I still am mystified.
But something that has helped me to deal with my mystification is to remember that Maths as we know it is not the be all and end all and only conceivable possible way. It could (theoretically anyway) have all happened completely differently.
How? I hear you cry.
Well, Maths is a language, just like other languages (despite having a few extra features and lower comprehensibility ratings than most). The fact that "banana" represents banana does not say something innate about the relationship between word and object (this is Wittgenstein and those guys, but watered down). It just happens to be the word that history and society and chance and whatever other factors have attached to our familiar yellow fruit.
Same with Maths. All of our Mathematical system is based on certain axioms (like the fact that a point has no dimensions). These axioms (stated most famously by Euclid) are assumed to be true in the proofs and ways of thinking about things which are so entrenched in us. It is very difficult to imagine them not being true. Yet in actual fact this is only the case because its convenient that it should be, or because we're used to it being so.
Plus, of course, in order to do anything interesting or useful we have to have a common frame of reference. So by all means lets argue about something more interesting than whether we should write 2x2 as 2^2 or 2_2; or whether 3ab really does mean 3 x a x b!
The other alternative is very much like saying that banana is a very bad word and difficult to pronounce, so from now on the yellow fruit shall be known only as Ba.
But something that has helped me to deal with my mystification is to remember that Maths as we know it is not the be all and end all and only conceivable possible way. It could (theoretically anyway) have all happened completely differently.
How? I hear you cry.
Well, Maths is a language, just like other languages (despite having a few extra features and lower comprehensibility ratings than most). The fact that "banana" represents banana does not say something innate about the relationship between word and object (this is Wittgenstein and those guys, but watered down). It just happens to be the word that history and society and chance and whatever other factors have attached to our familiar yellow fruit.
Same with Maths. All of our Mathematical system is based on certain axioms (like the fact that a point has no dimensions). These axioms (stated most famously by Euclid) are assumed to be true in the proofs and ways of thinking about things which are so entrenched in us. It is very difficult to imagine them not being true. Yet in actual fact this is only the case because its convenient that it should be, or because we're used to it being so.
Plus, of course, in order to do anything interesting or useful we have to have a common frame of reference. So by all means lets argue about something more interesting than whether we should write 2x2 as 2^2 or 2_2; or whether 3ab really does mean 3 x a x b!
The other alternative is very much like saying that banana is a very bad word and difficult to pronounce, so from now on the yellow fruit shall be known only as Ba.
Thursday, February 10, 2011
Prime
Prime numbers are far more interesting than I ever realised. When I was first introduced to them I really didn't see what all the fuss was about; and its taken many years to gradually get a very rudimentary idea of some of their awesomeness...
So you probably know that a prime number is one which only has two factors: 1 and itself. In other words, it can't be broken down into smaller numbers. An example of a prime number is 23.
When you compare 23 to 24, a composite (non-prime) number, you get two totally different pictures. There are lots of ways of breaking 24 down into smaller pieces: 8x3, 6x4, 12x2 are some examples. If you break it down into the very smallest possible pieces (in fact, into prime pieces or factors) you end up with 2x2x2x3.
By contrast, 23 can't be broken down into smaller factors at all!
Of course 23 is quite a boring example. But when you find an extremely large prime number, like 243,112,609 − 1 (which has 12,978,189 digits) well then the fact that it has absolutely no factors other than 1 and itself is really quite impressive.
According to Charlie Epps, this has amazing applications in computer coding. According to Scarlett Thomas (in PopCo, which you HAVE to read immediately if you haven't already) it has exciting applications in codes as in secret codes.
From my extremely lay-person's point of view, it seems the key to this is that when you multiply two very large prime numbers together to make an even larger composite number, it is very difficult to work backwards and work out what those original two prime numbers were. Hence some serious encryption...
However, if like me you don't really care too much about the details of applications, it's still incredible to think of such an enormous number, and then to try get your mind around the fact that it is not made up of any smaller pieces.
12,978,189 digits?
Wow.
So you probably know that a prime number is one which only has two factors: 1 and itself. In other words, it can't be broken down into smaller numbers. An example of a prime number is 23.
When you compare 23 to 24, a composite (non-prime) number, you get two totally different pictures. There are lots of ways of breaking 24 down into smaller pieces: 8x3, 6x4, 12x2 are some examples. If you break it down into the very smallest possible pieces (in fact, into prime pieces or factors) you end up with 2x2x2x3.
By contrast, 23 can't be broken down into smaller factors at all!
Of course 23 is quite a boring example. But when you find an extremely large prime number, like 243,112,609 − 1 (which has 12,978,189 digits) well then the fact that it has absolutely no factors other than 1 and itself is really quite impressive.
According to Charlie Epps, this has amazing applications in computer coding. According to Scarlett Thomas (in PopCo, which you HAVE to read immediately if you haven't already) it has exciting applications in codes as in secret codes.
From my extremely lay-person's point of view, it seems the key to this is that when you multiply two very large prime numbers together to make an even larger composite number, it is very difficult to work backwards and work out what those original two prime numbers were. Hence some serious encryption...
However, if like me you don't really care too much about the details of applications, it's still incredible to think of such an enormous number, and then to try get your mind around the fact that it is not made up of any smaller pieces.
12,978,189 digits?
Wow.
Wednesday, February 9, 2011
Proof and Imagination
A few years ago I saw a movie called Proof. It's about an elderly Maths professor and his daughter. He spends his whole life trying to prove a single theorem, and eventually gets Alzheimers or something like that. His daughter sacrifices her academic career to care for him... Anyway, I won't give the plot away, but the idea of the film centres around the great value of proof.
Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.
Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.
It's imagination that has a shape.
Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.
Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.
It's imagination that has a shape.
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