Mathematical Constants III

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):

 

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.

Draw and measure the diameter.

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!}

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 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 :-)

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.

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.

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):

Theorem = Claim

Definition = Name

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...

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.

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 2^43,112,609 − 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.

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.