Math Is Fun Forum

asebon

Guest

What is numbers?

I don’t know if it is right to ask this here but I’ll give it a try. Is there an answer to the question what is numbers? I have only empiric knowledge about them but I wonder if there is any definition or some logical ideas that can be used to define numbers.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

There are several ways to define numbers.  Most of them are very advanced.  The subject of defining numbers is included in what is called Abstract Algebra.  Here is a basic one.

Start off with 0 and 1.  We define them as such:

a + 0 = 0 + a = a, for any number a.
a * 1 = 1 * a = a, for any number a.

Now if we define "positive" very carefully, we can prove that 1 is positive and that if there is a least positive number, it must be 1.

Since we have addition, we can have 1 + 1, which is how we define 2.  And we define 3 by 1+1+1.  And so on.  So now we have all the positive integers.

Now if we have a number (positive integer) a, we can define another number b, such that:

a + b = 0.

This is the definition of a negative number, and we call b, "negative a".  So far, we have positive and negative integers, and 0.  And we can also say that if a + x = b, for any a and b, then we can find a solution for x and that it must be an integer.

But don't forget about multiplication.  ax = b does not have always have a solution with integers.  For example, 5x = 12.  So we need more numbers.  We introduce the term rational number:

A rational number is one that can be represented by a/b, where a and b are integers and b is not 0.

So if we have an equation ax = b, we know it must have a solution because x = b/a, and we know that b/a must be rational.  Note, that if a = 0 in the equation ax = b, then x can equal any number.

I'll stop there for now, there are also reals and complex numbers, but let me know if you understand this so far.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

Yes I understand. But I understand 1+1+1 empirically if want to define it I'll say that is 3 times 1 which is defined. My point is that 3 must be defined first. What is more you use "a" to define 1 and 1 to define numbers so you the concept of number to define numbers. and what is "+". Sorry for this tiring questions but please continue with complex and reels. If you can answer my questions ok but if no please contniue anyway

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

What you need to do is step away from the conception that numbers are things.  They aren't.  All they are, are names for things.  Why do we call gravity, "gravity"?  Would it change properties if we called it fuzzlewumps?  Absolutely not.  Numbers are just a name.

All 3 is, is a name for "1 + 1 + 1".  And then, as it turn out, by the way we define 1 and multiplication (as I said above), makes it so that 3 * 1 = 3

Does that make sense?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

All_Is_Number

Member

Registered: 2006-07-10

Posts: 258

Re: What is numbers?

Numbers are the alphabet of God.

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You can shear a sheep many times but skin him only once.

asebon

Guest

Re: What is numbers?

Yes. That was very helpfull.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

Ok, so now we get to the real and complex numbers.

We find that the rationals aren't quite good enough.  We see that there are many equations in nature which have exponents.  For example, the area of a square is s², where s is the lenght of a side.  But what happens when we have a sqare with area 2?  Then:

2 = s²
s = √2

As it turns out, we can prove that √2 cannot be rational.  But there are so many equations like this in nature, that we must include this as a solution.  Otherwise, we would have no solutions to a real problem.  So, we need a real solution.

In comes the real numbers.  The real numbers include all of the "radicals" which are just roots, such as the square root of 5 or the cube root of 9 or the 52.5th root of 2928.  But like I said before, the real numbers are supposed to be able to solve real equations.  Since pi and e come into everyday equations that we use, we must include them as well.

This is all fine an dandy, till you get to the 5th power.  Some equations such as:

Are not solvable by radicals.  We can approximate the solution, but we know of no way to write down an exact answer.  But these can be real equations as well, and so we must include their solutions in our real numbers.  So we have to include numbers we don't even know how to write!

So as you can see, the reals are pretty messy.  And this is just the tip of the iceburg.

But let's leave that mess behind, and assume it all works out properly (which you most certainly don't have to believe).  In come the complex numbers.  One thing the real numbers can't solve is:

And believe it or not, we can have equations that take on this nature.  So we introduce i, the imaginary number.  Now we can say that for every polynomial with complex coefficents, we can find at least 1 complex solution.  So we can have a solution to every equation.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

I believe it, I know their use in cosmology (in Hawking's theory).

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

Really?  I don't.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

It’s about the quantum theory of stories (I’m not sure if this is the English word). Every particle goes through every possible line and not only one. For every story there are two numbers one for the size of the wave and one for it's phase (particles are described as wave functions). The possibility for a particle to pass through a specific point is equal to the sum of all the waves of the stories, which go through the specific point. But there are many technical problems in the calculation of this sum. The problems disappear with the idea of imaginary time. That means that in calculations we count time with imaginary numbers. Further more imaginary time has as a result this: The difference between time and space has completely disappeared, they’re both dimensions of the same nature.  Sorry but I don't have more technical Knowledge about the subject. Tell me was that ok? Have you any questions?

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: What is numbers?

Ah, do you mean quantum theory of histories? As in the Feynman path integral?

Are you German? (Geschichte = story or history)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

Ahh, I think we have a miscommunication.  I thought you were talking about:

But let's leave that mess behind, and assume it all works out properly (which you most certainly don't have to believe).

And it appears now that you were talking about:

And believe it or not, we can have equations that take on this nature.

Anyone who has studied complex numbers and their uses in detail will tell you that they are in fact real.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

Yes that's what I mean. I'm Greek.

Ricky, what do you mean when you say they are reals?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

Real, as in they have a place/usefullness in our universe.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

Is this true?

0 = { }
1 = { {} }
2 = { {} , {{}} }

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

Surreals!

The question "is this true" has no answer.

Think of math as being a whole bunch of different universes.  Each universe can have different laws which you must follow.  What is true in one universe doesn't have to be true in another.  And you can create universes on a whim with any property you please.  For example:

In my universe X, 1 + 1 = 3.

And now it exists.  Pretty cool, huh?

The same applies to the surreal numbers (what you posted above).  In the surreal universe:

0 = {}
1 = { {} | } = {0 | }
...

And when you are inside the surreal universe, it is true.  But it doesn't hold outside the surreal numbers.  If we choose a different way to define numbers, we are in a different universe and so the same "laws" may not apply.  It turns out that the surreals are so cleverly defined that most, if not all, of the laws are the same.

Here is a great intro to surreals, if you're interested: http://www.tondering.dk/claus/surreal.html

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

Thanks. That's cool. It is Pretty abstract and I like it. Thanks again for your time and your answers:)

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: What is numbers?

Hahaha! The picture of Dali, appropriately placed on a page for surreal numbers, just cracks me up. Good document here.

asebon

Guest

Re: What is numbers?

[img]It's so beautiful, pure logic. I love it. I see that you know good sites. Can you give me a link for Mathematical logic? School mathematics is ok but not good enough. But this is very interesting. Thanks

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

I was thinking of writing something on mathimatical logic, notation, and methods of proving.  But I haven't gotten to it yet.  However, let me see if I can dig up some intro to proofs notes that I think I have.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is numbers?

Ok, here it is.  It is about 1mb:

http://www.sharebigfile.com/file/20551/IntroToProofs.zip.html

Note: it will only stay up there for about a week.  If you read this post and the link is dead, just reply to it and I will put it back up.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

asebon

Guest

Re: What is numbers?

ok I got it. Thanks