Concept

[Latios]

I have a friend who mention to me today that he thinks that all irrational numbers are no more than concepts, since they don't seem to have any physical representation to describe themselves.

 

If this is true, then what place does 0 and infinity fall in?

 

It seem 0 is got to be a number since it's not irrational, but he said that there is no representation for 0 either.

 

What's your opinion on the subject?

[jordan]

It all sounds ok to me. I would agree that infinity and irrational numbers are concepts. There would be a few more to throw in there with them. Zero, though. I would say zero is not a concept and can be represented in the world. It'll be interesting to see if there is some discussion or if there is a clear answer on this one. Either way, I'll have to wait till morning to see. Oh well.

[NSX]

I have a friend who mention to me today that he thinks that all irrational numbers are no more than concepts' date=' since they don't seem to have any physical representation to describe themselves.

 

If this is true, then what place does 0 and infinity fall in?

 

It seem 0 is got to be a number since it's not irrational, but he said that there is no representation for 0 either.

 

What's your opinion on the subject?[/quote']

 

What about the hypotoneuse of a right angle triangle with others sides of length 1 & 1?

 

I'm sure there's probably some set/number theory to explain formally what 0 and infinity are.

[MandrakeRoot]

I would even say all numbers are concepts, since none of them "exist" in the real world. But they have each been constructed (as concepts) to help us do something we couldn't do without. The natural numbers help us to count things, the fractions to solve equations of the type nx = m, where n and m are natural numbers, which are equations naturally arising in ratio type of questions. The irrationals help us to number even more things in the real world, such as the example above. And then finally the complex numbers are also a very usefull tool in solving polynomials equations, but also in differential equations and many other fields. All numbers can be constructed mathematically from the hypothesis that there exists an infinite set.

So all numbers are concepts and a world without is perfectible imaginable.

 

In set theory it is possible to join an element to an ordered set and to extend the ordering such that all elements in the original set are smaller then this "new" element. This procedure applied to N would give a set with a maximal element, which is denoted often with infinity. Zero is the algebraic "unit" element for the addition, i.e., the unique element such that a + 0 = a for all a in some set with an addition defined on.

 

Mandrake,

[jordan]

I agree that we have well defined concepts of irrationals and infinity, but I believe Latios was trying to say that since they can't be modeled in the realy world, they remain simply concepts. We can show that there are three of something, but I think you would have a difficult time showing there are exactly pi or i or infinity of something in our world.

[NSX]

but I think you would have a difficult time showing there are exactly pi ... of something in our world.

 

That's true, but only b/c we are limited by our instruments.

pi has been approximated by many over the vast years that geometry was here.

[jordan]

I noticed you didn't include infinity (or i) in your quote. My problem is that you seem to agree that infinity is not a measurable quantiny, so an irrational number (one by definition a non-repeating infinite decimal) wouldn't be measurable. They have infinite digits, and unless you explain how infinity is not a concept, but a concrete, observable, measurable quantity, I will hold that irrational numbers are concepts and not physical representations.

[Latios]

I do not think the exact value of pi is still unknown is because we have limited instruments. I believe there's a prove somewhere that put pi at the realm of irrationa numbers, which should mean that it will never end in digits after the decimal.

 

We don't know the exact value of pi is because its exact value does not actually exist, and its corresponding representation should therefore be remain imaginary.

[MandrakeRoot]

I still would say all numbers are concepts, because you cant say something as "there are three apples" without having a concept of what "three" exactly is.

They way i see it there are two separate worlds, one wherein you create you numbers mathematically and so in fact you define some sets with some properties. And the other world is the one in which we live wherein we will apply these concepts in order to facilitate our lives.

You can easily apply N to the real worlds once you have a concept of more or less. If you have some collection of appels and you can say that some collection contains more then another. You can basically order them from nothing to a lot of appels. Then you call no (entire) appels "zero appels", the next best thing being the least amount of (entire) appels more then no appels, you call it one etc;...

Like so you can also introduce fractions.

 

Since pi would be the surface of a circle with unit radius i would say that the concept "Pi" can be found in the real world. Since basically pi is historically defined as being the surface of the circle with unit radius.

 

By the way an irrational number is just that and its decimal representation is a representation and not the number !

0.49999999999(and so on and so on) represents the same number as

0.50000000000, but they are different representations of the same thing.

 

The point i am trying to make is that numbers are all concepts, but they are here since we have found them a use in average day live.

So they are all concepts that can be applied in every day life.

 

Mandrake

[jordan]

I still haven't seen anyone adress i or, more importantly, infinity.

 

Mandrake, you said all numbers are concepts because they have use in daily life, but the above two are good examples of ones that don't. I don't that they can be taken as anything more than concepts.

[Dave]

infinity is not a number - it is a term, and as such doesn't really come into the argument.

 

i has its applications, but it essenially just a definition like all other numbers. When we started doing the counting numbers, we defined one to be a single element, then went on to define 2, 3, 4, etc. These numbers are effectively derived from what we've experienced in the world. The same thing applies to the set of positive rationals - looking at half of a cake, etc.

 

Negative numbers (and indeed any other type of number) are just an extension of this. We've come to learn certain things about the way these numbers work, and so we've derived them ourselves through steps of logic and/or axiomatic methods.

 

I suppose what I'm trying to say here is that numbers don't necessarily have to have an interpretation in the real world; and at the end of the day, it's down to personal belief as to whether you think they are or not (like most things of this nature).

[jordan]

Should we refine our definition to abstract concepts instead of just concepts or call the thread dead?

 

I mean, everyone has said all numbers are concepts, but should we discuss the intent of the thread and ask if certain numbers do not fit in "normal concepts" and instead be labeled "abstract concepts" or am I misconstruing what's being said?

[MandrakeRoot]

Yeah i think i pretty much agree with dave, that it just depends on what you call a concept.

 

Since the complex right can be interpreted as the real plane i am sure there is some interpretation possible. For instance you can see multiplication with "i" as rotation over 90 degrees counterclockwise or something like so.

 

Mandrake

[MandrakeRoot]

Each set of numbers serves pretty much a certain goal. Naturals would allow you to count. Fractions to solve stuff like ax = b, when a and b are natural numbers. Irrationals are limits of fractions and complex numbers allow you to express roots of polynomials. So each expansion would sort out of the before ordinary, but it depends a little what you call ordinary i guess ?

 

Mandrake,

[jordan]

Alright, simple question:

 

Can we group irrationals, pi, i and infinity in the same group as natural numbers by the concept they represent?

[Dave]

If you define concept to be 'solving mathematical problems', then yes. If you define it to be 'derived from real-world observations', then I guess not.

 

As I said, it's all down to definition.

[jordan]

If you define concept to be 'solving mathematical problems', then yes.

 

Agreed

 

 

If you define it to be 'derived from real-world observations'' date=' then I guess not.

 

As I said, it's all down to definition.[/quote']

 

However, I thought this was the definition that the thread began on. That's why I've been fighting for this side.

[bloodhound]

I (personally) dont really see the point to need to visualise numbers. altough I admit its sometimes helpful..

 

I agree with Mandrake Root , that all numbers are concepts. Aliens on mars may do mathematics differently , but the concept will be the same

 

This post isnt very helpful is it

 

try drawing the graph of function f:R -> {0,1} defined by f(x)=1 if x is rational and f(x)=0 if x is irrational. Have some fun.

[fourier jr]

Alright' date=' simple question:

 

Can we group irrationals, pi, i and infinity in the same group as natural numbers by the concept they represent?[/quote']

 

Well those are all the same general concept I guess, but there are different kinds of numbers, so I would say no. Irrational numbers have a different definition from natural numbers.

 

Infinity is just the concept of something that's bigger than anything else. No matter how big a number you pick, infinity is bigger.