There exists one number N

1. Mar 27, 2008

arbol

The set N of natural numbers = {1, 2, 3, 4, ...}.

But there exists one (1) number N, such that

N = 12345678910111213... (where the unit's place is at infinity).

A good example of an irrational number then would be

1.234567891011121314...

Last edited: Mar 27, 2008
2. Mar 27, 2008

belliott4488

Doesn't that make N = infinity?

3. Mar 27, 2008

mathwonk

so, put a decimal in front of it.

oh he did that.

4. Mar 27, 2008

HallsofIvy

Staff Emeritus
No, there is no such number. All integers have only a finite number of digits. By the way, it is not at all a good idea by using "N" to represent the set of natural numbers and then say that "N" is a number.

Now THAT is a perfectly good irrataional number.

5. Mar 27, 2008

Hurkyl

Staff Emeritus
Are you sure that decimal string actually denotes a number? How can the unit's place be 'at infinity'? What digit is in the unit's place?

6. Mar 27, 2008

CRGreathouse

That's 10 times Champernowne constant.

7. Mar 28, 2008

HallsofIvy

Staff Emeritus
If it were possible to construct any irrational number by putting a decimal into some positive integer, that would imply that the set of irrational numbers is countable.

8. Mar 28, 2008

arbol

good question

9. Mar 28, 2008

arbol

It is necessary that N is not an interger, but it is one number.

10. Mar 28, 2008

arbol

you can call it anything you want

11. Mar 28, 2008

arbol

lim f(x) (as x approaches infinty) is infinity, but N is a single number (not a variable).

Last edited: Mar 28, 2008
12. Mar 28, 2008

arbol

a definition of an irrational number is a number that cannot be expressed in the form m/n, where m and n are intergers and n not equal to zero. such numbers are infinite to right of the decimal point and do not repeat. for example,

1.234567891011...

13. Mar 28, 2008

CRGreathouse

Yes, so together with Hallsofivy's statement you know that 123456789101112... is not an integer.

14. Mar 28, 2008

ramsey2879

All Numbers must have a meaning such that a rational number can be found to approximate the number within a chosen value, a expression that is an infinite string of numbers without any fixed decimal point does not have any meaning and is not a number.

Last edited: Mar 29, 2008
15. Mar 30, 2008

HallsofIvy

Staff Emeritus

Okay, what do you mean by "number". And my criticism was simply about using the same symbol, N, with two different meanings.

Thank you. But I do prefer to use standard terminology. If you did that, it might be easier to understand what you are trying to say.

??This is the first time you mentioned "f(x)". Where did that come from. Once again, the N you posit is NOT a "number" by any standard definition.

Yes, we know that- it is not necessary to state the obvious.

16. Mar 31, 2008

arbol

Yes. I think it does.

If f(x) = x, then

lim (of f(x) as x approaches infinity) = infinity = N. (but the unit's place of N is at infinity.)

Last edited: Mar 31, 2008
17. Mar 31, 2008

CRGreathouse

But since "infinity" is not an integer, you know that N isn't an integer.

18. Apr 2, 2008

LorenzoMath

1234567891011... is not a conventional way of representing real numbers, so unless you introduce your own convention, it doesn't mean anything. Whereas if you put a disimal point somewhere, it represents a real number in a conventional sense. Because, by convention, 1.234567... represents some real number to which the sequence, 1, 1.2, 1.23, 1.234, ... converges. This is what we call the completeness of R. If we agree to say that 1234567891011... represents where the sequence 1, 12, 123, 1234, ... go, then we may call it infinity, or more precisely, we introduce the concept of infinity.

19. Apr 2, 2008

CRGreathouse

Nitpicking: that's "a concept of infinity", not "the concept of infinity".

20. Apr 2, 2008

ramsey2879

No need to stay confused. Let go of your mindset which says it should be possible to put a decimal at a point of infinity of a string of numbers and have something meaningful. There can only be a finite string of numbers prior a decimal point to have anything resembling a number.