there exists one number N

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...

belliott4488

Doesn't that make N = infinity?

mathwonk

so, put a decimal in front of it.

oh he did that.

HallsofIvy

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.

Hurkyl

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?

CRGreathouse

That's 10 times Champernowne constant.

HallsofIvy

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.

arbol

good question

arbol

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

arbol

you can call it anything you want

arbol

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

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...

CRGreathouse

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

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.

HallsofIvy

Yes it was. Since it was about your post, do you have a good answer?

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.

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.)

CRGreathouse

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