there exists one number N


by arbol
Tags: exists, number
arbol
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#1
Mar27-08, 02:04 PM
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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...
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belliott4488
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Mar27-08, 02:06 PM
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Quote Quote by arbol View Post
N = 12345678910111213... (where unit's place is at infinity).
Doesn't that make N = infinity?
mathwonk
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Mar27-08, 04:09 PM
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so, put a decimal in front of it.

oh he did that.

HallsofIvy
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Mar27-08, 05:13 PM
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there exists one number N


Quote Quote by arbol View Post
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).
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.

A good example of an irrational number then would be

1.234567891011121314...
Now THAT is a perfectly good irrataional number.
Hurkyl
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Mar27-08, 06:05 PM
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Quote Quote by arbol View Post
But there exists one (1) number N, such that

N = 12345678910111213... (where the unit's place is at infinity).
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
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Mar27-08, 07:36 PM
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Quote Quote by arbol View Post
A good example of an irrational number then would be

1.234567891011121314...
That's 10 times Champernowne constant.
HallsofIvy
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Mar28-08, 06:54 AM
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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
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Mar28-08, 01:23 PM
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Quote Quote by Hurkyl View Post
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?

good question
arbol
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#9
Mar28-08, 01:24 PM
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Quote Quote by HallsofIvy View Post
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.
It is necessary that N is not an interger, but it is one number.
arbol
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#10
Mar28-08, 01:25 PM
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Quote Quote by HallsofIvy View Post
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.
you can call it anything you want
arbol
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#11
Mar28-08, 01:27 PM
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Quote Quote by belliott4488 View Post
Doesn't that make N = infinity?
lim f(x) (as x approaches infinty) is infinity, but N is a single number (not a variable).
arbol
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#12
Mar28-08, 01:32 PM
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Quote Quote by HallsofIvy View Post
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.
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
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Mar28-08, 01:57 PM
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Quote Quote by arbol View Post
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.
Yes, so together with Hallsofivy's statement you know that 123456789101112... is not an integer.
ramsey2879
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Mar28-08, 06:01 PM
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Quote Quote by arbol View Post
It is necessary that N is not an interger, but it is one number.
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
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Mar30-08, 06:39 AM
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Quote Quote by arbol View Post
good question
Yes it was. Since it was about your post, do you have a good answer?

Quote Quote by arbol View Post
It is necessary that N is not an interger, but it is one number.
Okay, what do you mean by "number". And my criticism was simply about using the same symbol, N, with two different meanings.

Quote Quote by arbol View Post
you can call it anything you want
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.

Quote Quote by arbol View Post
lim f(x) (as x approaches infinty) is infinity, but N is a single number (not a variable).
??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.

Quote Quote by arbol View Post
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...
Yes, we know that- it is not necessary to state the obvious.
arbol
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#16
Mar31-08, 04:53 PM
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Quote Quote by belliott4488 View Post
Doesn't that make N = infinity?
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
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Mar31-08, 06:12 PM
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But since "infinity" is not an integer, you know that N isn't an integer.
LorenzoMath
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#18
Apr2-08, 09:39 AM
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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.


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