there exists one number N

by arbol
Tags: exists, number
 P: 50 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...
P: 657
 Quote by arbol N = 12345678910111213... (where unit's place is at infinity).
Doesn't that make N = infinity?
 HW Helper Sci Advisor P: 9,371 so, put a decimal in front of it. oh he did that.
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there exists one number N

 Quote by 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).
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.
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 Quote by arbol 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?
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 Quote by arbol A good example of an irrational number then would be 1.234567891011121314...
That's 10 times Champernowne constant.
 PF Patron Sci Advisor Thanks Emeritus P: 38,424 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.
P: 50
 Quote by 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?

good question
P: 50
 Quote by 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.
It is necessary that N is not an interger, but it is one number.
P: 50
 Quote by 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.
you can call it anything you want
P: 50
 Quote by belliott4488 Doesn't that make N = infinity?
lim f(x) (as x approaches infinty) is infinity, but N is a single number (not a variable).
P: 50
 Quote by 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.
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...
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 Quote by 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.
Yes, so together with Hallsofivy's statement you know that 123456789101112... is not an integer.
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 Quote by arbol 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.
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 Quote by arbol good question

 Quote by arbol 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 by arbol 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 by arbol 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 by 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...
Yes, we know that- it is not necessary to state the obvious.
P: 50
 Quote by belliott4488 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.)

 HW Helper Sci Advisor P: 3,682 But since "infinity" is not an integer, you know that N isn't an integer.
 P: 41 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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