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...
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.
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?
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...
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.
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.
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.)
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.
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.