How come a number represented by an "infinite" string of digits is not an integer?

Question

What makes 9999... not an integer?
Adding 1 to any integer always results in an integer. Why is it when we add 1 an indefinite number of times, it stops being an integer?

Some explanations given to me are:

"An integer must have a countable number of digits."
However, countability is defined by the natural numbers which are a subset of integers. So the act of including a value with an "uncountable" number of digits will automatically make that number part of the countable set (and therefore it's number of digits part of the countable set). It's circular.

"Infinity is a concept, so any value with infinite value is not a number."
I agree infinity is a concept, but it's a concept to represent what we cannot specifically quantify. Much like how a variable x is used to represent a value we do not know. If we concede the set of integers is infinite, how can we say a value constructed with the mathematical operations valid within the set is not among those infinite number of elements?

If the set of integers include values that are written as an "infinite" sequence of digits, then there would be no need for irrational numbers. What we currently call irrational numbers can be expressed as a rational number with numerator and denominator that have an "infinite" number of digits.

Answer 1

"So the act of including a value with an "uncountable" number of digits will automatically make that number part of the countable set (and therefore it's number of digits part of the countable set)."



No. Any process that "counts" can be proven to reach (and therefore count) any finite integer. Such a process would never reach your number. It lies outside the countable set and is not an integer.



If you want to redefine all the terms of mathematics so that "integer" means this, then you're going to need a term for finite integers and a term for what are now rationals.



There is not just "one true" mathematics. There are lots of them. But the ones we study are the ones that are most useful in a wide range of situations. Is there anything that would make your redefinition of integers more useful?

Answer 2

Your explanation is full of errors and misunderstandings. I suggest that you look at the definition of the real numbers. See, e.g., http://en.wikipedia.org/wiki/Real_numbers#Definition

That should clear up some of your misunderstanding.



"An integer must have a countable number of digits."

No, it must have a finite number of digits. You also misuse the terms "countable" and "uncountable." Or at least I don't know what you mean by them. A set is "countable" if it is finite or if the members can be put into a 1-1 correspondence with the integers. Being countably infinite is a property of SETS. "Infinity" is not a number, only a "cardinal number" meaning it can be used to indicate the size of a set. (And there are many infinite cardinal numbers.) A set, like the real numbers, is "uncountable" if it is not finite and cannot be put into a 1-1 correspondence with the natural numbers. I don't know what you mean when you use the term "uncountable."



"...infinity is a concept, but it's a concept to represent what we cannot specifically quantify. Much like how a variable x is used to represent a value we do not know."

No, those aren't alike at all. Infinity is a concept of something being larger than any given value. A variable x is just a placeholder for a solution to be found. There's no similarity in the concepts at all.



If we concede the set of integers is infinite, how can we say a value constructed with the mathematical operations valid within the set is not among those infinite number of elements?

The mathematical operations are between two elements at a time, not an infinite number of operations. You can't add 1 and 2 and 3 at the same time. You can only add 1 and 2, then add 3 to the result. And you certainly can't add an infinite number of numbers at once. You can take the LIMIT of an infinite number of additions, but that's just the limit of the the partial sums. And if the limit goes to infinity, we say the limit doesn't exist. You should also study limits. http://en.wikipedia.org/wiki/Limit_%28mathematics%29





"If the set of integers include values that are written as an "infinite" sequence of digits, then there would be no need for irrational numbers. What we currently call irrational numbers can be expressed as a rational number with numerator and denominator that have an "infinite" number of digits."

All that falls apart when you realize integers must have a finite number of digits.



"The value of pi is basically whatever value we decide to terminate on when approximating it. One can never reach pi by iterating a function."

No, if you terminate, you just have an approximation. π is an actual real number, and is the limit of the rational numbers represented by the sequence of an increasing but finite number of decimals.



Dividing by "nothing" is not equivalent to dividing by "an infinitely small amount."

This is correct, but the real reason that you can't divide by 0 starts with the fact that there really is no such thing as division. Division is really just multiplication by the reciprocal of a number, usually written 1/x. The reciprocal of x is the number you multiply x by to get 1. There is no number n such that 0n = 1. So without a reciprocal for 0, you can't "divide" by it.



You need to acquire some solid mathematical theory to replace simplistic verbal descriptions that aren't rigorously based.

Answer 3

Interesting discussion. We have a few issues to address.



How come a number represented by an "infinite" string of digits is not an integer?

-Actually, a number represented by an infinite string of digits can still be an integer. For eg, 0.999....=1 is an integer. (It may be hard to accept here, pls read on.)



Adding 1 to any integer always results in an integer. Why is it when we add 1 an indefinite number of times, it stops being an integer?

-This is due to the idea of "infinity". I agree that infinity is a concept. In addition, I feel that infinity is the behaviour of increasing without bound. So, why is 0.999... an integer but 999... not an integer. We can see the difference is that 0.999... has an upper bound (more correctly the supremum) as 1. On the other hand, 999.... is increasing without bound, so we can describe this behaviour as infinity and i think u understand why infinity is not a number. In other words, the process of "adding indefinite 1" is one of the ways to 'obtain' the behaviour of increasing without bound.



An integer must have a countable number of digits."

-Any number (rational or irrational) can be represented by a countable number of digits. The way we write it tell us the number of digits is countable. That's y we can say pi has first digit is 3 , sec digit is 1, third digit 4...



If we concede the set of integers is infinite, how can we say a value constructed with the mathematical operations valid within the set is not among those infinite number of elements?

-Like i say above, now your construction with math operations is repeated indefinitely, so this process of construction gives infinity as the 'final outcome'. But if we stop anywhere while adding 1s, we will have an integer, say 10354. Then, we continue adding 1s and stop again, we get another integer, say 34056. Because the moment we stop and ask what is the value, we stop the 'process of adding 1s', so the behaviour of increasing without bound disappears. Thus, its not infinity, its an integer. However, if we continue adding again, the behaviour comes back and thus we get infinity.



If the set of integers include values that are written as an "infinite" sequence of digits, then there would be no need for irrational numbers. What we currently call irrational numbers can be expressed as a rational number with numerator and denominator that have an "infinite" number of digits.

-Some confusion with the definitions over here. A number is a rational number if it can be represented as a fraction, with finite number of digits. An irrational number cannot be represented by a fraction with finite number of digits. There is no intersection between the two. So," a number with numerator and denominator that have an "infinite" number of digits." will either simplify to be a rational number or not simplify which becomes a irrational number. If u r interested to know the fundamentals of real numbers, u can search for Dedekind's Cut or Cantor construction of real numbers.



Hi again. 2 corrections first. 1.Pi does not depends on wat ever value we terminate on when approximate it. Pi is a fixed value regardless of wat estimates we use. 2. The quotient of some number over zero need not be infinity. It is undefined.. meaning no value will ever fit e expression of 3/0.

The diff btw a irrational and 99999... is that irrationals have upper bound but 999.. doesnt.. one of the main tools Dedekind used in the construction of real numbers is supremum which means the least upper bound.

I get ur idea of expressing an irrational with infinite numerator and denominator. But there r probs to it. For eg, how do we compare 2 irrationals? Which is bigger ? 8273../384783... or 53433../26435..? If we have to divide and convert to the form 5.34344 that we have been using then this concept cant stand on its own. I m not sayin this way of construction is not possible. But it needs to come along with more properties to replace the version we have been using.

I dun understand wat u mean by " counting to infinity process cannot be terminated to retain its definition"

Answer 4

As would understand it, integers are inductive. That is, for any integer n you can construct a finite integer n+1.

You cannot do that with a number whose modulus is greater than 1 that is composed of an infinite sequence of digits.

Assuming positive numbers (it holds for negative numbers to0), adding 1 to a number such as

999..............., is meaningless as the number has diverged as the number of terms go to infinity.



In contrast, a rational number x such that 0 < |x| < 1 converges to a limit as the terms of t=its sequence of digits go to infinity. Further, these terms repeat in periodic cycles or end in an infinite number of zeros in the case of the reciprocals of integers of the form 2^p*5^q (where p and q are integers)



Irrational numbers do not have this property.



Since is infinity is not a number in the sense that it obeys the rules of addition (among other elementary operations), a numbers greater than 1 composed of an infinite sequence of terms, being divergent, likewise cannot obey the same rules. Hence, they cannot be called integers.



I don't know if that answers your question, but that's how I see it.