All Numbers on the Continuous Number Line With No Gaps
Gaps between numbers on the real number line
Solution 1
Well, the point being talked about is called denseness/density and it applies to rational numbers also. The case of rational numbers is simpler and ideally should be in the mind of a seventh grader. It is rather unfortunate that most textbooks don't emphasize this concept of density at the right time and later on students have to struggle while studying calculus.
We start with the following:
Theorem 1: Between any two (distinct) rational numbers lies another rational number.
This is an immediate consequence of the following:
Theorem 2: Given any positive rational number there is another smaller positive rational number.
This is easy to understand and prove as well. If $m/n$ is a positive rational number then $m/(n+1)$ is a positive and smaller rational number. So the whole thing is ultimately dependent on the existence of a larger positive integer $n+1$ given a positive integer $n$. Moreover this also shows that there is no least positive rational number.
Next we can use theorem 2 to prove theorem 1. If $a, b$ are two rationals with $a<b$ then the number $d=b-a>0$ and we just need to find a smaller positive rational number $d'$ (which exists via theorem 2) and we can take $c=a+d'$ as our rational number lying between $a, b$.
Both the theorems above can also be proved using midpoint technique. Thus if $d$ is a positive rational number $d/2$ is a smaller one. And clearly $(a+b) /2$ is a number which lies between $a, b$. It is important to note both the approaches towards these theorems. Note also that the result in theorem 1 can be repeated to get as many rational numbers as we please between any two given rationals.
Now the idea of a neighbor (successor or predecessor for the case of integers) breaks down for rational numbers. To put it more precisely given any rational number $r$, there is no least rational number which exceeds $r$ and there is no greatest rational number which is exceeded by $r$.
This is expressed informally by saying that given any rational number $r$ we can find a rational number as close (near) to $r$ and less (or greater) than $r$ as we please.
The definitions of key concepts (limits) in calculus use the ideas mentioned above and are crucially dependent on the fact that there is no least positive number and there is no largest positive integer. However the fact that there is no next neighbor for a given rational number is not a drawback. It's a feature which is used everywhere in the definitions in calculus. A consequence of these facts is that we have an infinite supply of smaller and smaller positive numbers and greater and greater positive integers. And calculus in general deals with and therefore needs such infinite things.
Another key and slightly difficult ingredient in calculus is what we call completeness and that deals with the fact that although rationals are dense, they do lack something and this inadequacy is fulfilled by creating real numbers. The idea of completeness is not given sufficient focus in most textbooks of calculus (like in this answer too, but that's only to keep the length of the answer in control and can be discussed in another answer if you wish) but one can summarize the picture as follows:
The idea of denseness is essential to formulate the concepts of calculus and the idea of completeness is necessary so that these concepts don't operate in a vacuum (ie they do have some non-trivial consequences).
Solution 2
Essentially, all this boils down to the special property of the real numbers that every possible gap is filled, so to say. This is what makes calculus possible, for it is this property of real numbers that assures us that we can be confident that the set of real numbers is compact.
The rational numbers do not have this property, despite being dense. That is, consider that between any two distinct integers, there are always a finite number of integers; however, between any two distinct rational numbers you can always find a rational number, so that between two different rational numbers there are infinitely many rationals.
But interesting as this is, which might lead one to expect that the rationals exhaust every possible gap, we find that some gaps still exist, and that is what the irrational numbers fill. Thus, the very definition of reals as being limits of sequences of rational numbers makes the set complete. This is why you can always get arbitrarily close to an irrational number without ever landing on it, so to say.
So, if there are no numbers between two integers, they are not necessarily distinct. But this is already satisfied by the rationals. The irrationals complete this by adding those points where no rational could possibly be.
Solution 3
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If we can find a neighboring number, then the basic rules of arithmetic have collapsed and been disproven. If this happened, the universe would be collapsing anyway. If $a,b$ are consecutive real numbers, with $a<b$, then set $c=\frac{a+b}{2}$. We have $$a=\frac{a}{2}+\frac{a}{2}<\frac{a}{2}+\frac{b}{2} = c < \frac{b}{2}+\frac{b}{2}=b$$ and we have disproven that $a,b$ are consecutive (since $c$ is between them, and is also a real number.
- Calculus is all about not single numbers, but sequences of numbers getting closer and closer to another number. In the interval $(3,5)$, we take a sequence $4,4.5, 4.9, 4.99, \ldots$ that are getting closer to the top of the interval but never reaching it.
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Comments
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I was reading a book called Calculus Basic Concepts for High Schools, and, under the topic limit, it was discussed that one cannot have two limits for a given sequence, provided the sequence has a limit. And the immediate implication of this is that there cannot be a neighboring number because one can always find a number between any selected numbers; however, close they may be. So in an open interval $(a,b)$, one cannot find the largest number.
I want someone to elaborate on this piece of text from "Calculus Basic Concepts for High Schools: L. V. Tarasov":
However, if there were a point neighboring 1, after the removal of the latter this "neighbor" would have become the largest number. I would like to note here that many "delicate" points and many "secrets" in the calculus theorems are ultimately associated with the impossibility of identifying two neighboring points on the real line, or of specifying the greatest or least number on an open interval of the real line.
- What would happen if we can find a neighboring number?
- How is calculus associated with the impossibility of identifying two neighboring points on the real line?
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I think the rationals do have that property. The problem with rationals is something else which is not related to denseness at all
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Your last paragraph is also wrong. If the difference between two numbers is zero then they are same no matter whether they are integers, rationals, reals or for that matter even complex.
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@ParamanandSingh What are you talking about. Where did I say the 'problem' of the rationals is that they aren't dense?
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That's precisely opposite to, and thus misrepresents, my answer.
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As for the last paragraph. That was a serious mix up. Thanks for pointing that out.
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You have clearly mentioned rationals do not have this property, despite being dense. But the property you are talking is only denseness ie finding numbers arbitrarily close to a given number.
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Oh, I see your point now. That's the tradeoff for trying to be informal and conversational. I wanted the answer to explain the completeness of the real number set intuitively. I'll do something about that.
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Explanations can avoid difficult aspects but not at the cost of correctness. I will wait for your fix and then reverse my downvote. Moreover the key point which OP wants to discuss is density so discussing completeness is optional. I have given it just a cursory glance in my answer at the end and if OP wishes to discusses completeness it can be done in a different question.
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@ParamanandSingh Done something. Does that convey the idea with little cost?
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Well i will remove the downvote, but it still needs some cleaning. Mainly the idea that you can get arbitrarily close to a real number without landing on it is related to density and not completeness.
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I inserted the last phrase, 'without landing on it' to roughly convey the idea that in a Dedekind cut of the real line, the only real numbers that truly partition the real line are the irrationals. Is there a better way you would say this? I agree it may be easily confused with density, but as I said I wanted to be as intuitive as I could be.
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Well your description about dealing with $\epsilon/2$ seems a bit contrived. If we agree to deal with a positive number $\epsilon$ then we have to accept the existence of $\epsilon/2$ which is smaller and positive as well. We don't need any of that sequence / limit stuff to understand this. It is nothing but a result of how the arithmetic operations and order relations work with numbers.
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@ParamanandSingh: With the integers, we do accept that there are some numbers we cannot divide by $2$, so it's not a big stretch to imagine that when extending the rational numbers with new ones, we might be willing to accept similar restrictions to those new numbers, if it helps with other issues (just as we accept for complex numbers that we no longer can order them). Well, it turns out it doesn't help; quite the opposite.
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But we do know that we are constructing the new system of reals as an ordered field. And the failure of ordering in the field of complex numbers is quite well known and advertised in most introductory textbooks. I think the concept of denseness has to be treated trivially and emphasized early on in math curriculum. It is the completeness which is subtle and needs to be discussed before introducing calculus.
Recents
Source: https://9to5science.com/gaps-between-numbers-on-the-real-number-line
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