A question on rationals and irrationals (numbers)

Original post by Maths Failure

I've wondered this : if a rational number always exists between any two irrationals and an irrational always exists between any two rationals how can it be that there are more irrationals than rationals on the real number line?

What makes you think there is always a rational number between any two irrationals (the implication of this theorem is that there isn't! Although this is rather abstract).

The concept of "more" can only really be defined in terms of "enumerability" (or "countability" if you like). The rational numbers can be enumerated (counted in an ordered way) using Cantor's diagonal argument which is basically a way to arrange all rational numbers such that they are in a grid with denominator changing along the columns and numerator along the rows. As such a grid clearly existed you can take a "zig-zag"-like path from the top left and extend to infinity in both direction (hence enumerating the set of all rational numbers). I suggest you look this up if you're interested.

For the set of all irrationals, no such enumeration is possible and so we say that it has a higher order of infinity than that of the rationals. i.e. therefore they are more "dense" on the real number line.

You may have made the initial assumption based on intuition which can often mislead you when it comes to advanced number theory problems such as this. This may be a tangent if you haven't done a lot of maths but the way I understand this theorem is...
Letting p be a number such that

a \leq p \leq a +\displaystyle\lim_{x\to \infty} ( \frac{1}{x} )

, where a is an irrational number such as

\sqrt{2}

. Here p is between two irrational numbers. But can p be rational?

Good question btw. Hope this sheds some light :smile:

Reply 2

It is intuitive that there are more real numbers than rational numbers by virtue of how the reals are constructed out of the rational numbers (if you don't know such a construction you will need to to make sense of this question).

The irrational numbers are the complement of the rational numbers in the reals i.e. the irrational numbers are the set

\mathbb{R}-\mathbb{Q}

. So if we take a countable number of elements away from an uncountable set, we must be left with something uncountable.

Reply 3

Original post by Jkn

What makes you think there is always a rational number between any two irrationals (the implication of this theorem is that there isn't! Although this is rather abstract).

It is a standard fact that there exists a rational number between any two irrational numbers.

For example, see: this for proof (or try it as an exercise).

Reply 4

Original post by Mark85

It is intuitive that there are more real numbers than rational numbers by virtue of how the reals are constructed out of the rational numbers (if you don't know such a construction you will need to to make sense of this question).

This is by virtue of the rational numbers being a subset of the real numbers. There exists no such construction as this would imply the real numbers are enumerable.

The irrational numbers are the complement of the rational numbers in the reals i.e. the irrational numbers are the set

\mathbb{R}-\mathbb{Q}

. So if we take a countable number of elements away from an uncountable set, we must be left with something uncountable.

Hence proving that the uncountability of the reals is a necessary and sufficient condition for the uncountability of the reals.

Original post by Mark85

It is a standard fact that there exists a rational number between any two irrational numbers.

For example, see: this for proof (or try it as an exercise).

This is proof that rational numbers exist between two reals..... ? If you think you can prove that a rational exists been every two irrationals then be my guest!

Reply 5

Original post by Jkn

This is by virtue of the rational numbers being a subset of the real numbers. There exists no such construction as this would imply the real numbers are enumerable.

Yes there is such a construction and no it doesn't imply that. See here for example.

Original post by Jkn

Hence proving that the uncountability of the reals is a necessary and sufficient condition for the uncountability of the reals.

You just said that the uncountability of the reals is equivalent to the uncountability of the reals. This is a tautology. What is your point? What did you mean to say?

Original post by Jkn

This is proof that rational numbers exist between two reals..... ? If you think you can prove that a rational exists been every two irrationals then be my guest!

Irrational numbers are in particular real numbers. In the proof I already posted, just specify that x and y are irrational.

Now stop posting and learn about the maths you want to talk about before you talk about it since you are embarrasing yourself and probably confusing the OP (and yourself no doubt).

Reply 6

Original post by Mark85

Yes there is such a construction and no it doesn't imply that. See here for example.

You just said that the uncountability of the reals is equivalent to the uncountability of the reals. This is a tautology. What is your point? What did you mean to say?

Irrational numbers are in particular real numbers. In the proof I already posted, just specify that x and y are irrational.

Now stop posting and learn about the maths you want to talk about before you talk about it since you are embarrasing yourself and probably confusing the OP (and yourself no doubt).

Sorry, I conflated construction with enumerability.

What I said doesn't imply the sets are equivalent. It is just a consequence of the fact that all non-irrational numbers that are a subset of the reals are enumerable.

Oh right, my mistake (that's what I get for going on tsr past midnight!) Is this not a paradox then?

Theres no need to be so aggressive. I saw this thread had had no replies and I thought that, even though I haven't learnt this properly, I had read about it in a book and thought the insight I could offer was better than nothing. Surely that's better than leaving a post unanswered

Reply 7

Original post by Jkn

Is this not a paradox then?

It depends what you mean. In the strict logical sense, no, definitely not.

If you mean in the sense that it seems counter intuitive, well, that depends on how well you understand the construction of the real numbers. For example when people find the fact that 0.99999....=1 counter-intuitive, it is generally because they don't understand the construction of the reals. Once you know that, it makes perfect intuitive sense and it is the same with this thing.

Theres no need to be so aggressive. I saw this thread had had no replies and I thought that, even though I haven't learnt this properly, I had read about it in a book and thought the insight I could offer was better than nothing. Surely that's better than leaving a post unanswered

I wasn't being aggressive; just emphasising that I had already pointed out the contradiction in what you had said.

Reply 8

Lord of the Flies

19

Original post by Jkn

This is proof that rational numbers exist between two reals..... ? If you think you can prove that a rational exists been every two irrationals then be my guest!

This is not a formal proof, but intuitively, any irrational

r

can be written as the limit of a Cauchy seq.

(q_n)

of rationals (think continued fractions). By picking any other irrational

r'

it is clear that since

|r-r'|\neq 0

there exists

m

such that

n\geq m\Rightarrow |r-q_{n}|<|r'-r|

. Hence there are infinitely many rationals between

r

and

r'

.

(edited 11 years ago)

Reply 9

around

13

Original post by Jkn

Sorry, I conflated construction with enumerability.

What I said doesn't imply the sets are equivalent. It is just a consequence of the fact that all non-irrational numbers that are a subset of the reals are enumerable.

Oh right, my mistake (that's what I get for going on tsr past midnight!) Is this not a paradox then?

Theres no need to be so aggressive. I saw this thread had had no replies and I thought that, even though I haven't learnt this properly, I had read about it in a book and thought the insight I could offer was better than nothing. Surely that's better than leaving a post unanswered

you sound like you know a lot, but actually you don't. just a tip.

Reply 10

Maths Failure

OP

I'm still very confused.

If what you're saying is true then I still don't understand how it's possible that there exists a rational number between any two irrational numbers and there exists an irrational number between any two rational numbers and that there are more irrational numbers than rational numbers in

[a,b]

where

a\ne b

. Surely there are just as many rational numbers than irrational numbers?

When picturing this visually I can only see a line where the numbers are (in ascending order) rational number, followed by irrational number, followed by rational number, followed by irrational number, etc (or vice-versa). How can it be that there are (infinitely) more irrational numbers than rational numbers?

Reply 11

Original post by Maths Failure

I'm still very confused.

If what you're saying is true then I still don't understand how it's possible that there exists a rational number between any two irrational numbers and there exists an irrational number between any two rational numbers and that there are more irrational numbers than rational numbers in

[a,b]

where

a\ne b

. Surely there are just as many rational numbers than irrational numbers?

When picturing this visually I can only see a line where the numbers are (in ascending order) rational number, followed by irrational number, followed by rational number, followed by irrational number, etc (or vice-versa). How can it be that there are (infinitely) more irrational numbers than rational numbers?

Your intuition is just wrong. You can't think of looking at the real line like a discrete set where you are listing all elements between any two. The fact is that every time you 'zoom in' to the real line, more detail appears that wasn't visible at the previous focus.

The only way you can start to make sense of it all is to actually learn how the real numbers are constructed and what cardinality is, especially in the non-finite sense. I think until you do though that - it is impossible to get a good view of what you are dealing with.

Even if you were just talking about discrete sets, your way of thinking is faulty. For example, in between every two integers, there are an infinite amount of rational numbers but there are exactly the same amount of rational numbers as there are integers.

(edited 11 years ago)

Reply 12

Maths Failure

OP

Original post by Mark85

Even if you were just talking about discrete sets, your way of thinking is faulty. For example, in between every two integers, there are an infinite amount of rational numbers but there are exactly the same amount of rational numbers as there are irrational numbers.

Really? So in the interval [0,1] there are just as many irrational numbers as rational numbers?

Reply 13

Original post by Maths Failure

Really? So in the interval [0,1] there are just as many irrational numbers as rational numbers?

Sorry, typo which I have just corrected. I meant to say that there are just as many rational numbers as integers.

There are countably many rational numbers and uncountably many irrationals.

(edited 11 years ago)

Reply 14

IrrationalNumber

12

Original post by Jkn

I saw this thread had had no replies and I thought that, even though I haven't learnt this properly, I had read about it in a book and thought the insight I could offer was better than nothing.

Actually, I think that this is one of the problems with the maths forum. People who know nothing at all about a subject come along and post something without stating that they know nothing about it. Then they end up confusing the OP.

Reply 15

Swayum

18

I don't understand why people are trying to prove statements like the irrationals are uncountable/rationals are countable/there exists a rational between two numbers/etc. This is not his question. He already knows all these facts. He's asking about the LINK between these facts, i.e. how is it that they don't contradict each other.

As far as I know (which isn't very far), there is no satisfying answer here, you just have to accept that infinity is weird.

Reply 16

Maths Failure

OP

That's the million dollar question: how is it that they don't contradict each other?

Reply 17

Original post by Maths Failure

That's the million dollar question: how is it that they don't contradict each other?

The problem is: why is it that you think that they do contradict each other?

I would say that this is because you don't know what real numbers are or understand what cardinality is and what uncountable means and think about it as you would a finite or discrete set and the simple answer then is that you just can't do things with the real numbers that you can do with discrete sets.

In particular, the way you visualise listing all elements between any two - it just doesn't work like that. That is the whole point of what it means for there to be an uncountable amount of elements.

Reply 18

Original post by IrrationalNumber

Actually, I think that this is one of the problems with the maths forum. People who know nothing at all about a subject come along and post something without stating that they know nothing about it. Then they end up confusing the OP.

Original post by around

you sound like you know a lot, but actually you don't. just a tip.

Sorry guys. I'd read about the diagonal argument which I thought would be helpful. I then succeeded in bungling in loads of other things which I thought I understood (though have now clearly been put in my place!)

I was just merely spending 20 or so minutes going through "answered maths threads" trying to help people who weren't getting any. The real problem is that a lot of the posts were old and so people don't end up replying and so I drifted into (sort of) writing things as a means of pondering them myself. This is a topic I'm interested in and so I thought I'd offer what I thought was correct. I was going to write for the last bit something like "though I'm sure someone who knows more will correct me" but simply forgot. The maths I wrote down wasn't intended to be rigorous, just a 'way of thinking about things' (which I now know is illogical).

I suppose the moral of the story is not to do post on tsr when you're tired (in my case I came across as arrogant or condescending though I was not trying to be!) Hope I haven't offended anyone by my post, I assure you that I am not ignorant to the true complexity of such problems!

Reply 19

davros