Why are there more irrational numbers than rational numbers?

[dathca]

can someone explain to me why there are always more irrational than rational numbers?

 

[pnaj]

Somewhat counter-intuitively, the rationals have a one-to-one relationship with the natural numbers (that is, you can list the rationals as you can the naturals)

It should be clear that there are more irrationals than naturals!

 

[matt grime]

by "more" we should clarify that you mean comparison of cardinalities, and is defined in terms of bijections (or maps in general) between sets, otherwise this is crank bait.

and since the reals are the (disjoint) union of the rationals and irrationals, if the irrationals were countable, as the rationals are, then the reals would be countable, when they aren't.

 

[pnaj]

Sorry Matt, I don't quite get what you're saying there.

There is a reasonably straightforward way of listing the rationals, in the same way as the naturals.

There are clearly more irrational numbers than natural numbers, in any sense of the word 'more'.

What's the crank bait?

 

[NateTG]

Sorry Matt, I don't quite get what you're saying there.

There is a reasonably straightforward way of listing the rationals, in the same way as the naturals.

There are clearly more irrational numbers than natural numbers, in any sense of the word 'more'.

What's the crank bait?

There are infinitely many of either. In some very real senses, that may mean that there is the same amount of both.

 

[pnaj]

Jeez, this forum amazes me sometimes.

People seem to be more interested in showing off or picking up minor technicalities, rather than answering the questions in a way that might forward the poster's understanding.

It seems to me that someone who is asking such a question might not know what 'cardinality' is, or even the technical term for a 1-1 relationship.

So, in reasonably simple terms ...

Take the closed interval [1, 2]. There are, er, 2 natural numbers in this interval. There are infinitely many irrational numbers in this interval.

The same holds for any interval [n, n+1].

The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.

In some very real senses, that may mean that there is the same amount of both.

In which real sense is that?

 

[fourier jr]

The real #s are uncountable but the rationals are countable. So the irrationals are uncountable, and there are "more" irrationals. it's not that hard

 

[NateTG]

Jeez, this forum amazes me sometimes.
People seem to be more interested in showing off or picking up minor technicalities, rather than answering the questions in a way that might forward the poster's understanding.

Matt Grime's point is entirely valid. In some situations it's very important to realize that what you cheerfully refer to as 'definitely more' is really a rather technical notion.

It seems to me that someone who is asking such a question might not know what 'cardinality' is, or even the technical term for a 1-1 relationship.

Odd, that you're the one that brought up 1-1 relationships (somewhat inaccurately, no less) and then accuse Matt Grime of being a crackpot when what he said is completely correct.

So, in reasonably simple terms ...

Take the closed interval [1, 2]. There are, er, 2 natural numbers in this interval. There are infinitely many irrational numbers in this interval.


The same holds for any interval [n, n+1].

Correct, but also that's really irellevant since the inital post was about rational numbers, not natural numbers, and has nothing to do with the conclusion that you reach.

The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.

Unfortunately, none of us has actually answered dathca's inital question:

can someone explain to me why there are always more irrational than rational numbers?

This problem was investigated by a mathematician named Cantor (http://en.wikipedia.org/wiki/Georg_Cantor) and has had profound effects on mathematics.

When dealing with infinite sets, traditional notions of more don't work, because, we run into questions like "Is infinity plus one more than infinity?" that the usual notions of more don't really handle well. Similarly, you can't count an infinite number of things.

So, let's say that to show two sets are the same size if we can put the elements into pairs, one from each set, so that each element is in only one pair, and each element is in a pair. This type of relationship is called a bijection.

Now, it's possible to show that there is a pairing even if there are infinitely many pairs.

For example, there is a bijection between the non-negative numbers (0,1,2,3,4...) and the integers (0,-1,1,-2,2,-3,3...) since we have the two lists, we can simply pair them off in order:
0 and 0
1 and -1
2 and 1
3 and -2
4 and 2
and so on.

Using the famous diagonal argument (http://en.wikipedia.org/wiki/Cantor's_diagonal_argument) Cantor proved that there is no such pairing between the rational numbers and the real numbers.

 

[matt grime]

When, pnaj, did "clearly more" become a mathematical term?

"clearly there more integers than positive integers." Don't let lax standards make you less than accurate.

 

[Gza]

I don't mean to play devil's advocate (and no pnaj, I'm not equating you with the devil by any means), I think that sometimes you must sacrifice a little accuracy in order to answer a question for a person untrained in the field. Name-dropping a bunch of fancy sounding terms may make you sound more impressive, but it does little to further one's understanding.

 

[pnaj]

This is just getting silly, now.

NateTG,
Please don't accuse me of saying things that I haven't said.

You keep saying I'm being inaccurate ... where?

And at least read what I did write. I said earlier that there is a 1-1 relationship between the rationals and the naturals. Is that wrong?

 

[pnaj]

Matt,

I just saw your reply.

What I was trying to say was that the person who asks this question might have just as much trouble with terms like cardinality and bijection, etc. as he/she does with the terms rational and irrational.

So, I tried to use terms that reflected that and you didn't. Please, let's just agree to disagree about it.

But, I stll am wondering what you meant by 'crank bait'.

 

[quantumdude]

NateTG: Read the posts by pnaj carefully. He never accused Matt of being a crackpot.

Jeez, this forum amazes me sometimes.

People seem to be more interested in showing off or picking up minor technicalities, rather than answering the questions in a way that might forward the poster's understanding.

In mathematics, definitions and deduction are not minor technicalities, they are everything. When people lose sight of that and start going off on reasoning larks with ill-defined concepts, then the likelihood of veering off into abject crackpottery drastically increases. That's what "crank bait" is: fodder for said crackpots to advance pet theories that in actuality make no sense whatsoever. Not to long ago, we had a rash of "0.9999...=1" threads that proved this in spades. And there's an active thread in Theory Development (I think I need not say whose it is) for another perfect example.

What I was trying to say was that the person who asks this question might have just as much trouble with terms like cardinality and bijection, etc. as he/she does with the terms rational and irrational.

So, I tried to use terms that reflected that and you didn't. Please, let's just agree to disagree about it.

But the thing is, you didn't (in your first post) explain anything to the poster at all!

You made a comment about how the rationals can be placed in a 1-1 correspondence with the naturals. OK, fine, that is equivalent to saying that the rationals have the same cardinality as the naturals. But it doesn't say a thing about the cardinality of the irrationals. When you addressed that (which was what the original question was, by the way), all you had to say in your first post was this:

It should be clear that there are more irrationals than naturals!

How exactly is that supposed to advance the poster's understanding?

Is it not clear why others felt the need to interject with some measure of detail?

 

[pnaj]

Tom,

Fair comment ... I will certainly be less casual in the future.
And I didn't know the 'crank' stuff was so heavy, so I'll watch out for that as well.

Paul.

 

[matt grime]

Yes, I threw in technical names. That is what the answer requires. If the OP doesn't know what they mean I am happy to explain them, or they can google for the definitions. The only thing I didn't explicitly and exactly give was the full and proper definition of cardinality for very sound technical reasons. If you want a hand wavy explanation then, yes, the rationals are listable (can be labelled exactly by the natural numbers), and the reals are not by Cantor's diagonal argument. Now if the irrationals were also listable, then we could form an alternating list of rationals and irrationals, and hence list the reals. Contradiction.
Is that reasonably sound? Though I've not proven you may list the rationals, and not list the reals, though they are proofs found in many places.
I don't see the OP having any trouble with the notion of rational or irrational.

 

[pnaj]

Matt,

I wasn't questioning your understanding at all. I asked you to point out where I was wrong. I do not want to mislead anyone and if I've got something wrong I want to be corrected.

The only problem I had with your post was the the use of the term 'crank bait' ... it sounded insulting to me but I didn't want to jump to the wrong conclusion so I asked you what it meant.

If you are going to throw out these rather cryptic comments, please don't be surprised when people are offended.

Paul.

 

[matt grime]

Oh, the crank bait comment was NOT meant to imply that you were a crank, sorry if that came across. It's just that anything involving the idea of "size" of infinite sets tends to get people with their own pet theories jumping in very quickly. (ie it baits the cranks into posting some garbage about aleph-0 being distinct from aleph-0 + 1, despite not understanding any of the terms they use. I didn't think that was a remotely cryptic comment.)

Having said that, your argument about why the irrationals were of a different cardinality from the naturals in post 6 was very wrong indeed.

 

[pnaj]

Matt,

Thanks for clearing up the 'crank bait' thing, I was getting worried.

You say I'm wrong (now I'm 'badly wrong over 6 posts'), but fail to say why. I really want to know. I'm not trying to provoke you.

If it's the word 'more', I've accepted that I used a bad word and could have definitely misled the OP.

The only other thing I can possibly imagine is that I didn't talk about the negative rationals? Is that it?

Paul.

EDIT: You edited 'badly wrong over 6 posts' to now read 'in post 6', but my question still stands.

 

[matt grime]

Take the closed interval [1, 2]. There are, er, 2 natural numbers in this interval. There are infinitely many irrational numbers in this interval.

The same holds for any interval [n, n+1].

The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.

Perhaps not wrong, then, but, so what that there are two naturals in that interval and infinitely many irrationals? there are also infinitely many rationals in that interval. You seem to be implying that, because there are infinitely many irrationals in each such interval, they must be "uncountable" (though we've yet to introoduce that term).
Moreover, why have you used the word "thus" as if there is somethin in what you say that implies the rationals are countable. And one need not say the naturals are countable, since that is the definitoin of countable.

I didn't edit my previous post as far as I can recall.

 

[NateTG]

Please don't accuse me of saying things that I haven't said.

Apparently I misread that

You keep saying I'm being inaccurate ... where?

You initial statement implies that a 1-1 function is sufficient to demonstrate that two sets have the same cardinality. This is incorrect, since, for example, the natural mapping of the rationals into the reals is 1 to 1.

Equal cardinality is demonstrated by the existence of a bijection (1-1 and onto). Admittedly, describing that use of 1-1 as inaccurate rather than incorrect assumes that you know that the appropriate relationship is a bijection. Similarly, the notion that the existence 1-1 relationship equivalent to having similar listings is inaccurate.

Somewhat counter-intuitively, the rationals have a one-to-one relationship with the natural numbers (that is, you can list the rationals as you can the naturals).

It should be clear that there are more irrationals than naturals!

Moreover, this is a rather poor explanation.

The following "argument" is really poor:

So, in reasonably simple terms ...

Take the closed interval [1, 2]. There are, er, 2 natural numbers in this interval. There are infinitely many irrational numbers in this interval.

The same holds for any interval [n, n+1].

The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.

This is an example of the post hoc ergo propter hoc falacy - you assume (probably unintentionally) that because the conclusion you reached is correct that the argument made for it is valid. However, the last sentence has very little, if anything to do prior claims.

It's almost as if it started as something about a finite number of rational numbers in an interval, and someone, upon realizing that that was false, and thoughtlessly substituted natural numbers rather than accepting that the argument did not hold water to begin with.

 

[pnaj]

Well, I've certainly learned a valuable lesson here.

 

[pnaj]

NateTG, I didn't see your last post until after my last one.

Thanks for acknowledging the 'misread' bit. I appreciate it.

But I have to come back because I'm afraid that you have 'put words in my mouth' again.

You said ...

You initial statement implies that a 1-1 function is sufficient to demonstrate that two sets have the same cardinality.

The trouble is, I most certainly did NOT imply equality of cardinality ... you have just wrongly assumed that, without actually carefully considering EXACTLY what I wrote.

I only considered listing the rationals 'as you can the naturals' and it IS sufficient for me to consider a 1-1 map if that's all I wanted to do, whether you think so or not.

So there's no room for mis-interpretation, take the map ...

$$\begin{array}{l} f:Q \to N \\ f(\frac{m}{n}) = 2^n 3^k \\ \textrm{where } k = \left. {\left\{ {\begin{array}{*{20}c} {2|m| \textrm{ if } m > 0} \\ {2|m| + 1 \textrm{ if } m \le 0} \\ \end{array}} \right.} \right\} \\ \end{array} \]$$

f is a bijection from Q to an infinite subset of N. Any infinite subset of N IS countable so I CAN make my list.
Note that f isn't onto N.
1-1 IS sufficient.


Your second point is also YOUR implication. It certainly wasn't mine. You've interpreted the post wrongly (probably unintentionally) and gone off on some fantasy.

The statement ...

The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.

... is perfectly fine on it's own, thanks.

Please don't make false implications on my behalf and then set about correcting them.

As I said earlier, I want to be corrected if I make mistakes. The actual mistakes I made (and won't make again, believe me) were the dreaded 'more' sentences.

Paul.

 

[master_coda]

Your second point is also YOUR implication. It certainly wasn't mine. You've interpreted the post wrongly (probably unintentionally) and gone off on some fantasy.

The statement ...

The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.

... is perfectly fine on it's own, thanks.

Please don't make false implications on my behalf and then set about correcting them.

When you write what looks like the start of the proof, and then immediately follow it with what looks like a conclusion of a proof, then when you have is a bad proof - no matter what the intention of your writing is. NateTG wasn't assuming that you were making an implication, he was reading the implication that you put there.

 

[pnaj]

It wasn't my implication, whether or not that's how it's being read.

 

[matt grime]

Intention or not, you've been hoist by your own petard on this one.
You can't get away with saying that someone else didn't answer the question (merely showed off) when you yourself have made little credible explanation, nor offered some hints as to what to look for in understanding the argument/proof.

Your claim that:

" The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't."


is not fine on it its own since you have not proven that the reals are not countable, and that the finite union of two countable sets is countable.

 

[pnaj]

Hi Matt,

Intention or not, you've been hoist by your own petard on this one.

You're right there!

But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals.

He has misquoted me on a number of occasions (and you have too, once). Now he assigns implications to me that I have NOT made and heads off on his flight of 'post hoc' fancy, trying to imagine my thought processes. I have acknowledged valid criticisms ... please refer to my earlier posts.

Paul.

PS.
You are forgetting your own advice. According to you, I have to prove every statement I make ...

Your claim that:

" The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't."


is not fine on it its own since you have not proven that the reals are not countable, and that the finite union of two countable sets is countable.

... but you don't have to!

Though I've not proven you may list the rationals, and not list the reals, though they are proofs found in many places.

Best make up your mind, Matt.

 

[matt grime]

"But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals."
i didn't insist that. clearly an injection from Q to N is sufficient to show they are countable (for several reasons, not least the obvious schroeder bernstein theorem, or is that showing off?)

and the last bit, yes, hyposcrisy abounds doesn't it! but you did accuse me (probably me) of not answering the question properly, and I'm pointing out you perhaps didn't either by your own standards.

 

[pnaj]

Matt,

I have acknowledged the mistakes I have made. I just will not acknowledge mistakes that have been made on my behalf.

So far, according to you, I've been 'wrong', 'badly wrong' and then 'oh, perhaps not wrong'. Not once have you acknowledged that YOU misunderstood my first statement and were WRONG in your assumption that I needed to show equal cardinality in order to list the rationals. Nor has NateTG.

It might have helped the situation somewhat if you and he had decided in the first place to just correct my mistakes, rather than throw out terse, rude little asides.

I am not sorry for insisting that you or NateTG be precise in your criticism. In fact, that was YOUR original criticism of me.

If you call that hyposcrisy (sic), that's up to you.

Paul.

 

[matt grime]

Could have sworn I posted this but: understood, my apologies.

 

[matt grime]

Could have sworn I posted this but: understood, my apologies for being terse and rude and misrepresenting you. However, if you may point out where I said you (and I presume that you is the specific "you", rather than a general "you") need to show equal cardinality in order to list the rationals? Listing the rationals clearly shows they are of equal cardinality, indeed is a proof they are countable. I have read back and can't see what it is makes you think that, though perhaps I misunderstand what you're saying.

Is it perhaps that you think I wanted you to prove them countable? No, that wasn't it, it really was the assertion you made that "clearly there are more irrationals than naturals" without any substantiation, as there was no indication of what you meant by "more" or "size" (post 2), and what led to your out burst about "jeez some people in this forum amaze me..."

When I say "wrong" etc, perhaps I should make it clear I meant, oh, not false, but, in some loose sense, not mathematical enough, though that is very subjective. I suppose I was just mystified by post 6 which apparently starts off with a premise and a has conclusion, but they aren't linked. That is how it appears, though we've clarified that isn't what was meant.

The reference to hypocrisy was aimed at me, not you.

 

[pnaj]

Matt,

Thanks for your last post. I accept everything you say.

As I said before, you've taught me a really good lesson in how NOT to answer questions. Hopefully, I won't be making the same mistake again.

Paul.

 

[NateTG]

Pnaj, I may have misinterpreted your posts, and I apologize for that.

Let's take a look at your original post:

Post #2:

Somewhat counter-intuitively, the rationals have a one-to-one relationship with the natural numbers (that is, you can list the rationals as you can the naturals.)

Post#22:

The trouble is, I most certainly did NOT imply equality of cardinality ... you have just wrongly assumed that, without actually carefully considering EXACTLY what I wrote.

The paranthetical remark indicates that it's possible to list the rationals in the same way that it's possible to list the naturals - a statement that's equivalent to saying that there's a bijection between the two sets - which is stronger than stating that there is a 1-1 relationship.

In the context of an explanatory answer, it might be good to define, or describe, what a 1-1 relationship is, which is what I assumed the paranthetical remark was supposed to be, and in that context the statement is inaccurate - perhaps it was intended for a different purpose.

The post also assumes that the reader is sufficiently familiar with cardinal numbers to understand that the existence of 1-1 mappings, or bijections is usefull in comparing the 'size' of sets, and it's unclear whether you want to show less than or equal to, or equal to.

"But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals."

A 1-1 map from the rationals to the natural numbers is sufficent to make a list of the rationals. As indicated in my prior post, it's not at all clear from the phrasing you initially used ",1-1 relation", whether you're referring to an injection in a specific direction. (Yes, I know that there's always an injection between two sets.) Moreover, invoking implied notions of countability is a really poor way to describe something to anyone who is unfamilar with cardinal numbers in the first place.

Consider, for example that the reals have a 1-1 relationship with the natural numbers - for example, the usual embedding of the naturals in the reals is an injection - but we both already know that the reals are not countable.

[NateTG] has misquoted me on a number of occasions (and [matt grime has] once).

Where? I cut and pasted the direct quotes.
I may have interpreted what you wrote differently than you, but that because the posts are ambiguous at best, and, in several places, I'm not the only one that finds your interpretation unusual. Could you list examples of what you consider to be me misquoting you?

 

[pnaj]

NateTg,

Well all I can really say to that is: fair comment.

Paul.

P.S.
NateTg ... if you really want me to list the misquotes, I will, but I'd rather not.

 

[NateTG]

NateTg ... if you really want me to list the misquotes, I will, but I'd rather not.

I don't really think that I've misquoted you, so I would really like to see what you consider to be misquotes.

 

[pnaj]

NateTG,

Well I didn't want to open it all up again.

But fair enough.


The 'Jeez' post was actually directed at your previous post. I had already asked Matt to explain his comments and he hadn't responded yet. You interpreted it as if I was further questioning Matt's comments.

Matt Grime's point is entirely valid. In some situations it's very important to realize that what you cheerfully refer to as 'definitely more' is really a rather technical notion.

On the same post ...

Odd, that you're the one that brought up 1-1 relationships (somewhat inaccurately, no less) and then accuse Matt Grime of being a crackpot when what he said is completely correct.

... I have no idea where you got this, and I'm afraid it coloured my judgement of you. It seemed to me that you weren't actually reading what I actually wrote, just what you thought I wrote.


We've cleared this one up already.

Correct, but also that's really irellevant since the inital post was about rational numbers, not natural numbers, and has nothing to do with the conclusion that you reach.

Here's another one ...

You initial statement implies that a 1-1 function is sufficient to demonstrate that two sets have the same cardinality. This is incorrect, since, for example, the natural mapping of the rationals into the reals is 1 to 1.

It didn't, as I showed earlier, but you interpreted it that way.


And you know what I think about this ...

This is an example of the post hoc ergo propter hoc falacy - you assume (probably unintentionally) that because the conclusion you reached is correct that the argument made for it is valid. However, the last sentence has very little, if anything to do prior claims.

It's almost as if it started as something about a finite number of rational numbers in an interval, and someone, upon realizing that that was false, and thoughtlessly substituted natural numbers rather than accepting that the argument did not hold water to begin with.

Paul.