can someone explain to me why there are always more irrational than rational numbers?
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!
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.
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 infintely many of either. In some very real senses, that may mean that there is the same amount of both.
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 which real sense is that?
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
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.
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.
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.
Unfortunately, none of us has actually answered dathca's inital question:
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.
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.
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.
This is just getting silly, now.
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?
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'.
NateTG: Read the posts by pnaj carefully. He never accused Matt of being a crackpot.
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.
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:
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?
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.
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.
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.
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.
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?
EDIT: You edited 'badly wrong over 6 posts' to now read 'in post 6', but my question still stands.
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.
Apparently I misread that
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 existance 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 existance 1-1 relationship equivalent to having similar listings is inaccurate.
Moreover, this is a rather poor explanation.
The following "argument" is really poor:
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.
Separate names with a comma.