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can someone explain to me why there are always more irrational than rational numbers?
There are infintely many of either. In some very real senses, that may mean that there is the same amount of both.pnaj said: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?
In which real sense is that?In some very real senses, that may mean that there is the same amount of both.
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.pnaj said: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.
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.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.
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.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].
Unfortunately, none of us has actually answered dathca's inital question:The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.
This problem was investigated by a mathematician named Cantor (http://en.wikipedia.org/wiki/Georg_Cantor) and has had profound effects on mathematics.Dathca said:can someone explain to me why there are always more irrational than rational numbers?
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.pnaj said: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.
But the thing is, you didn't (in your first post) explain anything to the poster at all!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.
How exactly is that supposed to advance the poster's understanding?pnaj said:It should be clear that there are more irrationals than naturals!
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).pnaj said: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.
Apparently I misread thatpnaj said:Please don't accuse me of saying things that I haven't said.
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.You keep saying I'm being inaccurate ... where?
Moreover, this is a rather poor explanation.pnaj said: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!
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.pnaj said: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.
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.You initial statement implies that a 1-1 function is sufficient to demonstrate that two sets have the same cardinality.
... is perfectly fine on it's own, thanks.The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.
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 said: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 ...
... is perfectly fine on it's own, thanks.The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't.
Please don't make false implications on my behalf and then set about correcting them.