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dathca
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can someone explain to me why there are always more irrational than rational numbers?
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.
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.
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.
Dathca said:can someone explain to me why there are always more irrational than rational numbers?
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.
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.
pnaj said:It should be clear that there are more irrationals than naturals!
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 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!
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.
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.
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 ...
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.
You're right there!Intention or not, you've been hoist by your own petard on this one.
... but you don't have to!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.
Best make up your mind, Matt.Though I've not proven you may list the rationals, and not list the reals, though they are proofs found in many places.
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.)
pnaj said: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.
pnaj said:"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."
pnaj said:[NateTG] has misquoted me on a number of occasions (and [matt grime has] once).
pnaj said:NateTg ... if you really want me to list the misquotes, I will, but I'd rather not.
On the same post ...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.
... 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.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.
It didn't, as I showed earlier, but you interpreted it that way.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.
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.