I see all kinds of "proof sketches" and "intuitions" about why there are more irrationals than rationals, but the same explanations could be used to argue that there are more rationals than whole numbers.
So which is it? How can there be more irrationals than rationals but not more rationals than whole numbers?
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>the same explanations could be used to argue that there are more rationals than whole numbers
no they can't
>So which is it?
you know which.
the proof is as follows. It's easy to show rationals are countable. so to show irrationals are uncountable it's enough to show the reals are uncountable. to show the reals are uncountable, it's enough to show that the set S of infinite strings of 0s and 1s are uncountable (as s -> 0.s gives an injection from these strings to the reals).
to prove this set is uncountable, assume it is countable. let f:N->S be a bijection. let s_i denote the ith digit of string s. then the string (1-f(1)_1)(1-f(2)_2)(1-f(3)_3)... is not in the image, a contradiction
Infinity is a weird beast.
>there are more rationals than whole numbers
well, you have an easy bijection between them so this should be obviously false.
the existence of a bijection is a fairly intuitive way to understand equal "size". if there exist injective but not surjective functions from A to B, then B must somehow be "larger" even when A and B are both infinite.
>equal "size"
Infinite sets fuck up intution.
But at least there will be more rational numbers than integers if we set a point that where |x|
what the fuck are you talking about
equal size means equal cardinality, and it's made completely formal by bijections
>But at least there will be more rational numbers than integers if we set a point that where |x|
Assuming we don't count doubles, 0es and negatives.
We'd have 1 + 2 + 3 + ... + n rationals minus n integers.
So how many unique rationals would we have? (Hmm. This reeks a bit of prime numbers somehow. Because a rational will at least eliminate every multiple of 2,3, and the others.)
what the fuck are you talking about
how many rationals are there in a compact interval? a countably infinite amount.
Did you miss the point where I cut the infinite set down to finite size, where the border is the same in both (
i don't understand the question. if you consider some finite real interval then you'll have a finite number of integers but a countably infinite number of rationals.
is that what you mean?
please stop using javascript notation for variables. use something like [-n,n].
see
>what the fuck are you talking about
possibly a bot
There are not more irrationals than rationals, because there are infinite rationals, so... Idc what people have to say about this, that's just the way it is
We all know this is a bait thread but now I got a reason to post black science man
I see now the points that I have overlooked or not specified fully in writing.
The idea was counting proper fractions in the interval [1,n] with a denominator in the same interval and comparing their count to the number of integers in the same interval.
It should follow loosely that there are more rationals than integers in those finite sets.
(So how many unique improper fractions with a denominator in that interval would there be in this interval.)
Infinite rationals if the set is limited to the interval [-x,x] and the denominator limited?
>The idea was counting proper fractions in the interval [1,n] with a denominator in the same interval and comparing their count to the number of integers in the same interval.
Your idea is correct. If you do this, you can get a function from Z to Q which is injective but not surjective.
You're making a common mistake in thinking this proves anything.
In order for Z and Q to have different cardinalities _EVERY_ function between them must fail to be bijective.
What you're doing is a more complicated form of the very simple argument
>look the function f(x) = x puts all the integers in the rationals but there's still a bunch left
we totally skipped functions and bijections in my intro to proofs class
we were proving like divisibility stuff and congruences most of the time
>muh RSA scheme
>muh gcd
that sounds like math for CS
this is covered in more of a set theory type class for math
Mind you. The statement was still limited to the finite sets and loosely sketched an idea (not a proof) that there are more rationals than integers IN THAT SAME LIMITED SET.
Those sets limited to the interval [-x,x] are not equivalent to Z and Q anyway.
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While everyone is being stupid, can someone explain to my dumb ass how I should visualized a family of functions? Specifically for example, the set of all real valued functions. I have no idea how to go about understanding this.
the set of functions f:X -> Y where Y is a metric space have a natural metric d(f,g) = sup{x}(d(f(x)-g(x)))
Long skirts look so great on women. Is that comic book good?
>why there are more irrationals than rationals
Holy fucking shit OP
That's what brought me here you fucking cunt, gas yourself
Second post in thread was the answer I was looking for
It's a book, and I don't know if it's good yet, it's on its way