Which is larger? This discussion started in my "Math." thread before it was deleted. Mort-Hog had a great explanation, too.

Mort said any integer in [0..1] can be represented by 1/n, where n is some integer. Detty asked how you would represent 0.75. I'd like to know, too.

Set of reals is technically larger. Set of integers are countably infinite, set of reals is uncountable. If you could create a bijection between the set of integers and the set of reals between 0 and 1, that would show they have the same cardinality, but you can't. If you try to map every integer to every real between 0 and 1, you'll still miss out on many (well, infinitely many) real numbers. Thus the reals between 0 and 1 have a larger cardinality.

Any self-respecting person with a math degree should know this. Anyone who's not a math major probably shouldn't even touch this discussion cause they'll have no idea wtf they're talking about.

You would represent 0.75 by counting all real numbers between 0 and 0.75. Specifically, 0.75 would be represented by the integer that is infinity*0.75.

Also. Your poll is flawed.

Another thing to consider, the set of all rational numbers between 0 and 1 has the same cardinality as the set of all integers. That accounts for some of the reals between 0 and 1, but then you have to account for the irrationals too in order to include all the reals, thus giving you a larger cardinality for the reals from 0 to 1.

this

Oops

If [subset=Z]N[/subset] and [subset=R]Z[/subset] then |R| > |Z|

Proof:
.......

There's a proof technique known as "diagonalization" that can show that there is a real number between 0 and 1 that cannot be counted by the set of real numbers. Basically, to say that they are the same size means there is a mapping in which the integer 1 corresponds to some decimal, 2 corresponds to another decimal (from 0 to 1)

So, according to the idea that they are the same size, there should be some mapping like the following. (note: it doesn't matter what the mapping is in this example, as I'm about to show that it can't be mapped)

1 = 0.1234567...etc
2 = 0.2455675...
3 = 0.3756756...
4 = 0.4856723...
5 = 0.5965698...
...

This goes on for all positive integers to infinity (side-note, if you wish to include one as a positive number, that doesn't change anything here)

Now, I've already mapped every single positive integer all the way to infinity to a decimal place, so if I can build a new number that hasn't been mapped, I have proven that there are more real numbers between 0 and 1 than there are positive integers. So, what if I were to build a new number, x, by taking the first decimal place from the first mapped integer and increasing it by 1 (if the number was 9, it goes to 0), followed by the second decimal from the second decimal place and increasing it by 1, etc...

1 = 0.1234567...etc
2 = 0.2455675...
3 = 0.3756756...
4 = 0.4856723...
5 = 0.5965998...
...

x = 0.25670...

As you can see, x differs from every single existing number by at least one decimal place, thus I have created a number that was not in the mapping. Hence, the size of all real numbers from 0 to 1 is larger than the size of all positive integers.

And now for a fun, quirky fact. This technique can also be used to show that there are programs that cannot be written

Technically the term "larger" doesn't work here because infinity is not a number. You could say that one set approached infinity faster, though.

This isn't funny, Emon. This is your last warning.

lawl but seriously I await the inevitable implementation of calculuzzz

Actually it does work because there are infact different sizes of infinity It's weird, I know. I mean, I did show that there was at least one more number in the reals from 0 to 1 than there were integers to infinity. One more is one more...

If you don't believe me, look up Georg Cantor (he's the one who developed the diagonalization technique I mentioned. Not to mention set theory itself... ;))

Uh?

But it does work, as demonstrated by |{x | 0 ≤ x ≤ 1 ^ x ⊂ R}| > |Z|.

Humor goes right past you!

Oh right, my sides are tearing now, I see.

this.

I was thinking, as best I can tell the difference between the two sets is also infinite. Would it be possible to find three infinite sets of differing sizes?
Ie Set X < Set Y < Set Z?
Is there some order of infinity used to describe these sets? Are these infinite sets limited between countably and uncountably infinite sets or do they have their own infinite number of magnitudes:psyduck:

I just felt like voting.

Does it make a difference? Infinity is not a number. I agree that one gets there faster, and I get the whole mapping thing (sorta). I mean lim(x -> 0) 1/x^2 is bigger than lim(x -> 0) 1/x in the sense that one gets there much faster. But "grows with out bound" is still "grows with out bounds" no matter how you get there. (Which you don't).

Obi, how many integers are there?

Doesn't it make sense that the answer to such a question ought to be a number? What else could the answer be?

Edit: I realize my post was vague. Short clarification: the answer to the question I posed involves a different meaning of "infinity" than is used in limits. google aleph number/Georg Cantor.

We're talking about sets, not functions, they're not "going" anywhere. There are no rates involved. Calculus doesn't really apply here. What really matters is this: |{x | 0 ≤ x ≤ 1 ^ x ⊂ R}| > |Z|

The cardinality of the set of rationals between 0 and 1 is the same as the cardinality of the set of all integers. But there are more than rationals between 0 and 1, there are also irrationals. Infinitely many. The cardinality of the set of reals between 0 and 1 is infinitely larger than the set of all integers.

Yeah, I realize that, but even so, infinity is just a concept that means "no limit". There is no limit to either the amount of counting numbers or real numbers. I mean, there are more numbers between 0 and 1 than there counting numbers, but saying one set is "larger" than the other, seems to be applying a finite concept to infinity.

I read that, but it's still not the same situation as 8 > 7. I'm not really disagreeing with you I'm just bickering over terms.

That's the thing, it kind of is the same situation.

Needs this pic:



Also, I have done some degree level maths and I have to admit that this question is far beyond me!

(Hooray for ordinary maths!)

You need a few courses in discrete math to understand it, that's about it. Most engineering and science majors don't get discrete unless they are computer science or software engineering.

Yeah, they basically taught us what we needed to know, and then let us go back to the pub.

Heh, I covered that first question during A-level maths. I think it was mostly just our teacher seeing which of us could work it out for ourselves, especially as quite a few were applying for maths degrees. This kind of thing doesn't really help you figure out structural stresses or machine dynamics.

Coincidentally, the same people who don't 'get' this are the same people who don't 'get' that 1=0.999 repeating

no, it clips through the ground

er....I can't remember which option I chose.

But yeah, for every integer N, there is a real number on the interval [0,1] represented by 1/N....and then some.

We aren't talking about the Sith2 universe.

sith2 never clipped through the ground it clipped through the edges between two pieces of ground, theres a difference

We're not comparing finite numbers. It's a bit different.

It is, but it also isn't.

Welcome to math.

Actually, it is the same situation as 8 > 7, or more aptly put, it's the same as 1 > 0

I showed there was at least 1 element that couldn't be counted (aka 1 more element in the set of reals from 0 to 1 than the positive integers)

Thus, compare x + 1 to x... it's bigger. Sure, x is infinity, but this just goes to show that one infinity can be bigger than another. Hence why there are "countable infinite sets" and "uncountable infinite sets"

Countable sets < uncountable sets.

Nah, I get that 0.999 recurring definitely equals one. I just don't know the difference between two infinities :s

If you'd pressed me for an answer, I would've probably guessed that both sets have equal sizes, i.e. infinity