Question

[Peegee]

Given that a line has infinite points (if it has less I'd like to know the number):

- which is larger, the amount of points on the line or the amount of points in a solid square which is the area line^2?

Or do they both have the same amount of points?

[Leeza]

*head explodes*

PG, at least let me finish my morning coffee before I have to ponder stuff like this. :)

[Tavrobel]

I would say a line because it doesn't end, unlike a square (or cube for that matter), which does have a specified limit as far as dimensions are concerned. Even if the square were big, the line would eventually get to a point where it surpasses the square, despite how long it would have to be.

However, for what you are asking, both of them are undefined, since you let the line go on forever, and are building a square around it. Since infinity times two is still infinity, neither are larger. Concept + numbers = lose.

[Peegee]

No the line has a fixed length.

[Tavrobel]

The square, then. Since it's line^2, which is obviously more than X. It's not infinite if it has a fixed length.

[Dr Unne]

There are an infinite number of points in a line, and an infinite number of points in a plane, whether the line / plane are bounded or not. So the only question is how do you compare their infiniteness to each other?

From what I remember of number theory, the set of real numbers is an uncountable infinity. The number of points in a bounded line maps to the set of real numbers. The number of points in a bounded plane seems (without giving it much thought) like it also maps to the set of real numbers. I would guess that they have the same sort of infiniteness: uncountable infiniteness.

This kind of thing can be counter-intuitive. Like, if you compare the set of all integers, with the set of all even integers, it turns out they have the same size; they are both countably infinite. "Size" is defined differently for infinities than it is for actual numbers.

("Countable set" means that the set can be mapped to the set of natural numbers. http://en.wikipedia.org/wiki/Countable_set )

[Psydekick]

My head just burst, i have no idea:confused:

[Cookie]

The square is infinite to the power of 4. So technically greater than the line but both would go on forever so technically they are the same.

[Peegee]

There are an infinite number of points in a line, and an infinite number of points in a plane, whether the line / plane are bounded or not. So the only question is how do you compare their infiniteness to each other?

From what I remember of number theory, the set of real numbers is an uncountable infinity. The number of points in a bounded line maps to the set of real numbers. The number of points in a bounded plane seems (without giving it much thought) like it also maps to the set of real numbers. I would guess that they have the same sort of infiniteness: uncountable infiniteness.

This kind of thing can be counter-intuitive. Like, if you compare the set of all integers, with the set of all even integers, it turns out they have the same size; they are both countably infinite. "Size" is defined differently for infinities than it is for actual numbers.

("Countable set" means that the set can be mapped to the set of natural numbers. http://en.wikipedia.org/wiki/Countable_set )

yay for Dr Unne

[Hambone]

I suck at math. Does that help?

[Yamaneko]

For practical purposes infinity is infinity no matter which way you look at it.

[Peegee]

However there are greater amounts of infinity as we have illustrated, though the metaphor is misleading: infinity isn't a number.

[I Took the Red Pill]

For practical purposes infinity is infinity no matter which way you look at it.
Right, it's like asking if there are more whole numbers (1,2,3,4...) or more even numbers.

[Peegee]

For practical purposes infinity is infinity no matter which way you look at it.
Right, it's like asking if there are more whole numbers (1,2,3,4...) or more even numbers.

There are more whole numbers than even numbers. Both have infinite amount of numbers though.

[KentaRawr!]

A line is more versatile, so I'll go with that. >_>

I sux at math.

[Dr Unne]

For practical purposes infinity is infinity no matter which way you look at it.

Actually, no. There are different kinds of infinities and the differences matter. They matter in practical ways like for computer programming.




For practical purposes infinity is infinity no matter which way you look at it.
Right, it's like asking if there are more whole numbers (1,2,3,4...) or more even numbers.

There are more whole numbers than even numbers. Both have infinite amount of numbers though.

It doesn't make any sense to say that there are "more" of one than the other, when looking at infinite sets. You could only say that about a finite subset.

[rubah]

This square you're mentioning, if it's the length of an infinite line squared, that's like a space, right?

[nik0tine]

I would guess that the points on the plane are a sort of 'less restricted' infinity. It's still an infinite number, but the plane allows them to be organized differently. I think.

[Yamaneko]

For practical purposes infinity is infinity no matter which way you look at it.

Actually, no. There are different kinds of infinities and the differences matter. They matter in practical ways like for computer programming.
I'm interested, could you give an example without confusing me too much?

[GhandiOwnsYou]

Have we really reverted back to 2nd grade?

IE: UR STOOPID!
UR STOOPIDER!
UR STOOPID TIMES A THOUSAND!
UR STOPPID TIMES A MILLION!
UR STOOPID TIMS A FINITY!
UR STOOPID TIME A INFINITY + 1!

[Peegee]

For practical purposes infinity is infinity no matter which way you look at it.

Actually, no. There are different kinds of infinities and the differences matter. They matter in practical ways like for computer programming.




For practical purposes infinity is infinity no matter which way you look at it.
Right, it's like asking if there are more whole numbers (1,2,3,4...) or more even numbers.

There are more whole numbers than even numbers. Both have infinite amount of numbers though.

It doesn't make any sense to say that there are "more" of one than the other, when looking at infinite sets. You could only say that about a finite subset.

It doesn't make any sense, which is why when I learned that there are 'more' points in a square than a line, I was confused to no end.

The metaphors don't apply.

[Dr Unne]

For practical purposes infinity is infinity no matter which way you look at it.

Actually, no. There are different kinds of infinities and the differences matter. They matter in practical ways like for computer programming.
I'm interested, could you give an example without confusing me too much?

Umm, let's see. There are a lot of algorithms for producing digits of pi. Pi has an infinite number of non-repeating digits (so far as we can tell now). The best we can do is approximate it as best as possible. For example some algorithms are an infinite series whose limit is pi as you take it out to infinity; so the more members of the series you calculate, the closer your result is to pi. But you'd need to do an infinite number of members of the series to get exactly pi.

However not all algorithms are equal. Some approach pi much faster than others. So if you do a day's worth of work, you'd get more digits of pi with one algorithm than the other. It depends how quickly they converge. So even though they are both infinite series, and both have the same limit (so both would give the same answer if you took an infinite number of members of the series), they are practically different in terms of time and productivity.

That may not have anything to do with anything we're talking about though. I may have just made that up.