It’s been more than a decade since I’ve taken a math class, so I’m not exactly an expert on the topic. But I remember from geometry (which I absolutely sucked at) something to the effect that for any given line there were an infinite number of points that could be plotted on it. I don’t understand how that’s possible. If I have a line like this, for example:
(imagine it’s not broken up), it’s obviously a finite length and can only contain a finite nuber of points. Right? (Yeah, I know I’m wrong; I just wanna know why)
THen I thought maybe they meant that on a graph you could plot points at, say, 5.0, then 5.01, then 5.001, etc. on and on to a billion+ decimal places, which must be where the infinite comes in. But if the line only takes up the space on a graph arbitrarily numbered, say, 1 to 6, I can see it taking up a billion+ decimal places after 1-4, but it seems to me that it’s missing all the decimal places after numbers 7- googol or whatever, so how can it still be considered infinite? I know I’m missing something (quite a lot of things, probably), but could someone explain this to me in language as non-technical as possible? Assuming my question’s not so poorly worded that it’s completely baffling, anyway. Thanks.
Just because the line is a finite length doesn’t mean that it doesn’t have an infinite number of points on it.
Which set of numbers is greater, the set of all posative integers, or the set of all even posative integers? Without thinking, you’d say that the set of all posative integers has double the numbers, right? Wrong, they’re both infinite. Same deal here, sorta…
Micahw is on the right track: some infinities are bigger than others.
Lets say that there are an infinite number of points between 0 and 1…call this infinity(a).
There are an infinite number of points between 0 and 10…
call this infinity(b).
Infinity(a)/infinity(b)=1/10
This cancelling out of infinities happens all the time in quantum mechanics…its called “renormalization”.
I am not a physicist and I know nothing of renormalization, but all I can say is that’s not the way mathematics normally works.
True, there are larger and larger infinities, but it’s easily proved that any two line segments are the same size (have the same cardinality).
A nice graphical way of showing this (which may get all screwed up since it’s hard to draw a picture here):
Picture any two line segments:
A_______________________________B
C______________D
.E
(E comes from this: Draw a line through A,C; draw a line through B,D; E is the intersection of these two lines).
Pick any point on the line segment AB. Connect that point to E–in doing so, you will select a unique point on CD (the point where you intersect CD). Similarly, picking a point on CD uniquely determines a point on AB. In other words, we have a 1-1 correspondence between the points of AB and the points of CD, so the two lines have the same cardinality (as collections of points).
A line 1 millimeter long has the same number of points as a line 5 miles long.
OK, I’m beginning to understand (actually, my GF helped me out with it, too). The numbers on the graph aren’t what’s important, it’s the points that matter. I guess I can see how an infinite number of points can be plotted on a given line, in a kind of illiogical logical way. It just trips me out, is all. Thanks, guys.
This whole thing reminds me of an argument I once had with my chemistry professor. He insisted that points had no mass–whereas I insisted that they -could-, but it would measure to be infintesimaly small, just as a point’s dimensions are infinately small.
I couldn’t prove him right. But he couldn’t prove me wrong, either. So I guess questions like these all depend on how you want to form your opinions. That’s what the nature of science and theory is all about.
Well it seems like you probably get it, but I think all you have to realize is that just like Mikahw said just because something seems like it would have a “bigger” set of values doesn’t mean that both aren’t infinte.
both lines have an infinite number of points and neither line has more points because infinity is just infinity there are no sizes to it, you can’t add one to infinity, it just doesn’t make sense
Points have mass? Points are a theoretical concept. They don’t have existance in the physical world, so of course they don’t have mass.
Since points have zero length, there can be an infinite number of them contained on even the shortest line segment. No matter how close two points are to each other, you can always in theory find a third half way between them. They can’t overlap because they have no length, remember?
Philosphers have been pointing out certain flaws with this idea for centuries. One famous example is the parable of Achilles and the Tortoise, which I won’t recount here, but you can look it up. In this century, quantum physics has theorized that there is a smallest possible distance. Something travelling this distance would pass from point A to point B without crossing the intervening gap. Needless to say this is some freaky s**t, which you don’t want to lay on the the already adled brain of the average high school geometry student.
My previous post was in response to Ashtar, of course.
Actually, as mentioned in the thread linked by Terminus Est, there are different sizes to infinity. You can add one to infinity, but the result will be the same, if you’re talking about the cardinalities of the sets. On the other hand, if we’re talking about the ordinals (basically, sets with certain ordering properties), consider this. Take the “smallest” infinite set, the positive integers {1,2,3,4,…}, which of course has the order
1 < 2 < 3 < 4 < …
Now introduce a new element to it, say “w”, and consider the set {1,2,3,4,…,w}, which has the order
1 < 2 < 3 < 4 < … < w (w is bigger than everything else}.
This is the ordinal “(smallest infinity) + 1”, for lack of a better way of labeling it.
Of course, the size/cardinality of these two sets is the same, but even so, there are sets with strictly larger cardinality than the positive integers (or any given infinite set, for that matter).
And a reminder (as Greg Charles already noted) that the line you have drawn on your piece of paper or electronically on this MB is a physical line in the real world. It is made up of a finite number of points, in the sense that the grains of pencil lead used to draw the line are made up of atoms that are of finite length (very very tiny, but finite).
The mathematical concept of line, however, has an infinite number of points on it in the sense that, whenever you have two points, there is always another one between them. So if you have one point at .000000001 distance from the end and another point at .000000002, then you can find .0000000015 which is halfway between them. Etc.
It is important to distinguish between the mathmematical concept of line (which doesn’t exist in the physical world) and an actual, physical line.