Line or Ray Longer?Date: 12/11/2001 at 21:01:57 From: Leslie Williamson Subject: Line vs ray Which is longer, a ray or a line? A ray has a definite starting point and goes for infinity, so therefore I think a line is longer. A line goes both ways for infinity, so in my opinion, it should theoretically be twice as long as a ray could ever be. Date: 12/12/2001 at 13:47:38 From: Doctor Ian Subject: Re: Line vs ray Hi Leslie, That's one way to look at it, but let's look at it from a different angle. Suppose you think of a line and a ray as collections of points. One would be 'longer' than the other if it has more points, right? So another way to ask 'which is longer' is to ask 'which has more points'. Suppose we start listing all the points on the ray, and label them a, b, c, d, etc. a b c d o-------------> Now, suppose we map those points onto alternating positive and negative points on the line. a b c d o-------------> <---------o-------------> d b a c That is, a(ray) -> a(line) b(ray) -> -a(line) c(ray) -> b(line) d(ray) -> -b(line) and so on. To determine whether two sets have the same size, we can pair up the elements. If we run out of elements in one set before we run out of elements in the other, then the set that runs out is smaller. Does that make sense? {bob ted fred barney} <--- | | | | |-- same size {carol alice wilma betty} <--- {john paul george ringo} <--- smaller | | | | {greg marsha peter jan bobby cindy} Okay, so the question is: If we pair up the elements on the line and the ray in this way, which set will run out of elements first? You can use the same kind of reasoning to show that the set of positive integers is the same size as the set of all integers... which is the same size as the set of rational numbers. Or that the set of points in the interval from 0 to 1 is the same size as the set of points in the interval from a to b, where a and b are any finite numbers. Weird, isn't it? What's trickier is to show that the set of real numbers is larger than the set of integers. You can find a nice introduction to these ideas at Infinity - Michigan Math and Science Scholars http://www.math.lsa.umich.edu/~mathsch/courses/Infinity/index.shtml I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 12/12/2001 at 14:09:43 From: Doctor Peterson Subject: Re: Line vs ray Hi, Leslie. I'd like to add something to what Dr. Ian said. You may object, as I do, that the length of a line is not the same thing as the number of points in it; for example, a line segment has finite length but infinitely many points. So instead of talking about points on the line and the ray, just cut each of them up into one- centimeter segments. Then we can match up SEGMENTS just the way he said to match up points: a b c d o---+---+---+---> <---+---+---o---+---+---+---> f d b a c e g You can match every piece of the line to a piece of the ray, so they are the same length! The point is, infinity doesn't behave the way numbers do; 2 times infinity is equal to infinity. So even though the line IS twice as long, the ray is the same length. In fact, one of the characteristics of infinite sets is just this: a subset (the ray) has the same size as the whole set (the line). Here is a similar discussion in our archives: Infinite Sets http://mathforum.org/dr.math/problems/gibbs9.24.97.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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