August 21st, 2014, 02:54 AM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18  Ray's is bigger
This is a question from the internet by user bpark1806 (not in this forum): "Would an infinite line and an infinite ray be equally long? I want to know too." I haven't changed a word of it. I think bpark1806 is basically asking which object (the line or the ray) has greater length. Personally speaking, both objects are infinite and neither is greater/lesser than the other. But you could easily superimpose the ray onto the line and see that the line is "obviously" smaller than the line. How would you answer this question? Thanks. 
August 21st, 2014, 06:14 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,080 Thanks: 698 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
It depends on how you draw the line. A ray typically has a source and travels in a straight line in one direction. If a line is drawn from the same point in one direction, in a similar manner to the ray, then the line and the ray have the same length. If the line is drawn by two pens travelling in precisely opposite directions then the line has twice the length of the ray. The fact that the length of each line is 'infinitely long' makes no difference for this property. This is why people shouldn't really interpret infinity as a number. It's a concept and it means different things in different situations. Mathematics avoids this by introducing new properties and ways of dealing with infinities, which are beyond me at the present time 
August 21st, 2014, 09:31 PM  #3 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18 
Thanks Benit13. I forgot to give the background on which this question was asked. Let me supply it below: A ray has a beginning point but no endpoint *> A line has neither beginning nor ending <> Which is longer, a ray or a line? 
August 21st, 2014, 09:55 PM  #4  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
a ray does have an endpoint. $\displaystyle \ \ $ It is what you labeled as the asterisk. This endpoint is the "beginning point" of the ray.  
August 21st, 2014, 10:56 PM  #5 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18 
Sorry for the mistake: A ray has one endpoint *> A line has no endpoints <> Which is longer? A ray or line? Does it even make sense to ask this question? 
August 21st, 2014, 11:01 PM  #6 
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Because neither is finite in length, you cannot state one is longer than the other.

August 23rd, 2014, 10:12 AM  #7 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 371 Thanks: 68 
An analogous question might be— Which set contains more elements: 1. The set of all positive integers? 2. The set of all even positive integers? Intuitively, it seems that there would be twice as many positive integers as even positive integers. But two sets have the same number of elements if a onetoone correspondence can be established between the two sets, and such a correspondence can be established. For we can match every element n in the set of positive integers with the element 2n in the set of even positive integers. When we do this, there are no positive integers left over which do not match, and therefore the number of elements is the same. The number of elements in each set is called "aleph subzero". Similarly, every point in a line could be matched to some point in a ray. Therefore there are the same number of points in the line as in the ray. I realize the question is not whether thee are the same number of points, but whether the infinite line is "longer". My understanding is that labelling the points with numbers would include not only the rationals, but also the real numbers, and their number would be aleph subone. The length of each would still be based on the number of points (in this case aleph subone). Thus as I see it, the line and the ray would be of equal length. Last edited by Timios; August 23rd, 2014 at 10:19 AM. 
August 23rd, 2014, 10:23 AM  #8 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 371 Thanks: 68 
Strangely enough to say, there are more real numbers than rational numbers. It can be proved that one CANNOT set up a onetoone correspondence between them. The number of rational numbers is called "aleph subzero" and the number of reals "aleph subone". 
August 23rd, 2014, 10:37 AM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,082 Thanks: 2360 Math Focus: Mainly analysis and algebra 
I would suggest that a better analogy would be to compare the size of the positive reals (possibly including zero) to all of the reals. It doesn't change your conclusion though. I would argue that neither have a well defined length (because they are infinite). However their lengths are both of cardinality $\aleph_0$ as you point out. I'd see this as being the 'order' of their length. It doesn't mean that they are of exactly the same length in the same way as two pencils might be. Rather, they are of the same order of length in the same way that all pencils are of the same order of length. 
August 23rd, 2014, 11:24 AM  #10  
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 371 Thanks: 68  Quote:
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