My Math Forum  

Go Back   My Math Forum > High School Math Forum > Elementary Math

Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion


Thanks Tree10Thanks
Reply
 
LinkBack Thread Tools Display Modes
August 21st, 2014, 02:54 AM   #1
Senior Member
 
shunya's Avatar
 
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.
shunya is offline  
 
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
Thanks from shunya
Benit13 is offline  
August 21st, 2014, 09:31 PM   #3
Senior Member
 
shunya's Avatar
 
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?
shunya is offline  
August 21st, 2014, 09:55 PM   #4
Banned Camp
 
Joined: Jun 2014
From: Earth

Posts: 945
Thanks: 191

Quote:
Originally Posted by shunya View Post
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?
shunya,

a ray does have an endpoint. $\displaystyle \ \ $ It is what you labeled as the asterisk.

This endpoint is the "beginning point" of the ray.
Thanks from shunya
Math Message Board tutor is offline  
August 21st, 2014, 10:56 PM   #5
Senior Member
 
shunya's Avatar
 
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?
shunya is offline  
August 21st, 2014, 11:01 PM   #6
Banned Camp
 
Joined: Jun 2014
From: Earth

Posts: 945
Thanks: 191

Quote:
Originally Posted by shunya View Post
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?
Because neither is finite in length, you cannot state one is longer than the other.
Thanks from shunya
Math Message Board tutor is offline  
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 one-to-one 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 sub-zero".

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 sub-one. The length of each would still be based on the number of points (in this case aleph sub-one). Thus as I see it, the line and the ray would be of equal length.
Thanks from shunya

Last edited by Timios; August 23rd, 2014 at 10:19 AM.
Timios is offline  
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 one-to-one correspondence between them.

The number of rational numbers is called "aleph sub-zero" and the number of reals "aleph sub-one".
Thanks from shunya
Timios is offline  
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.
Thanks from shunya and Timios
v8archie is offline  
August 23rd, 2014, 11:24 AM   #10
Senior Member
 
Joined: Jan 2014
From: The backwoods of Northern Ontario

Posts: 371
Thanks: 68

Quote:
Originally Posted by v8archie View Post
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.
I agree—because there are $\aleph_1$ elements in the reals rather than $\aleph_0$.

Quote:
I would argue that neither have a well defined length (because they are infinite).
Agreed.

Quote:
However their lengths are both of cardinality $\aleph_0$ as you point out.
Uh, wouldn't that be $\aleph_1$?
Thanks from shunya
Timios is offline  
Reply

  My Math Forum > High School Math Forum > Elementary Math

Tags
bigger, mine, ray



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Prove false: there is an even prime bigger than 2 shunya Elementary Math 9 April 9th, 2014 04:54 PM
how many times is a bigger then d gelatine1 Algebra 2 December 29th, 2012 01:09 AM
Which one is bigger? Albert.Teng Algebra 8 April 24th, 2012 05:17 AM
Which function is bigger nadroj Algebra 4 November 9th, 2009 09:39 AM
What mean s 'negative number is five times bigger'? W300 Algebra 2 October 22nd, 2009 10:11 AM





Copyright © 2017 My Math Forum. All rights reserved.