My Math Forum Ray's is bigger

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

 August 23rd, 2014, 11:13 AM #11 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra Sorry. Typo.
 August 23rd, 2014, 11:25 AM #12 Senior Member   Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 391 Thanks: 70 No problem, Archie. I would like to mention that I just can't get my mind around the fact that there is an infinite number of rationals ($\aleph_0$) and that there is a BIGGER infinite number of reals ($\aleph_1$). Psychologically, I would like to deny this, but I have studied the proof, and it is valid.
 August 24th, 2014, 07:00 PM #13 Senior Member     Joined: Oct 2013 From: Far far away Posts: 429 Thanks: 18 So, this question is not nonsensical. The ray and line have length of the order aleph-sub-one, using the analogy of comparing all the reals with the positive reals (including zero). So neither the ray nor the line is bigger or smaller in length, than the other. Am I correct?
August 24th, 2014, 10:25 PM   #14
Math Team

Joined: Dec 2013
From: Colombia

Posts: 7,674
Thanks: 2654

Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by shunya So neither the ray nor the line is bigger or smaller in length, than the other. Am I correct?
At the risk of repeating myself, neither has a (well defined) length so we can't compare their lengths.

 August 24th, 2014, 10:35 PM #15 Senior Member   Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230 Since both are infinite, both go endlessly and one cannot say which is bigger, but still, if both go till infinity, the rate of increase of both should be specified. If both increase by the same rate then I would say that a line is bigger. Thanks from shunya
 August 25th, 2014, 01:45 AM #16 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 Math Focus: Physics, mathematical modelling, numerical and computational solutions I'm not sure I agree that the line and the ray have the same length. I agree that a one-to-one correspondence can be applied between positive integers and positive even integers, so they must have the same number of elements, but the length of something isn't defined by the number of elements defining the course of the line, it is the sum of all intervals between adjacent elements over the total span of the object. Since the distance between adjacent integers is 1 and the distance between adjacent even integers is 2, I would argue that the latter is twice the length of the former. Just to be clear, my previous post used the idea of 'direction' to convey the differences in length between a line with an end-point and line without, because this is generally understandable by the lay-person. I guess a more specific (and clearer) way of conveying that is to say that if the ray is defined to be the set of positive integers (including 0), the line is defined to be the set of all integers and n is the number of non-zero positive integers, then: Number of elements of ray: n + 1 Number of intervals of ray: n Length of one interval: 1 Length of ray: n Number of elements of line: 2n + 1 Number of intervals of ray: 2n Length of one interval: 1 Length of line: 2n As $\displaystyle n\rightarrow \infty$, the lengths of both lines tend to infinity, but that doesn't magically make both lengths the same. I agree with the previous post stating that it doesn't really make sense to discuss the length of infinite objects, but the purpose of this exercise anyway is to demonstrate that $\displaystyle \infty$ is not really a number and must be considered carefully in every problem involving it. Last edited by Benit13; August 25th, 2014 at 01:47 AM.
August 25th, 2014, 07:46 AM   #17
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
 Originally Posted by shunya So, this question is not nonsensical. The ray and line have length of the order aleph-sub-one, using the analogy of comparing all the reals with the positive reals (including zero). So neither the ray nor the line is bigger or smaller in length, than the other. Am I correct?
Only assuming the continuum hypothesis.

There is a bijection between the points on a ray and the points on a line, so the two have the same cardinality. But this is only $\aleph_1$ under CH. It's unconditionally $2^{\aleph_0}=\beth_1.$

But it's not clear if the original question is asking this or something else.

 Tags bigger, mine, ray

,

,

# ray is bigger or line

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post shunya Elementary Math 9 April 9th, 2014 03:54 PM gelatine1 Algebra 2 December 29th, 2012 12:09 AM Albert.Teng Algebra 8 April 24th, 2012 04:17 AM nadroj Algebra 4 November 9th, 2009 08:39 AM W300 Algebra 2 October 22nd, 2009 09:11 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top