Length of arc paradox

Vinay080
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Main Question or Discussion Point

NdICu.jpg

Premise 1: Line is composed of points.
Premise 2: Each point is associated with specific co-ordinates (x,y).
Premise 3: Lines of equal length have equal number of points. Lines of greater length have greater number of points.
Premise 4: Each value of x in the function f(x) gives a single value of y.
Premise 5: Each point (x,0) between L and M is associated with a unique point of (x,y) according to the function y = f(x). So, each point on the length LM has a unique single partner on the arc PQ. Hence, number of points on the arc is equal to number of points on the line LM. Length LM = Length of arc PQ.

Conclusion: The above analysis seems to conclude length of LM to be equal to length of arc PQ, which is definitely wrong as evident from the figure itself.
So, is length of arc PQ greater than LM?

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The figure has been extracted from the book "Higher Engineering Mathematics" by Dr.B.S.Grewal.
 

Answers and Replies

jbriggs444
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What leads you to assert premise 3? Much trouble lurks there.
 
Vinay080
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What leads you to assert premise 3? Much trouble lurks there.
Sorry I thought it to be self evident. I don't have much math back ground. Can you be little more explicit?
 
jbriggs444
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Curves are infinitely divisible. They contain infinitely many points. It is not straightforward to compare "how many" points are in two sets when both sets are infinite. One way of comparing "how many" points are in two sets is to match the points up in a one to one correspondence, a "bijection". If such a correspondence can be found then the there are just as many points in the one set as in the other. This particular notion of how many is called "cardinality".

Note that if you take a curve and scale it up, the set of points in the new curve has the same cardinality as the original. It is easy to find a bijection. Just match each point in the original curve with its corresponding point in the new curve.

Obviously, this makes cardinality a poor yardstick for arc length.

Arc length is more usually defined in terms of a limiting process. The length of an arc is approximated by the sum of the lengths of a series of straight line segments connecting points on the arc, starting at one end and going to the other. The length of an arc is taken to be exactly equal to the "limit" of the sum of those lengths as the size of the little line segments decreases toward zero.
 
MarcusAgrippa
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jbriggs444 has said everything there is to say on this point - cardinality is not length. They are completely different concepts.

Here is a mind bender for you Vinay080: there are as many points on a plane as there are on a line. For that matter, there are as many points in three-dimensional space as there are on a line. And there are as many points .... - and so on, but NOT to infinitum!
 
jbriggs444
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@vinay80, If you Google for cardinality, you can find any number of pages. The wiki page http://en.wikipedia.org/wiki/Cardinality is decent, but uses terms like "injection", "surjection" and "bijection" which may be unfamiliar. To gain a proper understanding, you should click through those links to discover what those terms mean.
 
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The length of the interval [0, 2] is 2, and the length of the interval [0, 1] is 1. Nothing surprising there. However, both intervals have the same number of points, which might be surprising in light of your (@Vinay080) premise 3. When you start comparing infinite sets, the usual way of doing things doesn't apply any more. Instead, what happens is that you see if you can come up with a reversible pairing that matches each item in the first set with its counterpart in the second set.

For the two intervals I gave, if x is a number in the interval [0, 2], then x/2 will be its counterpart in the interval [0, 1]. This mapping is an isomorphism between the two sets, and shows that the sets have the same number of elements. So although the lengths of the two intervals are different, both sets have the same number of elements (points).
 
jbriggs444
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For the two intervals I gave, if x is a number in the interval [0, 2], then x/2 will be its counterpart in the interval [0, 1]. This mapping is an isomorphism between the two sets, and shows that the sets have the same number of elements. So although the lengths of the two intervals are different, both sets have the same number of elements (points).
For your information, Vinay080, an isomorphism is, roughly speaking, a bijection on steroids. Not only does it establish a reversible mapping between the elements of the two sets, it also maps one or more properties or relationships among the elements of the one set so that the corresponding elements of the other set have the corresponding relationships and properties.

In the case of mapping between [0,2] and [0,1] defined by x => x/2, one thing that is preserved is order. i.e. If x is less than y then x/2 is less than y/2.
 
Strilanc
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jbriggs444 has said everything there is to say on this point - cardinality is not length. They are completely different concepts.

Here is a mind bender for you Vinay080: there are as many points on a plane as there are on a line. For that matter, there are as many points in three-dimensional space as there are on a line. And there are as many points .... - and so on, but NOT to infinitum!
Doesn't ##R^\infty## have the same cardinality as ##R##? (Note that I am talking about the case where ##\infty## is the countable infinity, not larger spaces like the space of ##R \rightarrow R## functions where you have a real value for each real number instead of each natural number).

Here's an onto mapping from ##[0, 1]## to ##[0, 1]^\infty##: let the ##i##'th digit after the decimal of the ##n##'th component of the vector in ##R^\infty## be the ##(p_n)^i##'th digit of the input in ##R##, where ##p_n## is the ##n##'th prime number. Also, magically fix the issues related to ##0.0001 = 0.00009999...## ambiguities.
 
jbriggs444
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Doesn't ##R^\infty## have the same cardinality as ##R##? (Note that I am talking about the case where ##\infty## is the countable infinity,
Yes.
 

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