Length of arc paradox

In summary, the conversation discusses the concept of cardinality and how it differs from length. Premise 1 states that a line is made up of points, while premise 2 explains that each point has specific coordinates. Premise 3 states that lines of equal length have an equal number of points, and lines of greater length have more points. Premise 4 states that for a function f(x), each value of x corresponds to a single value of y. Finally, premise 5 explains that each point on the line LM has a unique partner on the arc PQ according to the function y = f(x), leading to the conclusion that the number of points on the arc is equal to the number of points on the line LM. The
  • #1
Vinay080
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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.
 
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  • #2
What leads you to assert premise 3? Much trouble lurks there.
 
  • #3
jbriggs444 said:
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?
 
  • #4
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.
 
  • #5
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!
 
  • #6
@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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  • #7
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).
 
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  • #8
Mark44 said:
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.
 
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  • #9
MarcusAgrippa said:
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.
 
  • #10
Strilanc said:
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.
 

1. What is the "Length of arc paradox"?

The "Length of arc paradox" is a mathematical paradox that challenges our understanding of the relationship between the length of a curve and its curvature. It states that as the curvature of a curve increases, the length of the curve also increases, but at a slower rate. This leads to the paradoxical conclusion that a curve of infinite curvature can have a finite length.

2. Who first discovered the "Length of arc paradox"?

The "Length of arc paradox" was first discovered by the Swiss mathematician Johann Bernoulli in the 17th century. He posed the question to his fellow mathematicians, challenging them to find a solution to this paradox.

3. How does the "Length of arc paradox" relate to real-life situations?

The "Length of arc paradox" has applications in various fields such as physics, engineering, and computer graphics. In real-life situations, it can be seen in the design of roller coasters, car racing tracks, and even DNA strands. It also has implications for understanding the behavior of light and sound waves.

4. Is there a solution to the "Length of arc paradox"?

While there is no definitive solution to the "Length of arc paradox", mathematicians have proposed different approaches to address this paradox. Some have argued that it is a result of our limited understanding of infinity, while others have proposed alternative ways of measuring the length of a curve. However, there is still ongoing research and debate on this paradox.

5. Why is the "Length of arc paradox" important?

The "Length of arc paradox" challenges our understanding of fundamental mathematical concepts and forces us to think critically about seemingly paradoxical situations. It also has practical applications in various fields and contributes to the development of new mathematical theories and techniques. Additionally, studying this paradox can enhance problem-solving skills and promote critical thinking in mathematics and other disciplines.

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