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infinity1<>infinity2?

Tue 31 May, 2011 06:25 pm
here's something I've thought of (no books!). take a line segment. cut it somewhere not in the center. is one side larger? now divide each segment in half infinite times. both have infinite segments, but one has larger segments.
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north

1
Tue 31 May, 2011 07:01 pm
@hamilton,
hamilton wrote:

here's something I've thought of (no books!). take a line segment. cut it somewhere not in the center. is one side larger?

yes

Quote:
now divide each segment in half infinite times. both have infinite segments, but one has larger segments.

sure , so your point

contrex

2
Wed 1 Jun, 2011 12:10 am
@hamilton,
hamilton wrote:

here's something I've thought of (no books!). take a line segment. cut it somewhere not in the center. is one side larger? now divide each segment in half infinite times. both have infinite segments, but one has larger segments.

Did you really think of this yourself? or read about it in a book? I think the latter, given the number of dumb posts you keep making.

hamilton

1
Wed 1 Jun, 2011 06:07 am
@north,
none, really, besides whether this was right or not.
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hamilton

1
Wed 1 Jun, 2011 06:08 am
@contrex,
gee thanks. being bad at history is not being bad at math.
0 Replies

Fil Albuquerque

1
Wed 1 Jun, 2011 08:27 am
@contrex,
According to maths there are infinity's larger then others...just ask the "Omega man"....
raprap

1
Wed 1 Jun, 2011 09:18 am
@hamilton,
Cardinality and limits.

Cardinality is a measure of the number of elements of the two sets of line segments http://planetmath.org/encyclopedia/Cardinality.html. By your description the cardinality of the two sets is the same.

The segment limit http://en.wikipedia.org/wiki/Limit_(mathematics) is effectively the length of the sub-segments after being subdivided an infinite number of times.

let l1 & l2 be the length of each line segment and 2^n be the number of divisions.

then the limiting length of each sub-segment L1 & L2 is given by

lim L1=l1/2^n & lim L2=l2/2^n

so the limit of L1 & L2 as n goes to infinity is 0 (zero)

As the cardinality (# of segments) of the two segments is the same and the limit of the elements is the same both are equivalent. The major difference between the two sets is the rate convergence; however, that does not alter the equivalence of the sets.

Good observation though, one that isn't intuitively obvious.

Rap
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raprap

1
Wed 1 Jun, 2011 09:20 am
@Fil Albuquerque,
Not all infinite sets have the same cardinality (# of elements)

Rap
hamilton

1
Wed 1 Jun, 2011 11:29 am
@raprap,
this is a theoretical line segment im talking about.
raprap

1
Wed 1 Jun, 2011 11:38 am
@hamilton,
All line segments are theoretical. It still has to do with a one-to-one mapping. If the cardinality of the sets are the same, and from your description they are, the mapping is one-to-one

Rap
hamilton

1
Wed 1 Jun, 2011 11:44 am
@raprap,
i dont understand. could you explain that?
(i really need to read more books, i guess...)(please dont comment on that last thought, because the only responce is probably along the lines of 'duh'.)
raprap Selected Answer

2
Wed 1 Jun, 2011 12:20 pm
@hamilton,
I guess it has more to do with sets. If A & B are sets and A is {2,4,5} and B is {10,11,12} then both have 3 elements and a cardinality of 3. By a one-to-one mapping you can define two functions f and g that map all elements in A to B (f(2)=10 and so forth), g is the inverse (g(10)=2).

With your two sets of different length line segments they've the same cardinality (infinity) and you can define a function and an inverse to go from one segment to the other and back. Now for the length, infinity is a odd number (like zero) it really has special properties, it is neither prime or composite, dividing by infinity isn't defined, and multiplying by infinity is infinity. However, limits do give some clues.

For instance the limit of any number divided by n as n goes toward infinity approaches zero. So the limit of 8/n=800/n both go to zero as n goes to infinity. This is what is happening to your two line segments when they are subdivided an infinite number of times, the length goes to zero and they're the same. Interestingly if you were to divide the line segments by any very large but finite number you're conjecture would be absolutely correct. Infinity is something else.

BTW the inequality of the cardinality of infinite sets comes from Cantor who proved that there was not a one to one mapping between the infinite sets or rational numbers and real numbers.

Rap
hamilton

1
Wed 1 Jun, 2011 06:20 pm
@raprap,
ok... thanks. i understand now.
0 Replies

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