Although there are an infinite number of points within a one inch line?

Question

Is it not obvious that is an incomparably greater amount of space between those points? Obviously there are an infinite number of rational numbers between zero and one--1/2, 1/4, 1/8..., 1/3, 2/9, 457/2397, etc.. They say there are even many more irrational numbers on the real number line as well. Here's my point though. We know that any finite number divided by infinity is equal to zero. So, if you divide a one inch line into an infinite number of parts, each part will have zero length. And an infinite number of zeros added together equals zero. In other words dividing any finite thing by infinity results in its no longer existing. So, those infinite number of cardinal points between zero and one actually don't take up any of the space in that real one inch line, because each of those cardinal points has the dimensions of an abstract geometric point which has no length, width, or depth. Therefore since althought there are an infinite number of cardinal points in a one inch line, they are actually not occupying any of the space of the real line. So, how much more real dimension is there in a one inch line than the sum total of the infinite number of zero dimensional cardinal points added together? You can't compare to zero, because you can't divide by zero. Seems paradoxical in a way, doesn't it?

Answer 1

Actually, this sounds like one version of Zeno's paradox.



http://en.wikipedia.org/wiki/Zeno's_paradoxes



To me, though, all you've shown is that length isn't captured by the points in the length.



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@jman: Two things:

First off, if we consider cardinality, then there definitely is more irrationals than rationals. This just follows by cardinal arithmetic.

Second, the statement that "every number between two rationals is irrational" is far from true.

Answer 2

Infinity divided by infinity is not always 0. There are not technically "more" irrational numbers than rational, but rather the irrational numbers are more "dense". This can be seen because all rational numbers can be written as p/q were p and q are integers.Every number in between two rational numbers is therefore irrational.



Example of inf/inf not being equal to 0:

Limit as x approaches infinity of x/(2x) is equal to one half (Even though it is inf/inf)

Answer 3

in case you have been to devise the size of the line as against the type of divisions, you're able to discover the graph to be asymptotic with the size of the line = 0 at limitless divisions. i think of the challenge you're confronted with is which you're quantifying infinity as a mathematical quantity. Infinity is a mathematical theory that has no length or fee. Infinity is a theory because of the fact it could by no skill be almost reached, it rather is why for sensible applications, in case you preserve dividing the only-inch line in 0.5, you will continuously have a finite denominator. besides the incontrovertible fact that, in case you enable the size of the line be equivalent to a million/(sort of divisions) and additionally you enable the type of divisions to be limitless, the size of the line could be 0. besides the incontrovertible fact that, multiplying a million/(infinity) by utilising infinity would not recommend that the size of the line would be a million. when you consider which you're multiplying an indeterminate mathematical theory by utilising yet another indeterminate mathematical theory, there is not any thank you to make certain what the fee is different than an indeterminate mathematical theory.