Ice_Phoenix87 Posted October 23, 2004 Posted October 23, 2004 is the numbers between 0 and 1 infinite?
Callipygous Posted October 23, 2004 Posted October 23, 2004 because. there is .1 and .01 and .001 and .0001... and .2 and .02... there is no limit to how precise you can get.
timo Posted October 23, 2004 Posted October 23, 2004 The amount of integers between 0 and 1 is zero. The amount of rational numbers between 0 and 1 is countable (infinite). The amount of real numbers between 0 and 1 is uncountable (one could say: Even more infinite). There is no "between 0 and 1" for complex numbers unless further specification. The question is not valid for imaginary numbers because 1 isn´t an imaginary number. What numbers do you talk about? I´m drunk.
Callipygous Posted October 23, 2004 Posted October 23, 2004 well... all of those (rational numbers, real numbers, complex, etc.) fall under the catagory of "numbers." so since his question asks about how many numbers between them, its infinite. and whats this tomfoolery about countable vs uncountable?
bloodhound Posted October 23, 2004 Posted October 23, 2004 countable infinity is where you can find a bijection between the set S and the set of natural numbers, or equivalently vice versa. uncountable is when you cannot do that. its easy to show that the rationals are countable. while irrationals are not
timo Posted October 23, 2004 Posted October 23, 2004 Countable and uncountable are mathematical terms describing a 1st approx of the cardinality of a set. A set is countable if there´s on a bijection on a subset of the natural numbers. It´s uncountable if there´s not. Might sound stupid to you but it´s elementary (1st semester) math and also quite important for integration in the sense of Lebesque-integration (sets with a measurement of zero can be left out of the integration - countable sets usually have a measurement of zero) and other scientific areas (Poincare´s theorem about the return of orbits would come to my mind, spontaneously). Still drunk
Callipygous Posted October 23, 2004 Posted October 23, 2004 "still drunk" and still managing to put that post together? impressive. "Might sound stupid to you but it´s elementary (1st semester) math " actually i just wanted to use the word tomfoolery. but where is that first semester math? you mean in college?
gib65 Posted October 23, 2004 Posted October 23, 2004 proof of countability of rational numbers: All rational numbers can be expressed in the form n/d where n and d are integers. Then you plot a matrix of all the rational numbers such that the columns represent all values for n from 0 to infinity in increasing order, and the rows represent all values for d from 1 to infinity in increasing order (or visa-versa). Then for any arbitrary rational number, there is an entry for it in the matrix. To count them, begin in the corner (0/1) and enumerate diagonally. To take into account the negative numbers, for every positive rational number you count, also count its negative counterpart. Pretty neat, huh?
bloodhound Posted October 23, 2004 Posted October 23, 2004 i think showing irrationals are uncountable is much nearter (Cantor's diagonal method)
matt grime Posted October 23, 2004 Posted October 23, 2004 Countability is taught in the first term at both Cambridge and Bristol Universities to mathematicians. Surely the most elegant proofs are the simplest: send a/b to 3^a.5^b if it is a positive rational 2.3^a.5^b if it is negative. Clearly this shows the rationals are countable, and is a very powerful technique that works on many such 'show something is countable' questions.
MolecularMan14 Posted October 23, 2004 Posted October 23, 2004 .1 .11 .111 .1111 .11111 .111111 ~~~~~~~~~infinite You get the picture
bloodhound Posted October 23, 2004 Posted October 23, 2004 those are only rationals, and they are countable
Guest Doron Shadmi Posted November 4, 2004 Posted November 4, 2004 From a non-standard point of view about the Infinite concept, we first have to distinguish between two kinds of this concept, which are: 1) Actual Infinity. 2) Potential Infinity. Please look at http://www.geocities.com/complementarytheory/RiemannsLimits.pdf There are indeed (from the abstract point of view) infinitely many levels of infinitely many collections of elements. But I wish to add that according to my point of view Alef0+1 > Alef0, 2^Alef0 < 3^Alef0, etc... To understand why I do not accept the Cantorian point of view about infinitely many elements, please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf http://www.geocities.com/complementarytheory/Anyx.pdf http://www.geocities.com/complementarytheory/9999.pdf http://www.geocities.com/complementarytheory/Russell1.pdf http://www.geocities.com/complementarytheory/No-Naive-Math.pdf Thank you.
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