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In emanuela's post above, about the real numbers being of a higher order of infinity than the integers, it should be noted that the numbers represented by, e.g., 2.999999999... do not exhaustively represent the real numbers. A rational number is one that can be represented as the ratio of two integers. Thus, 0.3333333... can be represented as the ratio of 1 to 3. A feature of the rational numbers is that they can be represented as a repeating decimal, i.e., one in which a pattern of some finite number of digits repeats indefinitely at the end. 1/11=0.090909..., for example and 1/13=0.076923076923076923.... In some cases the repeating digit(s) will be 0, e.g., 1/8=0.12500000... In this case there is a non-repeating string of digits before we get to the repeating part. The repeating part of one of this type doesn't have to be zero; 2/15=0.13333333...
There is another type of real number, the irrational. Examples are pi, the square root of two, and e, the base of the natural logs. These cannot be represented as repeating decimals.
The the order of infinity of the rational numbers is the same as the order of infinity of the integers. Bean or emanuela could prove this for us. Add in the irrational numbers and *that's when you get the higher order of infinity.
This notwithstanding the fact that, as Alex pointed out, between any two rational numbers you can put more rational numbers, even an infinite number of rational numbers; there are no adjacent pairs of rational numbers.
Entire Thread Subject Posted By Posted dr rav 01/21/03 05:27 PM Re: dr wentworth 01/21/03 05:36 PM Re: dr Faldage 01/21/03 05:42 PM Re: dr Wordwind 01/21/03 06:04 PM Re: dr wwh 01/21/03 06:17 PM Re: Dr or Dr. wwh 01/21/03 06:20 PM Re: dr Faldage 01/21/03 06:33 PM Re: dr wwh 01/21/03 06:54 PM Re: dr Faldage 01/21/03 07:23 PM Re: dr wwh 01/21/03 07:43 PM Re: dr AnnaStrophic 01/21/03 11:25 PM Re: dr rav 01/22/03 01:55 PM Re: dr TheFallibleFiend 01/21/03 09:43 PM Re: dr johnjohn 01/21/03 11:25 PM Re: dr wwh 01/22/03 01:34 AM Re: dr Faldage 01/22/03 11:05 AM The RhubarbCommando speaks out for Britain. RhubarbCommando 01/22/03 12:24 PM Re: dr boronia 01/22/03 02:03 PM Post deleted by Wordwind Wordwind 01/22/03 02:10 PM Re: dr rkay 01/23/03 09:19 AM Re: dr Capfka 01/23/03 11:19 AM Re: dr Bean 01/22/03 02:55 PM Re: dr TheFallibleFiend 01/22/03 03:14 PM Re: dr Bean 01/22/03 04:05 PM Re: dr TheFallibleFiend 01/22/03 07:02 PM Re: dr Faldage 01/22/03 07:32 PM Re: dr TheFallibleFiend 01/22/03 08:23 PM Re: dr Faldage 01/22/03 08:29 PM Re: dr Bean 01/23/03 11:34 AM math emanuela 01/28/03 04:54 PM answer emanuela 01/28/03 04:55 PM Re: answer Faldage 01/28/03 06:38 PM Re: answer wwh 01/28/03 06:46 PM Re: answer Faldage 01/28/03 07:08 PM Re: answer wwh 01/28/03 07:45 PM Re: answer Faldage 01/28/03 07:53 PM Re: answer Alex Williams 01/28/03 10:38 PM Re: answer Bean 01/28/03 11:46 PM Re: answer tsuwm 01/29/03 12:11 AM Post deleted by Wordwind Wordwind 01/29/03 12:16 AM e Alex Williams 01/29/03 02:04 AM Re: e Faldage 01/29/03 11:05 AM Re: e Bean 01/29/03 11:10 AM Re: e Bean 01/29/03 11:14 AM Re: e TheFallibleFiend 01/29/03 11:50 AM Re: e Alex Williams 01/29/03 01:48 PM Re: e TheFallibleFiend 01/29/03 02:10 PM Re: e Faldage 01/29/03 02:36 PM Re: e Bean 01/29/03 02:49 PM Re: one = non-one tsuwm 01/29/03 02:58 PM Re: one = non-one Bean 01/29/03 03:12 PM Re: e Faldage 01/29/03 03:16 PM Re: e Bean 01/29/03 03:54 PM Re: e Alex Williams 01/29/03 04:13 PM Re: e tsuwm 01/29/03 04:18 PM Re: e Faldage 01/29/03 04:33 PM Re: e Alex Williams 01/29/03 05:04 PM Re: e Buffalo Shrdlu 01/29/03 05:32 PM Re: e Alex Williams 01/29/03 06:04 PM Re: e Faldage 01/29/03 06:06 PM infinity emanuela 01/30/03 06:36 AM Re: infinity Alex Williams 01/30/03 10:40 AM Re: infinity Bean 01/30/03 11:24 AM Re: numbers real and rational Faldage 01/30/03 12:03 PM Re: numbers real and rational Alex Williams 01/30/03 01:01 PM Re: numbers real and rational emanuela 01/30/03 01:56 PM Re: numbers real and rational Faldage 01/30/03 02:00 PM . Alex Williams 01/30/03 03:22 PM Re: . Faldage 01/30/03 03:31 PM Re: e Bean 01/29/03 04:32 PM Re: e Bean 01/29/03 02:45 PM math is not an exact science wofahulicodoc 01/30/03 03:31 PM numbers emanuela 01/30/03 04:54 PM Re: math Faldage 01/28/03 05:15 PM Re: math wwh 01/28/03 06:00 PM Re: math musick 01/28/03 06:16 PM parallel lines emanuela 01/29/03 01:31 PM 0,99999..... emanuela 01/29/03 01:17 PM Re: dr vika 01/28/03 04:42 PM Re: dr rav 01/22/03 08:51 PM Re: dr modestgoddess 01/23/03 08:32 PM Re: dr Faldage 01/23/03 08:40 PM Re: dr modestgoddess 01/23/03 08:44 PM Re: dr Bean 01/24/03 12:09 AM Re: dr Capfka 01/24/03 08:46 PM Re: dr Faldage 01/27/03 07:56 PM Re: dr AnnaStrophic 01/27/03 07:58 PM Re: duhing … and three dots Faldage 01/27/03 08:08 PM Re: duhing … and three dots wwh 01/27/03 08:15 PM Re: duhing … and three dots Faldage 01/27/03 08:18 PM Re: dr tsuwm 01/27/03 08:04 PM
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