1

Being a high school student, It's obvious to me that there are both an infinite number of rational and irrational numbers. However I don't really see if there is more rational than irrational, irrational than rational, or if they are of the same percentage of the reals. All I know is between two rationals there is a irrational, and between two irrational a there is a rational, along with my experience that I usually work more with rationals. How would we calculate the percentage of irrationals and the percentage of rationals to the reals. And how could some infinities be larger than others, doesn't seem to make sense to me.

Ahmed S. Attaalla
  • 17,875
  • 4
  • 38
  • 68
  • 1
    This has been discussed many times on MSE, if you type "cardinality irrationals" into the search box you will find plenty of posts to read. If you still have queries after that, please post another question - I'm sure many people will be happy to help. – David Jul 24 '15 at 00:25
  • 1
    Maybe easiest to think about it in terms of decimal expansions. If you built a number between 0 and 1 by randomly choosing digits to make it's decimal, there is no chance at all that you'd get a rational, right? To get a rational you'd need your sequence to start repeating at some point and that has probability 0 (whatever one might mean by that). This suggests that there are a whole lot more irrationals, no? – lulu Jul 24 '15 at 00:26
  • As lulu says, with probability 1, you will choose an irrational number (probability 0 of choosing a rational), thus , you could say 100% of numbers are irrational. 100% or 0% chance of getting something is different than you would normally think when there are infinitely many things. – Mark Jul 24 '15 at 00:30
  • There is an uncountable number of irrationals between two rationals, and a countable number of rationals between two irrationals. – Jonathan Hebert Jul 24 '15 at 00:57

0 Answers0