Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Rational and irrational numbers

  1. Dec 23, 2014 #1
    Are numbers which tend to an integer for example 1, that is 0.9999... And 1.00....01 irrational?

    [where I got the doubt: There was a question in my textbook--->
    $$f(x) = x^2+ax+1 \text{ if x is rational}$$
    $$f(x) = ax^2+bx+1 \text{ if x is irrational}$$
    If f(x) is continuous at x=1 and 2 find a and b.
    Why I had the doubt: Since if numbers that tend to 1 and 2 are irrational then 1 and 2 are the roots of the equation ##x^2+ax+1=ax^2+bx+1##]
     
  2. jcsd
  3. Dec 23, 2014 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Your question makes no sense. 0.999... by which I assume you mean "the limit of the sequence 0.9, 0.99, 0.999, ..." is 1. There is NO such number as [itex]0.000...1[/itex] because you cannot have a number with an "infinite number of 0s ended with a 1"- and infinite sequence has NO end. If you mean, as you should, "the limit of the sequence 0.01, 0.001, 0.0001, ...", then that number is 0. No, 1 and 0 are NOT irrational.

    But I have no idea what you mean by "if numbers that tend to 1 and 2 are irrational". Given any number, whether rational or irrational, there exist sequences of rational number and sequences or irrational number that converge to it.
     
  4. Dec 23, 2014 #3

    lavinia

    User Avatar
    Science Advisor

    Every number is the limit of both rationals and irrationals.

    What then does continuity at 1and 2 tell you then about the two expressions for f at 1and 2?
     
    Last edited: Dec 23, 2014
  5. Dec 23, 2014 #4
    Il rather post the question in the homework forum. Maybe I'm totally wrong. You can close the thread.
     
  6. Dec 23, 2014 #5

    lavinia

    User Avatar
    Science Advisor

    Think about my reply.
     
  7. Dec 24, 2014 #6
    All repeating/terminating decimals are rational.
    Also, as pointed out earlier, 0.99999999999..... is equal to 1.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Rational and irrational numbers
Loading...