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

What number is rational and irrational

  1. Aug 30, 2004 #1
    When is x rational and irrational?

    Also When is r positive and negative? :confused:
     
  2. jcsd
  3. Aug 30, 2004 #2

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    By definition an irrational number is one that is not rational. Therefore, it is not possible for a number to both rational and irrational. If you're thinking of the number zero (0) then you're mistaken since 0 is clearly rational - it can be expressed as a ratio of whole numbers.

    Also, a positive number is one that is greater than zero and a negative number is one that is less than 0. The only alternative to those two conditions is a number that is neither greater than zero nor less than zero - which is zero itself and it is neither positive nor negative.
     
  4. Aug 31, 2004 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The definition of "rational number" is that it can be written as a fraction.
    (1/2, 3/5, 2/3, etc.)
    It is also true that rational numbers are those that, in decimal notation, are either terminating or repeating. (1/2= 0.5 is terminating, 3/5= 0.6 is terminating, 2/3= 0.66666... is repeating)

    An irrational number is any number that is NOT rational. [itex]\sqrt{2}[/itex] for example, as well as [itex]\pi[/itex]. Irrational numbers tend to be written in "funny" ways exactly because they are not as simple to write as rational numbers.

    Are you really asking what a negative number is? That seems very elementary for someone who is also working with rational and irrational numbers. About the best answer is that positive numbers are greater than 0 and negative numbers are less that 0. That negative sign in front of a number is a pretty good clue!

    Since you specifically said "when is r positive or negative", I wonder if you are not thinking of polar coordinates. If that is the case the answer is easy: r is never negative, it can only be 0 or positive.

    I just realized that the question was NOT "what numbers are rational or irrational" or "what numbers are positive or negative" as I must have thought when I wrote this.

    Hurkyl is completely correct- there are NO numbers that are both rational and irrational and there are NO numbers that are both positive and negative.
     
    Last edited: Mar 7, 2010
  5. Aug 31, 2004 #4

    Alkatran

    User Avatar
    Science Advisor
    Homework Helper

    A = not A?
     
  6. Aug 31, 2004 #5

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Tina (welcome to PF, BTW!),
    It would be best if you give the context of your questions:

    Is it for example that you wonder:
    What can I assume in some calculation/proof, and what must I prove?
    Or just: how do I KNOW this?
     
  7. Aug 31, 2004 #6
    Aren't all irrational numbers just sequences of rational numbers?
     
  8. Aug 31, 2004 #7

    Alkatran

    User Avatar
    Science Advisor
    Homework Helper

    A sequence of rational numbers would be rational.

    1 then 2 -> 1.2
     
  9. Aug 31, 2004 #8
    Not all the time. The real numbers have the property that they are dense, i.e. for any real number a there is a sequence of rational numbers {r_n} so r_n--->a. So say for the irrational number pi

    r_1=3.0
    r_2=3.1
    r_3=3.14
    r_4=3.141

    and so on you get the idea. If a is irrational you can just choose r_n to be the rational numbers of the first n terms of the decimal expansion of a followed by zeroes. If r has its decimal expansion that agrees with the expansion of a to the mth place then the number differs from a less than 10^-m. So obviously the sequence of rationals {r_n} converges to a.
     
  10. Aug 31, 2004 #9

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Anyways, you're missing a subtlety.

    Any irrational number is equal to the limit of some sequence of rational numbers.


    Except for a few particular models of the real numbers, it is incorrect to say that an irrational number is a sequence of rational numbers.


    Also a correction:

    You meant to say

    The rational numbers are a dense subset of the real numbers.

    Or more compactly,

    The rationals are dense in the reals.
     
    Last edited: Aug 31, 2004
  11. Aug 31, 2004 #10
    Is there a proof of this somewhere?
     
  12. Aug 31, 2004 #11

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Yes. Gravenworld's last post (#8) is a good demonstration of this fact.

    A rigorous proof would require use of the completeness axiom, though.
     
    Last edited: Aug 31, 2004
  13. Aug 31, 2004 #12
    Ah yes thanks for catching my mistake. The rational numbers are dense in the real numbers.
     
  14. Sep 1, 2004 #13

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Ooops! I will edit that. Thanks.
     
  15. Sep 1, 2004 #14

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Hmm, couldn't edit- the post is too old. Of course, what I meant was that "an irrational number is any number that is not RATIONAL".

    By the way, the fact that, for any real number there exist a sequence of rationals that converge to it can be used as a definition of "real number".

    Let X be the set of all increasing, bounded, sequences of rational numbers (convergent or not). We say that two such sequences, {an} and {bn}, are equivalent if and only if the sequence {an- bn} converges to 0. It is easy to show that that is an "equivalence relation" and so partitions the set X into equivalence classes. We identify the real numbers with those equivalence classes (addition, multiplication are defined by using "representative" sequences). That definition makes it easy to prove that any increasing bounded sequence of real numbers converges to a real number and the "completeness axiom" follows from that.
     
  16. Sep 1, 2004 #15
    ... contrary to the rational numbers, the real numbers are dense in themselves :laughing:

    But when it goes to :
    ... the real numbers are not compact in themselves :tongue2:

    Sorry about that :blushing:
     
  17. Sep 1, 2004 #16

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I beg to differ; I think the rationals are also dense in themselves!
     
  18. Sep 1, 2004 #17
    I was just joking. I thought a discrete set cannot be dense at all, because all the subset of a discrete set are open, thus no subset can be closed, thus no subset can be dense. Now that I think about it seriously, I think you must be right. Is it true that any set is dense in itself ? What's the definite answer ! :confused: :uhh:

    Just had a thought : the fact that Q is (partially) ordered. It is sufficient to show that for any two rationals, another lies between them. Q is dense in itself. I do not delete the begining, since I would like to know what happens in the case of a set without partial order.
     
  19. Sep 1, 2004 #18

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Being dense is a topological property...

    Of course, any totally ordered space has a natural topology that respects the ordering (let the neighborhoods be open intervals), so it makes sense to speak of being dense in an ordered set.

    A set S is dense in a topological space T if the closure of S is all of T. So, clearly, T is dense in T.


    The rational numbers is not a discrete space: the property it lacks is being complete.

    Also, there's nothing stopping a set from being both open an closed. For any topological space, both the empty set and the whole thing are both open and closed. And, in general, any set that is isolated from the rest of the space (such as any set in a discrete topology) is both open and closed.



    As an addendum, the term "dense order" means something else entirely: an ordering is dense if, for any A and B with A < B there is a C so that A < C < B.


    edit: changed an erroneous use of the word "set" to "space"
     
    Last edited: Sep 2, 2004
  20. Sep 2, 2004 #19
    Thanks Hurkyl ! Much clearer in my head now... I should never have quit math for physics :cry:
     
  21. Mar 7, 2010 #20
    So any real number is the limit of a sequence of rationals. Is any real number also a limit of a sequence of irrationals? I found this proof of the latter statement...

    http://answers.yahoo.com/question/index?qid=20080921221913AAsHFAp

    it seems contradictory to have that every real number is both the limit of a sequences of rationals as well as the limit of a sequence of irrationals.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: What number is rational and irrational
Loading...