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Show that these numbers are irrational

  1. Oct 3, 2013 #1
    Greetings ,
    Im taking an online course on mathematical thinking, and this question has me stumped.

    r is irrational:

    Show that r+3 is irrational

    Show that 5r is irrational

    Show that the square root of r is irrational.

    Im sorry if i posted this in the wrong forum, but im not sure which category these questions fall under. I seriously dont even know where to start.

    Btw these questions are under the proofs section of the course.
  2. jcsd
  3. Oct 3, 2013 #2


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    Use "indirect proof". If r+ 3= x, a rational number, what can you say about x- 3? If 5r= x is a rational number, what can you say about x/5? If [itex]\sqrt{r}= x[/itex] is a rational number, what can you say about [itex]x^2[/itex]?
  4. Oct 3, 2013 #3
    Is it valid to say that a number is rational XOR irrational. My apologies as it over a decade since i attended high school. I also have only recently been doing proofs and so far only completed one proof in my life , so i am not very good at proofs.
  5. Oct 3, 2013 #4
    Do you mean proof by contradiction ( supposing the negation of the statement , then disproving the negation) ?
    We have just learned this type of proof in the last lecture

    If so, would (there exists r)(for any p )(for any q)[(r not equals to p/q) ^
    ( p not equals to q)] , and ( theres exist x[x=r+3]) implies (for any x )(there exists a )(there exists b)[(x=a/b) ^
    (p not equals to q) ]be a first step , then i negate the above
    and try to disprove the negation, and thus prove the original statement?
    Last edited: Oct 3, 2013
  6. Oct 3, 2013 #5


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    Yes, proof by contradiction. And yes, a number is either rational or irrational but not both - but that should be pretty clear from the definitions isn't it?
  7. Oct 3, 2013 #6
    Well im not sure that there could be other types of number other than rational or irrational, which is why i asked. In other words im not sure that (~rational implies irrational). Is it valid to say that ~rational if and only if irrational and
    ~irrational if and only if rational? If i appear dense, guilty as charged lol as its been some time since high school.
    Last edited: Oct 3, 2013
  8. Oct 3, 2013 #7


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    A rational number is a number that can be expressed as the quotient of two integers. An irrational number is one that can't. I would say the definition of irrational is ~rational.
  9. Oct 3, 2013 #8
    Thanks Dick , your help is much appreciated.
  10. Oct 5, 2013 #9
    Is this proof valid? I seen this proof from my study group in this course:

    if r + 3 is rational, then r is rational
    assume r + 3 is rational. then r + 3 can be expressed as follows
    where p and q are integers
    subtract 3 from both sides
    p - 3q (since q is an integer) as well as q is an integer, therefore r is rational
    Contrapositive proof: if r is irrational, then r + 3 is irrational (~ (r+3 is rational) implies ~(r is rational)).
  11. Oct 5, 2013 #10
    Is this proof valid?

    Suppose 5r is rational
    5r is p/q (p and q are intergers with no common factors)
    r= p/5q
    p and q are integers and 5 is an integer , and 5q is an integer since integers multiplied by intergers are always integers.
    Since the r= integers/ integers , r is a rational number.

    5r is rational implies r is rational.

    Contrapositive of this is:

    r is not rational implies 5r is not rational.
  12. Oct 5, 2013 #11


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    Yes, those look fine.
  13. Oct 6, 2013 #12
    Thanks Dick and Hallsofivy.
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