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Is there a direct proof…

February 6, 2008 pm29 4:24 pm

that the square root of 2 is irrational?

A kid asked, and I wasn’t sure, but didn’t think so.


(we did the standard proof by contradiction. Assume rational. write as p/q, (p,q) = 1, square it and show that 2|p. Then show 2|q. Contradiction…)

Anyone know off hand?

9 Comments leave one →
  1. February 6, 2008 pm29 7:32 pm 7:32 pm

    I haven’t had time to read through them all but does one of the proofs here give you what you need? A quick scan indicates that most of them rely on a contradiction though


  2. Brent permalink
    February 6, 2008 pm29 7:39 pm 7:39 pm

    I’m going to guess the answer is “no”. Since irrationality itself is defined indirectly (an irrational is any number which ISN’T rational) I’m not sure what a direct proof would even look like. But it probably depends on what your definition of a “direct” proof is, and what you think counts as a proof of sqrt(2)’s irrationality.

  3. Joshua Zucker permalink
    February 6, 2008 pm29 8:16 pm 8:16 pm

    There’s a nice proof by paper folding in Conway/Guy _Book of Numbers_. Depending on your definition of irrational, you might consider it a direct proof. But in some sense it’s the usual proof-by-contradiction infinite descent kind of thing: If there’s a fraction with smallest denominator, then there’s still one with a smaller denominator.

    I think I agree with the above, that there’s unlikely to be any really nice direct proof of irrationality without knowing a better definition of irrational.

    Maybe “has a nonterminating continued fraction” could be the definition of irrational? And then finding the CF for sqrt(2) suffices.

  4. February 9, 2008 pm29 5:43 pm 5:43 pm

    A google search for “direct proof irrational” provides (in hit #2, no less) a link to the Ask Dr Math thread on this very question.

  5. February 9, 2008 pm29 9:29 pm 9:29 pm

    Thank you!
    Each example has a bit of an indirect flavor… It looks like the answer should probably be “no” but with some room for interpretation.

    I reported what Brent and Joshua suggested, along with the comment by Mike, to my class. Didn’t think of just googling it, though.

    I like when I need to tell my class “I don’t know” and then tell them that I found an answer.

    The authority belongs to the mathematics, not to me.

  6. February 11, 2008 am29 4:13 am 4:13 am

    Dear jd2718,

    As you may have noticed, I kinda of replied to your post in my blog. Sorry for doing that, but it would be difficult to write here all the details (especially the LaTeX bits!).

    The direct link is:

    I found your blog very recently, but I’ll definitely follow it. I’ll link it from mine as well!

    All the best,

  7. February 11, 2008 am29 4:30 am 4:30 am

    Very slick answer, well worth paying João a visit. Warning, not for the faint of math, but not overwhelming for the math-inclined either.

    I need to update my blogroll but João, a Portuguese PhD candidate at Nottingham, and definitely Johnny on the spot, has earned a … spot.

  8. February 11, 2008 pm29 1:55 pm 1:55 pm

    Thanks Jonathan! I’m glad you appreciate it :-) We’ll keep in touch.


  1. Direct proofs : Joao Ferreira

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