that the square root of 2 is irrational?

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

$\sqrt{2}$

(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?

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

http://www.cut-the-knot.org/proofs/sq_root.shtml

Cheers,
Mike

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.

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!
http://mathforum.org/library/drmath/view/55839.html
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!).

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

All the best,
Joao

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.