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 October 27th, 2017, 09:54 AM #1 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 Square root of multiplication of 2 primes is irrational **Before you start reading I must say I don't know how to use math symbols so sorry if it's really messy, and if you have a guide somewhere on how to write maths here I'd appreciate ------------------------------------------------------------------------------- I'm trying to prove that sqrt(ab) is irrational, if a and b are different primes Using contradiction: Assume sqrt is rational: sqrt(ab) = x/y, where x/y is irreducible fraction then I do power of 2 to both sides to get: ab = (x^2)/(y^2), then: (y^2)*ab=x^2, so ab is divisor of x^2, and thus divisor of x. so there is a k that is k*ab = x, put that back in: (y^2)*ab=(k*ab)^2, divide both sides by (ab): (y^2)=(k^2)*ab, so ab is also divisor of y^2, and also of y! So we get that ab is divisor of x and y, contradiction to the assumption that x/y is irreducible fraction. Q.E.D ------------------------------------------------------------------------------- Is this OK? October 27th, 2017, 10:23 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,942 Thanks: 2210 Why does "ab is divisor of x²" imply that ab is a divisor of x? Thanks from Mathmatizer October 27th, 2017, 10:29 AM #3 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 I was thinking about Euclid's lemma but not sure if it applies here too because this lemma speaks of a single prime number and here we have multiplication of 2 primes. So I guess I am wrong  October 27th, 2017, 10:41 AM #4 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 Sorry, I was using Euclid''s lemma wrong! October 28th, 2017, 05:29 AM #5 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 Anyone? I can't find out October 28th, 2017, 08:13 AM #6 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond I've seen proofs that the square root of any non-negative integer that isn't itself a perfect square is irrational. Unfortunately, I can't recall it but I don't think they were particularly difficult. Can you prove that the square root of 2 is irrational and then extend that to all non-square integers? October 28th, 2017, 10:31 AM #7 Newbie   Joined: Oct 2017 From: Here Posts: 19 Thanks: 0 I think I did it finally  Tags irrational, multiplication, primes, root, square Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Gigabitten Real Analysis 4 May 11th, 2017 11:20 AM Spook Math 12 January 30th, 2015 11:42 AM shunya Elementary Math 4 July 2nd, 2014 06:21 AM shunya Elementary Math 2 March 20th, 2014 04:06 AM

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