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October 12th, 2014, 12:32 AM  #1 
Newbie Joined: May 2014 From: Singapore Posts: 11 Thanks: 0  Exercises on irrational numbers
Dear all, I have done question 1 of exercise 2.1 from the book Alan F beardon, Abstract Algebra and Geometry. Please answer some of my doubts. Q1. a) Show that √(2/3) is irrational. b) Use the prime factorization of integers to show that if √p/q is rational, where p and q are positive integers with no common factors, then p = r^2 and q = s^2 for some integers r and s. ANS: a) √2 is known to be irrational √3 is irrational because 3 is a prime number and its only factors are 1 and 3?? or it can be proved. p/q is the simplest fraction after cancelling common factors. p and q are integers. √3=p/q 3=p^2q^2 3∗q^2=p^2 Please check if my observation below is correct?? Observation: the multiple is 3 on LHS so if q is odd then p will also be odd. if q is even p will also be even. if both p and q were even numbers then p and q are not in their simplest terms therefore it violates √3 being a rational number. because √ of any number could be written as p/q but in simplest terms. simplest terms means both cannot be even and both cannot be odd. both can be odd if they are prime numbers e.g 7/3 or 9/5 etc.. In the above case, if q is odd (integer or prime number) p will be a odd number also with a factor 3 (because of the multiple 3) therefore it will not be in its simplest form. Therefore, √3 is an irrational number?? √2/3 is irrational because √2 and √3 are irrational!! but how to prove that an irrational number divided by another irrational number is also irrational???? b) Please check my answer to the second part of the question??? I have written in words can anyone translate to math notation?? Thanks.. square root of a prime number is always irrational because primes only have factors 1 and themself. Any integer number can be broken down into its prime factors. After cancelling all common primes in numerator and denominator. we are left with noncommon primes in numerator and denominator. As square root of prime number is always irrational. the irrational numerator and denominator produce an irrational outcome. so numerator p and denominator q should be squared numbers to produce integer fractions (after square root operation) which are rational. 
October 12th, 2014, 04:15 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
You have the right method. Just write $$\frac23 = \frac{p}{q}$$ 

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