Math 117: Honours Calculus I Fall, 2013 Assignment 1 September 6, due September 18 1. Prove that √ 3 is irrational. Suppose that there existed integers p and q such that p 2 = 3 q 2 . Without loss of generality we may assume that p and q are are not both divisible by 3 (otherwise we could cancel out the common factor of 3). We note that p 2 is divisible by 3. Express p = 3 n + r where r = 0, 1, or 2. Then p 2 = 9 n 2 + 6 nr + r 2 . If r = 1 or r = 2, then p 2 is not a multiple of 3. The only way that p 2 can be divisible by 3 is if p is itself a multiple of 3. (Alternatively, consider the prime factorization of p . Since 3 is prime, the only way it can be a factor of p 2 is if it is also a factor of p .) Hence 9 n 2 = 3 q 2 , or 3 n 2 = q 2 . Replacing p by q in the above argument, we see that q is also divisible by 3. This contradicts the fact that p and q are not both divisible by 3. 2. Let r and s be rational numbers. (a) Is r + s necessarily a rational number? (Prove or provide a counterexample.) Yes, let r = p/q and s = m/n , where p,m ∈ Z and q,n ∈ N ; we may always write p q + m n = pn + mq qn . (b) Is r − s necessarily a rational number? Yes, we may always write p q − m n = pn − mq qn . (c) Is rs necessarily a rational number? Yes, we may always write p q m n = pm qn . (d) Is r/s necessarily a rational number? No, consider r = 1, s = 0. There is no rational number r/s . 3. Let x and y be irrational numbers. Prove or provide a counterexample: (a) Is x + y necessarily an irrational number?