Question

Problem VI.(15 pts.) Suppose that is an irrational number. 1. Prove that j + cannot be a rational number 9 with gl < 2. 2. Can j + be a rational number whose absolute value is greater than 2? Why or why not?

Answer 1

If possible q = 1 + 1 when & u Irrational lal = 0+1 = 22+I +2 2 e Wow. Al the metic Meany, Geometare mean If j2 + 1 = 2 =) (8² 1,² = 0 => a = + which is Contradiction = get 1 / 2 2 2 => 1912 = 2² +1 +27 2+2=4 => 19 L q= m gcd (min) = L. If a s e al conal m = G²+1) = (1 ²+1) K = m na ki - Now gcd (min) = 1 => K=1

=> DT. D 6+1) = m I UN j = nett But which in huna W um her In other wes Irrational Contradiction e Can't be eational word it is noi a ational Number q with lqca Ho As we seen itt is always Irrational de Irresked tive és absolute Same