Question

Homework Problems Problem 1.10. The fact that that there are irrational numbers a, b such that a' is rational was proved in Problem 1.5. Unfortunately, that proof was non constructive it didn't reveal a specific pair, a, b, with this property. But in fact, it's easy to do this: let a ::= V2 and b::=2 log23. We know 2 is irrational, and obviously ab = 3. Finish the proof that this a, b pair works, by showing that 2 log 3 is irrational. 1.10. GOOD PROOFS IN PRACTICE

Answer 1

Solution - let's go in for contradiction. Let's assume that a log23 is rational then 2 log 3 can be unritter in the form of a where P and q chave no other common factor than 1. -> 200g 3= a where a and b are cope - 2 reg2 b integers zblog, 3 = A must be true (26) - ä must be true 29 = 32b must be true. but we can see that L.H.S is an enen number as ato and Ronis is an odd no. for byo as both a and b are coprime integers hence enen number cannot be equal to odd number hence our conta assumption was wrong therefore 2 log23 is an virational number

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