Proving a number is irrational.

  • Thread starter cragar
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  • #1
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Homework Statement


Prove that [itex] log_2(3) [/itex] is irrational.

The Attempt at a Solution



This is also equivalent to [itex] 2^x=3 [/itex] from the definition of logs.
Proof: For the sake of contradiction lets assume that x is rational and that their exists integers P and Q such that x=P/Q .
so now we have [itex] 2^{\frac{P}{Q}}=3 [/itex]
now I will take both sides to the Q power .
so now we have [itex] 2^P=3^Q [/itex]
since P and Q are integers, there is no possible way to have 2 raised to an integer to equal 3 raised to an integer, because 2^P will always be even and 3^Q will always be odd. so this is a contradiction and therefore x is irrational.
 

Answers and Replies

  • #2
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Looks good :)
 
  • #3
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sweet ok , I'm new to writing proofs so just want some confirmation.
 
  • #4
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I can't imagine that you would lose points for this, but for the sake of pedantry you might want to point out that P and Q would have to both be positive integers. Just because 2^0=3^0 and 2^P, 3^Q aren't even and odd respectively when P and Q are negative.
 

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