Science:Math Exam Resources/Courses/MATH312/December 2009/Question 06 (a)
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Question 06 (a) 

Show that is irrational. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Proceed by a contradiction. Suppose that the given number is rational. Square both side and simplify. 
Hint 2 

It might help to as a lemma prove that is irrational. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Assume towards a contradiction that
where . Now, squaring both sides yields
and isolating for the radical yields
Now the right hand side is rational so now it suffices to show that is irrational (this will simplify the computations to treat this as its own case). Suppose that
where . Squaring both sides and cross multiplying yields
Now, 2 divides the left hand side so writing and substituting yields . Dividing by 2 yields . Now, 2 divides the right hand side so . This contradicts the fact that since 2 divides both numbers. Thus is irrational and hence so is by the above argument as required. 