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

MATH312 December 2009

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Question 06 (a)
Show that $\displaystyle {\sqrt {3}}+{\sqrt {2}}$ 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 $\displaystyle {\sqrt {6}}$ 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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Assume towards a contradiction that $\displaystyle {\sqrt {3}}+{\sqrt {2}}={\frac {p}{q}}$ where $\displaystyle \gcd(p,q)=1$. Now, squaring both sides yields $\displaystyle 3+2{\sqrt {6}}+2={\frac {p^{2}}{q^{2}}}$ and isolating for the radical yields $\displaystyle {\sqrt {6}}={\frac {p^{2}-5q^{2}}{2q^{2}}}$ Now the right hand side is rational so now it suffices to show that $\displaystyle {\sqrt {6}}$ is irrational (this will simplify the computations to treat this as its own case). Suppose that $\displaystyle {\sqrt {6}}={\frac {a}{b}}$ where $\displaystyle \gcd(a,b)=1$. Squaring both sides and cross multiplying yields $\displaystyle 6b^{2}=a^{2}$ Now, 2 divides the left hand side so $\displaystyle 2\mid a$ writing $\displaystyle a=2A$ and substituting yields $\displaystyle 6b^{2}=4A^{2}$. Dividing by 2 yields $\displaystyle 3b^{2}=2A^{2}$. Now, 2 divides the right hand side so $\displaystyle 2\mid b$. This contradicts the fact that $\displaystyle \gcd(a,b)=1$ since 2 divides both numbers. Thus $\displaystyle {\sqrt {6}}$ is irrational and hence so is $\displaystyle {\sqrt {3}}+{\sqrt {2}}$ by the above argument as required.

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