318 /www/nrich/html/content/00/06/15plus3/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><p>Prove that if $a$ is an integer and not a square number then $\sqrt{a}$ is irrational.</p> <p> </p> <div class="framework">NOTES AND BACKGROUND<br></br> If you have seen a proof that the square root of 2 is irrational the interactivity <a href="http://nrich.maths.org/public/viewer.php?obj_id=1404&amp;part=index">Proof Sorter</a> may help you to understand the proof better. You can look this proof up in textbooks written for students in their last year of school mathematics or first year of university mathematics.</div> <br></br> <br></br> <br></br></mdoxml> <mdoxml version="1.0"><br></br> Good solutions were submitted by Hyeyoun from St Paul's Girls' School, London, Michal from Daramalan College, and Ling Xiang Ning from Raffles Institution, Singapore. <p>This is Hyeyoun's solution.</p> <p>Suppose that $\sqrt{a}$ is rational. Therefore, where $x$ and $y$ are coprime integers and $y \ne 0$, we have \[ a = \frac{x^2}{y^2} \]</p> <p>We can always write the rational number so that x and y are coprime, that is they have no common factors except 1 so, as $\sqrt a$ is rational, it follows that $x$ and $y$ are coprime.</p> <p>If $a$ is an integer, $y^2$ must be a factor of $x^2$ and so $y=1$.Therefore $a$ is a square number.</p> <p>It follows (as the contrapositive) that if $a$ is an integer and is not a square number then $\sqrt{a}$ is irrational.</p> <br></br></mdoxml> <mdoxml version="1.0"><span style="font-weight: bold;">Why do this problem?</span><br></br> It is instructive as an example of proof using the contrapositive of the statement that must be proved.<br></br> <br></br> <span style="font-weight: bold;">Possible approach</span><br></br> The class might discuss the proof in pairs before having a class discussion of the proof. It is very short but needs to be carefully argued.<br></br> <br></br> <span style="font-weight: bold;">Key question</span><br></br> What if the statement we are trying to prove were not true?<br></br> <br></br> <span style="font-weight: bold;">Possible support</span><br></br> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1404&amp;part=index"> Proof Sorter</a><br></br> <br></br> <br></br> <br></br> <br></br></mdoxml> <mdoxml version="1.0">What if the statement you are trying to prove is not true?<br></br></mdoxml> 2 3 0 0 0 0 1 Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.) Numbers and the Number System Rational and irrational numbers Numbers and the Number System Square numbers Using, Applying and Reasoning about Mathematics Theorem and contrapositive Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Numbers and the Number System Powers & roots