You may have seen a proof that the square root of 2 is irrational and you can look this up in most textbooks written for students in their last year of school mathematics or first year of University mathematics. It is just as easy to prove that if 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>a</mi></mrow></math> is not a square number then  
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><msqrt><mrow><mi>a</mi></mrow></msqrt></mrow></math> is irrational.
<p/>
Prove that if 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>a</mi></mrow></math> is a natural number and  
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><msqrt><mrow><mi>a</mi></mrow></msqrt></mrow></math> is rational then it is a square number (an integer 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow>
</msup>
</mrow></math> for some integer 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math>.) 

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