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Adding, Subtracting, Dividing or Multiplying Irrational Numbers to Obtain Rational Numbers

We can find irrational numbers that add, times, divide or multiply to give rational numbers.
\[2+ \sqrt{5}\]
  and  
\[2- \sqrt{5}\]
  add to give a rational number.
\[(2 + \sqrt{5}) + (2 - \sqrt{5}) = 4\]

Note that the  
\[{} + \sqrt{5}\]
  cancels the  
\[- \sqrt{5}\]
.
\[ \sqrt{5}\]
  and  
\[2 \sqrt{5}\]
  multiply to give a rational number.
\[ \sqrt{5}+ \times 2 \sqrt{5} =2 \times \sqrt{25} = 2 \times 5 = 10\]

\[ \sqrt{5}\]
  and  
\[2 \sqrt{5}\]
  divide to give a rational number.
\[ \frac{\sqrt{5}}{ 2 \sqrt{5}} =\frac{1}{2}\]

\[3+ \sqrt{5}\]
  andhj  
\[2 + \sqrt{5}\]
  subtract to give a rational number.
\[(3 + \sqrt{5}) - (2 + \sqrt{5}) =3 + \sqrt{5} - 2 - \sqrt{5} =1 \]

Note that the  
\[{} + \sqrt{5}\]
  cancels the  
\[{}+ \sqrt{5}\]
.

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