An irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction \(m/n\), where \(m\) and \(n\) are integers, with \(n\) non–zero. The most well known irrational numbers are \(\pi \) and\(\sqrt 2 \).  

1. Sum or difference of two irrational numbers may not be irrational.
2. Product or division of two irrational numbers may not be irrational.
3. Product of one rational and one irrational number may or may not be irrational. For example \(3 \times \sqrt 2 \)is irrational but \(0 \times \sqrt 3 \) is rational. Note that 0 is rational.

The constant \(\pi \) is irrational. 

Example 4: If \(A{\rm{ }} = {\rm{ }}0.{a_1}{a_2}{a_1}{a_2} \ldots  \ldots  \ldots  ..\infty \) and \(B{\rm{ }} = {\rm{ }}0.{b_1}{b_2}{b_3}{b_1}{b_2}{b_3} \ldots  \ldots  \ldots  \ldots ..\infty \), find the minimum positive integer  that should be multiplied to \((A + B)\) so that product is always an integer.

Solution:

\(A\) and \(B\) both are repeating decimals

\(A =\frac{{{a_1}{a_2}}}{{99}}\)

\(B = \frac{{{b_1}{b_2}{b_3}}}{{999}}\)

Now the minimum positive integer that should be multiplies to \((A + B)\) to make it an integer will be LCM of 99 and 999 = 10989.