Rational numbers:

A rational number is a number which can be expressed as a ratio of two integers. Non–integer rational numbers (commonly called fractions) are usually written as the fraction \(\frac{a}{b}\), where \(b\) is not zero, \(a\) is called the numerator, and \(b\) the denominator.

Any repeating number with a constant cycle or numbers with terminating digits are also known as rational numbers.

Some of the examples of rational numbers are:
\(\frac{2}{3}\), \(0.3333......\), \(2.25\) etc.

The decimal representation of a real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic. For example, the decimal representation of \(\frac{1}{3} = 0.3333333..\) becomes periodic just after the decimal point, repeating the single–digit sequence "3" indefinitely. Similarly the fraction \(\frac{1}{7} = 0.142857142857.......\) is repeating with a period of 6 digits. All the repeating numbers are rational numbers. Repeating decimal can be represented using bar lines on the digits. \(\frac{1}{7} = 0.142857142857.... = 0.\overline {142857} \)

Some interesting examples of repeating decimal:

\(\cfrac{1}{7} = 0.\overline {142857} \), \(\cfrac{2}{7} = 0.\overline {285714} \)

\(\cfrac{3}{7} = 0.\overline {428571} \), \(\cfrac{4}{7} = 0.\overline {571428} \)

\(\cfrac{5}{7} = 0.\overline {714285} \), \(\cfrac{6}{7} = 0.\overline {857142} \)

We see that all the six repeating digits are same in all the fractions given above.

Conversion from Repeating Numbers to Fraction:

Note that every repeating decimal can be converted to fraction (\(p/q\) form). Suppose the number is \(0.12\overline {345} \), this number is equal to 0.12345345345345……up to infinite. In this number 12 is non repeating number, 345 is repeating number, number of repeating digits is 3 and number of non repeating digits is 2.  The \(p/q\) form is given by: \[\left[ {\frac{{{\rm{Entire}}\,\,\,\,{\rm{Number}}\; - \;{\rm{Non}}\,\,\,{\rm{repeating}}\,\,\,{\rm{number}}}}{{\mathop {99999..}\limits_{{\rm{(up}}\,{\rm{to}}\,\,{\rm{number}}\,\,{\rm{of}}\,\,{\rm{rep}}{\rm{.}}\,\,{\rm{digits)}}} \mathop {000000000}\limits_{\,\,\,\,\,\,\,\,\,\,{\rm{(upto}}\,\,{\rm{number}}\,\,{\rm{of}}\,\,{\rm{non}}\,\,{\rm{rep}}\,\,{\rm{digits)}}} }}} \right]\]

For example the number \(0.2\overline {34} \), 34 is repeating and 2 is non repeating. The equivalent fraction of this number is \(\frac{{234 - 2}}{{990}} = \frac{{232}}{{990}} = \frac{{116}}{{495}}\)

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Answer Solution

Irrational numbers:

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\(\sqrt 2 \).  

• Sum or difference of two irrational numbers may not be irrational.
• Product or division of two irrational numbers may not be irrational.
• 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.