Decimal Representation Of Rational Numbers
Conversion of decimal numbers into rational numbers of the form
Case-1 : When decimal number is of terminating nature.
step 1 : Write the given decimal.
Step 2 : Count the number of digits in its decimal part.
Step 3 : Remove decimal point from the number. Write it as numerator. Write '1' in the denominator and put as many zeros on the right side of '1' as the number of digits in the decimal part of the given decimal number.
Step 4 : We get rational number and convert it to lowest form by dividing numerator & denominator by the common divisor.
Ex : Convert 0.15 into \(
\frac{p}
{q}
\) form
\(
0.15 = \frac{{15}}
{{100}} = \frac{3}
{{20}}
\)
Case 2 : Conversion of a pure recurring decimal into \(
\frac{p}
{q}
\) form
Ex : Convert \(
23.\overline {43}
\) into \(
23.\overline {43}
\) form
Let \(
x = 23.\overline {43}
\)
x = 23.434343................1
Multiplying both sides by 100, we get
100x = 2343.4343 ..............2
2 - 1
100x = 2343.4343.....
- x = 23.4343.....
99x = 2320
\(
x = \frac{{2320}}
{{99}}
\)
Case 3 : Conversion of a mixed recurring decimal into \(
\frac{p}
{q}
\) form
Ex : Convert \(
0.12\overline 3
\) into \(
\frac{p}
{q}
\) form
Solution : Let x = \(
0.12\overline 3
\)
No. of digits after decimal point without balance = 2
\(
100x = 12.\overline 3
\)So, multiply with \(
10^2
\) on both sides.
\(
= 12 + 0.\overline 3
\) (repeating decimal is on the right side of the decimal point)
\(
\begin{gathered}
= 12 + \frac{3}
{9} \hfill \\
= \frac{{108 + 3}}
{9} \hfill \\
= \frac{{111}}
{9} \hfill \\
\therefore x = \frac{{111}}
{{900}} = \frac{{37}}
{{300}} \hfill \\
\end{gathered}
\)
Note : 1. If we can express the denominator of a simplified rational number in the form \(
\text{2}^\text{p} \text{5}^\text{q} \text{ or 2}^\text{p} \text{,5}^\text{q} \text{ where p, q } \in N
\) , then the number has a terminating decimal expansion.
2. If we cannot express the denominator of a simplified rational number in the form \(
\text{2}^\text{p} \text{5}^\text{q} \text{ or 2}^\text{p} \text{,5}^\text{q} \text{ where p, q } \in N
\) then the number has a non terminating decimal expansion.
Ex : 1. 0.375 = \(
\frac{{375}}
{{1000}} = \frac{3}
{{2^3 }}
\)
\(
2.\frac{7}
{8} = \frac{7}
{{2^3 }}or\frac{{7 \times 5^3 }}
{{2^3 \times 5^3 }}or\frac{{75}}
{{10^3 }}
\)