10th Class Mathematics Real Numbers Decimal Representation of Numbers

         Decimal Representation of Numbers

There are three different types of decimal representation of numbers.

i. Terminating

ii. Non terminating and Repeating

iii. Non terminating and Non-Repeating  

            Terminating Decimal

If the decimal representation of \[\frac{a}{b}\] comes to an end then it is called terminating decimal.

If the prime factor of denominator having 2, 5 or 2 and 5 only then decimal representation of \[\frac{a}{b}\] (which is in the lowest form) must be terminating.

As for example\[2\frac{4}{5}\] is a terminating decimal i.e. \[2\frac{4}{5}=\frac{14}{5}=2.8\]  

Check whether \[\frac{39}{24}\] is terminating or non-terminating

Solution:

To convert \[\frac{39}{24}\] into the lowest form, we get \[\frac{39}{24}=\frac{13}{8}\], here denominator of \[\frac{13}{8}\] is 8 whose prime factor is \[~\text{2}\times \text{2}\times \text{2}\] which contains only 2 as a factor.

Therefore, it is a terminating decimal. \[\frac{13}{8}=1.625\]  

            Non-terminating and Repeating Decimals

A decimal in which digit or set of digits repeated in a particular fashion is called repeating decimal.  

(1) 2.3333333333333..............

(2) 0.123123123123123....... etc.

In the above given two examples we observe that 3 repeats itself in example 1 and 1, 2, 3 repeat itself in example 2.

The above decimal is also written as \[\text{2}.\overline{\text{3}}\] and \[0.\overline{\text{123}}\]  

Method of Conversion of Non-terminating and Repeating Decimal into a Fraction

Step 1: Suppose the given decimal as any variable like \[x,\text{y},.......\] etc.

Step 2: Multiply the given decimal with 10 or power of 10 in such a way that only repeating digits remain on the right of the decimal or all non-repeating terms which are on the right come to left of the decimal.

Step 3: Multiply the decimal obtained in step 2 with 10 or powers of 10 in such a way that repeated digit or a set of digit comes to the left of the decimal. i.e. We multiply by 10 if there is only one digit is repeated, multiply by\[\text{1}{{0}^{\text{2}}}\] or 100 if two digits repeated and so on.

Step 4: Now subtract the decimal obtained in step 2 from the decimal obtained in step 3.

Step 5: Solve the equation whatever get in step 4 and the value of variable in simplified form is the required fraction.  

Convert the following into fraction.

(i) \[0.\overline{\text{123}}\]                                    

(ii) \[0.\text{23}\overline{\text{41}}\]

Solution: (i) \[0.\overline{\text{123}}\]  

Step 1: Suppose \[x=0.\overline{123}\].  

Step 2: Here, there is no non-repeating term on the right of the decimal. Therefore, there is no need to multiply by 10 or powers of 10. \[x=0.\text{123123123123}\]                                     .....(i)  

Step 3: Now there are three digits on the right of the decimal which are repeated, that is why\[x\] is multiplied by 103 or 1000. \[x=0.\text{123123123123 }.....)\times \text{1}000\] \[\text{1}000x=\text{123}.\text{123123123123}\]                             .....(ii)  

Step 4:                 \[\begin{align}   & \text{1}000x=\text{123}.\text{123123123123}....... \\  & \underline{-\,\,\,\,\,\,\,\,\,x=~\,\,\,\,\,\,0.\text{123123123123}.......} \\  & \,\,\,999x=123 \\ \end{align}\]  

Step 5:   or,\[x=\frac{123}{999}\],or \[x=\frac{41}{333}\]

Therefore, \[0.\overline{123}=\frac{41}{333}\]

(ii) \[x=0.\text{2}\overline{\text{341}}\]

\[x=0.\text{23414141}\]                                                              .....(i)

Multiply both sides of (i) by 100 (because there are two digits on the right of the decimal), we get

\[\text{1}00x=\text{23}.\text{414141}\]                                               .....(ii)

Multiply the equation (ii) by 100 so that repeated digits 4 and 1 come to the left of decimal. Now (ii) will be converted into

\[\text{1}0000x=\text{2341}.\text{414141}\]                                   .....(iii)

Subtract (ii) from (iii), we get

\[\begin{align}   & \text{1}0000x=\text{2341}.\text{414141}..... \\  & \underline{\,\,\,\,\,\,\text{1}00x=\text{ }\,\,\,\,\,\text{23}.\text{414141}....} \\  & \,\,9900x=2318 \\ \end{align}\]

\[x=\frac{2318}{9900}=\frac{1159}{4950}\]

Therefore, \[0.23\overline{41}=\frac{1159}{4950}\].    

Non-terminating and Non-repeating

A decimal in which no one digit on the right of decimal repeated periodically is called non-terminating and non-repeating decimal. For example 0.101001000100001000001............ is a non-terminating and non-repeating decimal.  

            Characteristics of Rational Number

i.   Rational numbers are either terminating or a repeating decimals.

ii.  There are infinite rational numbers between two given rational numbers.

iii.  The sum of two rational numbers is rational.

iv.  The difference of two rational numbers is rational.

v.   The product of two rational numbers is rational.

vi.  The division of a rational number by a non-zero rational number is rational.