## Arithmetic Chapter 9 Recurring Decimal Number

**Arithmetic Chapter 9 Recurring Decimal Numbers**

- To convert a vulgar fraction into a decimal fraction, in some cases, it is observed that the operation of division comes to an end i.e., there is no remainder left and the decimal fraction obtained contains a finite number of digits after the decimal point.
- These decimal fractions are called
**Finite Decimal Fractions.** - Again in some cases, it is observed that the operation of division never comes to an end i.e., there is an infinite number of digits after the decimal point, these decimal fractions are called
**Infinite Decimal Fractions.** - For infinite decimal fractions, in some cases, it is observed that the same remainder or a set of the same remainders occurs repeatedly in the operation, of division.
- So a figure (digit) or a set of figures (digits) appears repeatedly in a definite order in the quotient during the operation of division in some cases of infinite or non-terminating decimal fractions.
- Such a type of decimal number is called a
**Recurring Decimal Number.**

## Arithmetic Chapter 9 Conversion of Vulgar fraction into Recurring Decimal Fraction

- Let us consider the Vulgar fraction 1/3∴ 1/3 = 0.333….

= 0.3. - The given fraction is 1/3 Due to the presence of 1 in the numerator and 3 in the denominator of the given vulgar fraction, when it is converted into a decimal fraction a non-terminating decimal fraction is obtained.
- In the process of division, the dividend i.e., the numerator 1 which is less than the divisor 3 (i.e., the denominator), we put a decimal point in the quotient and put one zero after 1 in the dividend so that the new dividend becomes 10 and dividing it by 3, we get 3 as the quotient and it is placed after the decimal point in the quotient.
- The remainder obtained is 1.
- We put one zero on the right side of the remainder every time of division.
- Thus the new remainder is 10 after putting one zero after 1 and it is divided by 3, the quotient obtained is 3.
- But every time the same remainder 1 occurs i.e., the same remainder 1 occurs repeatedly and the division never comes to an end.
- The quotient is 0-333 and we write this as 0-3. We place a dot above 3 after the decimal point and it indicates that after the decimal point, 3 recurs endlessly in the quotient.
- Such a type of decimal fraction is called a
**Recurring Decimal Fraction.** - Again, we consider the vulgar fraction 2/7 and it is to be converted into a decimal fraction.∴ 2/7 = 0.285714285714….

= 0.285714. - We have to divide 2 by 7 (here numerator is 2 and the denominator is 7) in the vulgar fraction 2/7.
- We see that the remainder of 2 occurs in the seventh step.
- Thus the repetition starts at the seventh step. The successive quotients after the decimal point are 2, 8, 5, 7; 1, 4, 2, 8, 5, 7, 1, 4, 2,

2/7*∴*0.285714.*=* - Two dots are placed one is above the digit 2 and the other is above the digit 4.
- This indicates that all the digits present between 2 and 4 will recur in the same order as they occur in 285714.
- The decimal conversion of the vulgar fraction 2/7 is 0.285714, which is a recurring decimal number.
- Similarly, we get,35/12 = 2.91666….= 2.916.
∴ 421/330 = 0.127575….

= 0.1275 - From the above discussions, we conclude that the order to convert a vulgar fraction into a decimal fraction, the numerator of the given vulgar fraction is divided by its denominator if the operation of division never comes to an end
- i.e., Always there is a remainder left, the same remainder or a set by the same remainders occurs repeatedly then a digit or a set of digits recurs repeatedly or endlessly in a definite order in the quotient, and the fraction thus obtained is a non-terminating decimal fraction which is called a
**Recurring Decimal Fraction.**

## Arithmetic Chapter 9 Recurring Period

- In a recurring decimal fraction, some or all the digits recur.
- The portion of the decimal fraction which recur is called the
**Recurring period or Recurring part.** - The portion of a recurring decimal fraction after the decimal point which does not recur is called it’s
**Non-recurring Part.** - For example, we consider the recurring decimal fraction 2.43725, portion 725 of it is the Recurring Period, and portion 43 of the given recurring decimal fraction (after the decimal point) is the Non-recurring part.

## Arithmetic Chapter 9 Pure Recurring Decimal Fraction And Mixed Recurring Decimal Fraction

- A recurring decimal fraction that does not contain any non-recurring part is called a
**Pure Recurring Decimal Fraction.** - A recurring decimal fraction that contains a non-recurring part is called a Mixed Recurring Decimal Fraction.
- For example, 0.36, and 021578 are Pure recurring decimal fractions,s and 0.231, and 0.065217 are Mixed recurring decimal fractions.

## Arithmetic Chapter 9 Conversion of a few special vulgar fractions into recurring decimals

∴ 1/7 = 0.142857

2/7 = 0.285714

3/7 = 0.428571

4/7 = 0.571428, etc.

- From above, we get, when the vulgar fractions having 7 as the denominator are converted into decimal fractions, then the converted decimal fractions are pure recurring decimal fractions and they obey some rules.
- In the recurring part of the decimal conversion of a vulgar fraction whose denominator is 7, six digits 1, 2, 4, 5, 7, 8 (excluding the digits 3, 6, 9) are arranged in some special definite order.
- For example, in the recurring part of the decimal conversion of 1/7, six digits 1, 2, 4, 5, 7, 8 recur in the order 1, 4, 2, 8, 5, 7.
- We place the digits 1, 4, 2, 8, 5, and 7 along the circumference of a circle in a clockwise direction.
- Then we can get equivalent recurring decimal fractions of the vulgar fractions

1/7, 2/7, 3/7, 4/7,….1/7 = 0.142857

2/7 = 0.285714

3/7 = 0.428571

4/7 = 0.571428

5/7 = 0.714285

6/7 = 0.857142. - There are 6 digits in the recurring part of all these pure recurring decimal fractions.
- Now we shall discuss the decimal conversion of vulgar fractions whose denominator is 13.
- Here we get two types of periods.
- In the periods of decimal conversion of 1/13, 3/13, 4/13, 9/13, 10/13, and 12/13, we get the digits 0, 7, 6, 9, 8, and 3 in the cyclic order.∴ 1/13 = 0.076923∴ 1/13 = 0.076923

3/13 = 0.230769

4/13 = 0.307692

9/13 = 0.692307

10/13 = 0.769230

12/13 = 0.923076. - There are 6 digits in the periods of all these pure recurring decimal fractions.
- Again in the periods of decimal conversion of six vulgar fractions 2/13, 5/13, 6/13, 7/13, 8/13, and 14/13, we get the digit 1, 5, 3, 8, 4, 6 in the cyclic order.

∴ 2/13 = 0.153846

∴ 2/13 = 0.153846

5/13 = 0.384615

6/13 = 0.461538

7/13 = 0.538461

8/13 = 0.615384

11/13 = 0.846153.

## Arithmetic Chapter 9 Number of digits in the Non-recurring part of a Mixed recurring decimal

- The decimal conversion of a vulgar fraction is a mixed recurring decimal if the denominator of the vulgar fraction when it is expressed in its lowest term has a factor of 2 or 5 or both.
- The number of digits in the non-recurring part will be the highest index of 2 or 5 present in the denominator of the vulgar fraction.
- Observe the following examples:

**Example 1:** **\(\frac{1}{12}=\frac{1}{2^2 \times 3}\)**

The index of 2 in the denominator is 2.

So the number of digits in the nonrecurring part of the mixed recurring decimal fraction = The index of 2 in the denominator = 2.

1/12 = 0.083.

Non-recurring part = 08.

**Example 2: \(\frac{1}{15}=\frac{1}{3 \times 5}\)**

The index of 5 in the denominator = 1.

So the number of digits in the nonrecurring part of the mixed recurring decimal fraction = The index of 5 in the denominator = 1.

1/5 = 0.06.

Non-recurring part = 0.

∴ The number of digits in the non-recurring part = 1.

**Example 3: \(\frac{1}{24}=\frac{1}{2^3 \times 3}\)**

The index of 2 in the denominator = 3.

So the number of digits in the nonrecurring part = The index of 2 in the denominator = 3.

1/24 = .0416.

Non-recurring part = 3.

**Example 4: \(\frac{1}{60}=\frac{1}{5 \times 2^2 \times 3}\) **

The indices of 2 and 5 in the denominator are 2 and l respectively.

So the number of digits in the non-recurring part = The greater of the indices of 2 and 5 = The greater of 2 and 1=2.

∴ 1/60 = 0.016.

No-recurring part = 01.

∴ The number of digits in the non-recurring part = 2.

## Arithmetic Chapter 9 Conversion of Recurring Decimal into Vulgar Fraction

- There are two types of recurring decimals pure recurring decimals and mixed recurring decimals.
- First, we shall discuss about the conversion of pure recurring into a vulgar fraction.

1. Conversion of pure recurring decimal into vulgar fraction

**Example 1: Convert 0.1 into vulgar fractions.**

** Solution **:

**Given**

0.1

0.2 x 10 = (.2222 ) x 10

= 2.2222 ….. (1)

0.2x 1 = (0.2222 ) x 1 = 0.2222………… (2)

(Multiplying 0.2 by 1)

Subtracting (2) from (1), we get,

0.2 (10 – 1) = (2.2222 ) – (.2222 ) = 2

or, 0.2 x 9 = 2

or 0.2 = 2/9

0.1 into vulgar fractions = 2/9

**Example 2: C onvert 0*35 into a vulgar fraction.**

** Solution**:

**Given**

0*35

0.35 = 0.353535

Multiplying both sides by 100 and 1 respectively, we get,

0.35 x 100 = (0.353535 ) x 100 = 35.353535…………(1)

0-35 x 1 = (0.353535 ) x 1 = 0.353535………(2)

Subtracting (2) from (1), we get,

0.35 (100 – 1) = 35

or, 0.35 x 99 = 35

or, 0.35 = 35/99.

From the above examples, we get the following rule of conversion of pure recurring decimal into vulgar fraction

For example 0.54632 = \(\frac{54632}{99999}\)

0.205 = \(\frac{205}{999}\)

0.51

= 51/99

= 17/13.

0.35 into a vulgar fraction 35/99

**2. Conversion of mixed recurring decimals into vulgar fractions: **

**For this observe the following examples**

**Example 1: Convert 0.1275 into a vulgar fraction.**

**Solution : **

**Given**

0.1275

0-1275 = 0-12757575……..(1)

Multiplying both sides by 10000, we get,

0.1275 x 10000 = (0.12757575 ) x 10000 = 1275.757575………(2)

Multiplying both sides of (1) by 100, we get,

0.1275 x 100 = (0.12757575 ) x 100

= 12.757575……….(3)

Subtracting (3) from (2), we get,

(10000 – 100) x 0.1275 = 1275 – 12

or, 9900 x 0.1275 = 1275 – 12

or, 0.1275 = \(\frac{1275-12}{9900}\)

= \(\frac{1263}{9900}\)

= \(\frac{421}{3300}\)

0.1275 into a vulgar fraction = \(\frac{421}{3300}\)

**Example 2. Convert 0-26321 into a vulgar fraction.**

**Solution:**

**Given**

0-26321

0.26321 = 0.26321321321…………(1)

Multiplying both sides of (1) by 100000, we get,

0.26321 x 100000 = (0.26321321321 ) x 100000 = 26321.321321

Again multiplying both sides of (1) by 100, we get,

0.26321 x 100 = (0.26321321321……) x 100 = 26.321321321

Subtracting (3) from (2) we get,

0.26321 (100000 – 100) = 26321 – 26

or, 0.26321 x 99900 = 26321 – 26

or, 0.26321 = \(\frac{26321-26}{99900}\)

= \(\frac{26295}{99900}\)

= \(\frac{8765}{33300}\)

0-26321 into a vulgar fraction = \(\frac{8765}{33300}\)

**Example 3. Convert 3.128 into a vulgar fraction.**

** Solution**:

**Given**

3.128

3.128 = 3.1282828.. (1)

Multiplying both sides of (1) by 1000, we get,

3.128 x 1000 = (3.1282828 ) x 1000 = 3128.282828..

Again multiplying both sides of (1) by 10, we get,

3.128 x 10 = (3.1282828 ) x 10 = 31.282828…… . .

Subtracting (3) from (2), we get,

(1000 – 10) x 3.128 = 3128 – 31 or, 990 x 3.128 = 3128 – 31

or, 3.128 = \(\frac{3128-31}{990}\)

= \(\frac{3097}{990}\)

= \(3 \frac{127}{990}\)

∴ 3.128 = \(3 \frac{127}{990}\)

From the discussions of the above examples, we get the following rule

**Some examples are given below:**

1. 0.02028 = \(\frac{2028-2}{99900}\)

= \(\frac{2026}{99900}\)

= \(\frac{1013}{49950}\)

2. 10293 = \(\frac{10293 – 102}{9990}\)

= \(\frac{52424}{9990}\)

= \(5 \frac{2474}{9990}\)

= \(5 \frac{1237}{4995}\)