## Decimal Representation Of Rational Numbers

Example 1:    Express $$\frac { 7 }{ 8 }$$ in the decimal form by long division method.
Solution:    We have,

∴ $$\frac { 7 }{ 8 }$$ = 0.875

Example 2:    Convert $$\frac { 35 }{ 16 }$$ into decimal form by long division  method.
Solution:    We have,

Example 3:    Express $$\frac { 2157 }{ 625 }$$ in the decimal form.
Solution:    We have,

Example 4:    Express $$\frac { -17 }{ 8 }$$ in decimal form by long division method.
Solution:    In order to convert $$\frac { -17 }{ 8 }$$ in the decimal form, we first express $$\frac { 17 }{ 8 }$$ in the decimal form and the decimal form of $$\frac { -17 }{ 8 }$$ will be negative of the decimal form of $$\frac { 17 }{ 8 }$$
we have,

Example 5:    Find the decimal representation of $$\frac { 8 }{ 3 }$$ .
Solution:    By long division, we have

Example 6:    Express $$\frac { 2 }{ 11 }$$ as a decimal fraction.
Solution:    By long division, we have

Example 7:    Find the decimal representation of $$\frac { -16 }{ 45 }$$
Solution:    By long division, we have

Example 8:    Find the decimal representation of $$\frac { 22 }{ 7 }$$
Solution:    By long division, we have

So division of rational number gives decimal expansion. This expansion represents two types
(A) Terminating (remainder = 0)

So these are terminating and non repeating (recurring)
(B) Non terminating recurring (repeating)
(remainder ≠ 0, but equal to devidend)

These expansion are not finished but digits are continusely repeated so we use a line on those digits, called bar $$(\bar{a})$$.
So we can say that rational numbers are of the form either terminating, non repeating or non terminating repeating (recurring).

## Representing Irrational Numbers On The Number Line

Represent  √2 & √3 on the number line:

Greeks discovered this method. Consider a unit square OABC, with each side 1 unit in lenght. Then by using pythagoras theorem

$$OB=\sqrt{1+1}=\sqrt{2}$$
Now, transfer this square onto the number line making sure that the vertex O coincides with zero

With O as centre & OB as radius, draw an arc, meeting OX at P. Then
OB = OP = √2 units
Then, the point represents √2 on the number line
Now draw, BD ⊥ OB such that BD = 1 unit join OD. Then

OD = $$\sqrt{{{(\sqrt{2})}^{2}}+{{(1)}^{2}}}=\sqrt{3}$$ = units With O as centre & OC as radius, draw an arc, meeting OX at Q. Then
OQ = OD = √3 units
Then, the point Q represents √3 on the real line
Remark: In the same way, we can locate √n for any positive integer n, after $$\sqrt{n-1}$$ has been located.

Existence of √n for a positive real number:
The value of √4.3 geometrically : –
Draw a line segment AB = 4.3 units and extend it to C such that BC = 1 unit.
Find the midpoint O of AC.
With O as centre and OA a radius, draw a semicircle.

Now, draw BD ⊥ AC, intersecting the semicircle at D. Then, BD = √4.3 units.
With B as centre and BD as radius, draw an arc, meeting AC produced at E.
Then, BE = BD = √4.3 units

## What is a Rational Number?

A rational number is a number which can be put in the form $$\frac { p }{ q }$$, where p and q are both integers and q ≠ 0.
p is called numerator (Nr) and q is called denominator (Dr).

You can also go through https://cbselibrary.com/ncert-solutions-for-class-10-maths-chapter-1/ for more solved exercises on Rational Numbers

• A rational number is either a terminating or non-terminating but recurring (repeating) decimal.
• A rational number may be positive, negative or zero.
Examples:
• The sum, difference and the product of two rational numbers is always a rational number.
• The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.

Results
Since every number is divisible by 1, we can say that :

1. Every natural number is a rational number, but every rational number need not be a natural number.
For example, 3 = $$\frac { 3 }{ 1 }$$ , 5 = $$\frac { 5 }{ 1 }$$, 9 = $$\frac { 9 }{ 1 }$$ and so on.
but, $$\frac { 7 }{ 9 } ,\frac { 11 }{ 13 } ,\frac { 5 }{ 7 }$$ are rational numbers but not natural numbers.
2. Zero is a rational number because $$\left( 0=\frac { 0 }{ 1 } =\frac { 0 }{ 2 } =…. \right)$$.
3. Every integer is a rational number, but every rational number may not be an integer.
For example $$\frac { -2 }{ 1 } ,\frac { -5 }{ 1 } ,\frac { 0 }{ 1 } ,\frac { 3 }{ 1 } ,\frac { 5 }{ 1 }$$, etc. are all rationals, but rationals like $$\frac { 3 }{ 2 } ,\frac { -5 }{ 2 }$$ etc. are not integers.
4. Rational numbers can be positive and negative.
5. Every positive rational number is greater than zero.
6. Every negative rational number is less than zero.
7. Every positive rational number is greater than every negative rational number.
8. Every negative rational number is smaller than every positive rational number.

### Equivalent Rational Numbers

∵  Rational no. can be written with different Nr and Dr.

Such rational number that are equal to each other are said to be equivalent to each other.
Example: Write $$\frac { 2 }{ 5 }$$ in an equivalent form so that the numerator is equal to –56.
Solution:
Multiplying both the numerator and denominator of $$\frac { 2 }{ 5 }$$ by –28, we have

### Lowest Form of a Rational Number

A rational number is said to be in lowest form if the numerator and the denominator have no common factor other than 1.

Example: Write the following rational numbers in the lowest form :

Solution:

### Standard Form of a Rational Number

A rational number $$\frac { p }{ q }$$ is said to be in its standard form if
(i) its denominator ‘q’ is positive
(ii) the numerator and denominator have no common factor other than 1.
For example : $$\frac { 3 }{ 2 } ,\frac { -5 }{ 2 } ,\frac { 1 }{ 7 }$$, etc.

Example: Express the rational number $$\frac { 14 }{ -21 }$$ in standard form.
Solution:
The given rational number is $$\frac { 14 }{ -21 }$$.
1. Its denominator is negative. Multiply both the numerator and denominator by –1 to change it to positive, i.e.,

2. The greatest common divisor of 14 and 21 is 7. Dividing both numerator and denominator by 7, we have

### Equality of Rational Numbers

Method-1: If two or more rational numbers have the same standard form, we say that the given rational numbers are equal.
Example: Are the rational numbers $$\frac { 8 }{ -12 }$$ and $$\frac { -50 }{ 75 }$$ equal?
Solution: We first express these given rational numbers in the standard form.
The first rational number is $$\frac { 8 }{ -12 }$$.
(i) Multiplying both the numerator and denominator by –1.

(ii) Dividing both the numerator and denominator by the greatest common divisor of 8 and 12, which is 4.

Again, the second rational number is $$\frac { -50 }{ 75 }$$.
(i) The denominator is positive.
(ii) Dividing both numerator and denominator by the greatest common divisor of 50 and 75, which is 25.

Clearly, both the rational numbers have the same standard form.
Therefore, $$\frac { 8 }{ -12 }$$ = $$\frac { -50 }{ 75 }$$

Method-2: In this method, to test the equality of two rational numbers, say $$\frac { a }{ b }$$ and $$\frac { c }{ d }$$, we use cross multiplication in the following way : $$\frac { a }{ b } =\frac { c }{ d }$$
Then a × d = b × c
If a × d = b × c, we say that the two rational numbers $$\frac { a }{ b }$$ and $$\frac { c }{ d }$$ are equal.
Example: Check the equality of the rational numbers $$\frac { -7 }{ 21 }$$ and $$\frac { 3 }{ -9 }$$.
Solution:

### Comparison of Rational Numbers

Comparing fraction. We compare two unequal fractions, each is written as another equal fraction so that both have the same denominators. Then the fraction with greater numerator is greater.
Example : To compare $$\frac { 7 }{ 6 }$$ and $$\frac { 5 }{ 8 }$$, find the L.C.M. of 6 and 8 (it is 24) and

To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.

Note :
Every positive rational number is greater than negative rational number.

### Representation of Rational Numbers on Number Line

We know that the natural numbers, whole numbers and integers can be represented on a number line. For representing an integer on a number line, we draw a line and choose a point O on it to represent ‘0’.
We can represent this point ‘O’ by any other alphabet also. Then we mark points on the number line at equal distances on both sides of O. Let A, B, C, D be the points on the right hand side and A’, B’, C’, D’ be the points on the left of O as shown in the figure.

The points on the left side of O, i.e., A’, B’, C’, D’, etc. represent negative integers –1, –2, –3, –4 whereas, points on the right side of O, i.e., A, B, C, D represent positive integers 1, 2, 3, 4 etc. Clearly, the points A and A’ representing the integers 1 and –1 respectively are on opposite sides of O, but at equal distance from O. Same is true for B and B’ ; C and C’ and other points on the number line.
(1) Natural Numbers

(2) Whole Numbers

(3) Integers

Negative numbers are in left side of zero (0) & positive numbers are in right side.
∵  negative numbers are less than positive numbers
∴  If we move on number line from right to left we are getting smaller numbers.
Also OA = distance of 1 from 0
OD’ = distance of –4 from 0
D’A = distance between –4 and 1. etc.
(4) Rational Numbers
(a) If Nr < Dr:
We divide line segment OA (i.e. distance between 0 & 1) in equal parts as denominator (Dr).

(b) If Nr > Dr:
Example: Represent $$\frac { 13 }{ 3 }$$ and $$-\frac { 13 }{ 3 }$$ on number line.
Solution:

Therefore, from O mark OA, AB, BC, CD and DE to the right of O such that
OA = AB = BC = CD = DE = 1 unit.
Clearly,
Point A,B,C,D,E represents the Rational numbers 1, 2, 3, 4, 5 respectively.
Since we have to consider 4 complete units and a part of the fifth unit, therefore divide the fifth unit DE into 3 equal parts. Take 1 part out of these 3 parts. Then point P is the representation of number $$\frac { 13 }{ 3 }$$ on the number line. Similarly, take 4 full unit lengths to the left of 0 and divide the fifth unit D’E’ into 3 equal parts. Take 1 part out of these three equal parts. Thus, P’ represents the rational number $$-\frac { 13 }{ 3 }$$.

### Rational Number Example Problems With Solutions

Example 1:    Is zero a rational number? can you write it in the form , where p and q are integers and q ≠ 0?
Solution:

Example 2:   Find five rational numbers between  3/5 and 4/5.
Solution:

Example 3:    Find six rational numbers between 3 and 4.
Solution:

Example 4:    Find two rational & two irrational numbers between 4 and 5.
Solution:

Therefore the 3 rational numbers between 4 & 5 are $$\frac { 9 }{ 2 }$$, $$\frac { 19 }{ 2 }$$

The 2 irrational numbers between 4 and 5 are as follows:
a = 4.101001000….
b = 4.20002000….

## What Is Surd Or Radical

Real Number Line:
Irrational numbers like √2, √3, √5 etc. can be represented by points on the number line. Since all rational numbers and irrational numbers can be represented on the number line, we call the number line as real number line.

Surds:
√2, √3, √5, √21, ……………. are irrational numbers, These are square roots (second roots), of some rational numbers, which can not be written as squares of any rational number.

1. If a is rational number and n is a positive integer such that the nth root of a is an irrational number, then a1/n is called a surd or radical.
Example: √2, √3, √5 etc.
2. If is a surd then ‘n’ is known as order of surd and ‘a’ is known as radicand.
3. Every surd is an irrational number but every irrational number is not a surd.

A surd of order 2 is called a quadratic surd.
Example:  √3 = 31/2 is a quadratic surd but  √9= 91/2 is not a quadratic surd, because  √9= 91/2 = 3 is a rational number. So,  √9 is not a surd.

Cubic Surd:
A surd of order 3 is called a cubic surd.
Example:  The real number $$\sqrt[3]{4}$$ is a cubic surd but the real number $$\sqrt[3]{8}$$ is not a cubic surd as it not a surd.

A surd of order 4 is called a biquadratic surd. A biquadratic surd is also called a quadratic surd.
Example:  $$\sqrt[4]{5}$$ is a biquadratic surd but $$\sqrt[4]{81}$$ is not a biquadratic surd as it is not a surd.

1. For any positive integer ‘n’ and a positive rational number ‘a’.
2. A surd which has unity only as rational factor is called a pure surd.
3. A surd which has a rational factor other than unity is called a mixed surd.
4. Surds having same irrational factors are called similar or like surds.
5. Only similar surds can be added or subtracted by adding or subtracting their rational parts.
6. Surds of same order can be multiplied or divided.
7. If the surds to be multiplied or to be divided are not of the same order, we first reduce them to the same order and then multiply or divide.
8. If the product of two surds is a rational number, then each one of them is called the rationalising factor of the other.
9. A surd consisting of one term only is called a monomial surd.
10. An expression consisting of the sum or difference of two monomial surds or the sum or difference of a monomial surds and a rational number is called binomial surd.
Example: $$\sqrt{2}+\sqrt{5},\,\sqrt{3}+2,\,\,\sqrt{2}-\sqrt{3}$$ etc. are binomial surds.
11. The binomial surds which differ only in sign
(+ or –) between the terms connecting them, are called conjugate surds.  binomial surd.
Example: $$\sqrt{3}+\sqrt{2}$$ and $$\sqrt{3}-\sqrt{2}$$ or $$2+\sqrt{5}$$ and $$2-\sqrt{5}$$ are conjugate surds.

## Surd Or Radical Example Problems With Solutions

Example 1:    State with reasons which of the following are surds and which are not
(i) √64      (ii) √45       (iii) √20 × √45
$$(\text{iv})\text{ }8\sqrt{10}\div 4\sqrt{15}\text{ (v) }3\sqrt{12}\div 6\sqrt{27}\text{ (vi) }\sqrt[3]{5}\times \sqrt[3]{25}$$
Solution:    (i) √64 = 8
8 is a rational number, hence √64 is not  a surd.
(ii) $$\sqrt{45}=\sqrt{9\times 5}=3\sqrt{5}$$
Because the rational number 45 is not the square of any rational number, hence √45 is a surd.

Which is an irrational number.
Because the rational number 8/3 is not the square of any rational number, hence the given expression is a surd.

Example 2:    Simplify the following
$$\text{(i) }{{\left( \sqrt[3]{5} \right)}^{3}}\text{ (ii) }\sqrt[3]{64}$$
Solution:

Example 3:    Find the value of x in each of the following:
$$\text{(i) }\sqrt[3]{4x-7}-5=0\text{ (ii) }\sqrt[4]{3x+1}=2$$
Solution:

Example 4:    Simplify each of the following:
$$\text{(i) }\sqrt[3]{3}\times \sqrt[3]{4}\text{ (ii) }\sqrt[3]{128}$$
Solution:

Example 5:    Simplify each of the following:
$$\text{(i) }\sqrt[3]{\frac{8}{27}}\text{ (ii) }\frac{\sqrt[4]{3888}}{\sqrt[4]{48}}$$
Solution:

Example 6:    Siplify each of the following
$$\text{(i) }\sqrt[4]{\sqrt[3]{3}}\text{ (ii) }\sqrt[2]{\sqrt[3]{5}}$$
Solution:

Pure And Mixed Surds:
(i) Pure Surd:
A surd which has unity only as rational factor, the other factor being irrational, is called a pure surd.
Example: $$\sqrt{3},\,\,\sqrt[5]{2},\,\,\sqrt[4]{3}$$ are pure surds.
Example: $$\sqrt[{}]{6},\,\,\sqrt[3]{12}$$ are pure surds.
(ii) Mixed Surd:
A surd which has a rational factor other than unity, the other factor being irrational, is called a mixed surd.
Example: $$2\sqrt{3},\,\,5\,\sqrt[3]{12},\,\,2\,\sqrt[4]{5}$$ are mixed surds.

TypeI: On expressing of mixed surds into pure surds

Example 7:    Express each of the following as a pure surd.
$$\text{(i) 2}\sqrt{3}\text{ (ii) 2}\text{.}\sqrt[3]{4}$$
$$\text{(iii) }\frac{3}{4}\sqrt{32}\text{ (iv) }\frac{3}{4}\sqrt{8}$$
Solution:

Example 8:    Expressed each of the following as pure surds
$$\text{(i) }\frac{2}{3}\sqrt[3]{108}\text{ (ii) }\frac{3}{2}\sqrt[4]{\frac{32}{243}}$$
Solution:

Example 9:    Express each of the following as pure surd
$$\text{(i) a}\sqrt{a+b}\text{ (ii) }a\sqrt[3]{{{b}^{2}}}\text{ (iii)2ab}\sqrt[3]{ab}$$
Solution:

TypeII: On expressing given surds as mixed surds in the simplest form.

Example 10:    Express each of the following as mixed surd in its simplest form:
$$\text{(i) }\sqrt{80}\text{ (ii) }\sqrt[3]{72}\text{ (iii) }\sqrt[5]{288}$$
$$\text{(iv) }\sqrt{1350}\text{ (v) }\sqrt[5]{320}\text{ (vi) 5}\text{.}\sqrt[3]{135}$$
Solution:

Example 11:    Express $$\sqrt[4]{1280}$$ as mixed surd in its simplest form
Solution:

## Conversion Of Decimal Numbers Into Rational Numbers Of The Form m/n

Case I: When the decimal number is of terminating nature.
Algorithm:

• Step-1: Obtain the rational number.
• Step-2: Determine the number of digits in its decimal part.
• Step-3: Remove decimal point from the 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 rational number.
• Step-4: Find a common divisor of the numerator and denominator and express the rational number to lowest terms by dividing its numerator and denominator by the common divisor.

Case II: When decimal representation is of non-terminating repeating nature.
In a non terminating repeating decimal, there are two types of decimal representations

1. A decimal in which all the digit after the decimal point are repeated. These type of decimals are known as pure recurring decimals.
For Example: $$0.\overline{6},\,\,0.\overline{16},\,\,0.\overline{123}$$ are pure recurring decimals.
2. A decimal in which at least one of the digits after the decimal point is not repeated and then some digit or digits are repeated. This type of decimals are known as mixed recurring decimals.
For Example: $$2.1\overline{6},\,\,0.3\overline{5},\,\,0.7\overline{85}$$ are mixed recurring decimals.

## Conversion Of A Pure Recurring Decimal To The Form p/q

Algorithm:

• Step-1: Obtain the repeating decimal and pur it equal to x (say)
• Step-2: Write the number in decimal form by removing bar from the top of repeating digits and listing repeating digits at least twice. For sample, write x = $$0.\overline{8}$$ as x = 0.888…. and x = $$0.\overline{14}$$ as x = 0.141414……
• Step-3: Determine the number of digits having bar on their heads.
• Step-4: If the repeating decimal has 1 place repetition, multiply by 10; a two place repetition, multiply by 100; a three place repetition, multiply by 1000 and so on.
• Step-5: Subtract the number in step 2 from the number obtained in step 4
• Step-6: Divide both sides of the equation by the coefficient of x.
• Step-7: Write the rational number in its simplest form.

## Conversion Of A Mixed Recurring Decimal To The Form p/q

Algorithm:

• Step-1 : Obtain the mixed recurring decimal and write it equal to x (say)
• Step-2 : Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point
• Step-3 : Multiply both sides of x by 10n  so that only the repeating decimal is on the right side of the decimal point.
• Step-4 : Use the method of converting pure recurring decimal to the form p/q and obtain the value of x

## Conversion Of Decimal Numbers Into Rational Numbers Example Problems With Solutions

Example 1:    Express each of the following numbers in the form p/q.
(i) 0.15                        (ii) 0.675                       (iii) –25.6875
Solution:

Example 2:    Express each of the following decimals in the form p/q.
$$\text{(i) 0}\text{.}\overline{\text{6}}\text{ (ii) 0}\text{.}\overline{\text{35}}\text{ (iii) 0}\text{.}\overline{\text{585}}$$
Solution:

The above example suggests us the following rule to convert a pure recurring decimal into a rational number in the form p/q.

Example 3:    Convert the following decimal numbers in the form p/q.
$$\text{(i) }5.\bar{2}\text{ (ii) }23.\overline{43}$$
Solution:

Example 4:    Express the following decimals in the form
$$\text{(i) }0.3\overline{2}\text{ (ii) }0.12\overline{3}$$
Solution:

Example 5:    Express each of the following mixed recurring decimals in the form p/q
$$\text{(i) }4.3\overline{2}\text{ (ii) }15.7\overline{12}$$
Solution:

Example 6:    Represent 3.765 on the number line.
Solution:    This number lies between 3 and 4. The distance 3 and 4 is divided into 10 equal parts. Then the first mark to the right of 3 will represent 3.1 and second 3.2 and so on. Now, 3.765 lies between 3.7 and 3.8. We divide the distance between 3.7 and 3.8 into 10 equal parts 3.76 will be on the right of 3.7 at the sixth mark, and 3.77 will be on the right of 3.7 at the 7th mark and 3.765 will lie between 3.76 and 3.77 and soon.

Example 7:    Visualize $$4.\overline{26}$$ on the number line, upto 4 decimal places.
Solution:    We have,   $$4.\overline{26}$$ = 4.2626
This number lies between 4 and 5. The distance between 4 and 5 is divided into 10 equal parts. Then the first mark to the right of 4 will represent 4.1 and second 4.2 and soon. Now, 4.2626 lies between 4.2 and 4.3. We divide the distance between 4.2 and 4.3 into 10 equal parts 4.2626 lies between 4.26 and 4.27. Again we divide the distance between 4.26 and 4.27 into 10 equal parts. The number 4.2626 lies between 4.262 and 4.263. The distance between 4.262 and 4.263 is again divided into 10 equal parts. Sixth mark from right to the 4.262 is 4.2626.

Example 8:    Express the decimal $$0.003\overline{52}$$ in the form p/q
Solution:    Let x = $$0.003\overline{52}$$
Clearly, there is three digit on the right side of the decimal point which is without bar. So, we multiply both sides of x by 103 = 1000 so that only the repeating decimal is left on the right side of the decimal point.

Example 9:    Give an example of two irrational numbers, the product of which is
(i) a rational number
(ii) an irrational number
Solution:    (i) The product of √27 and √3 is √81= 9, which is a rational number.
(ii) The product of √2 and √3 is √6, which is an irrational number.

Example 10:     Insert a rational and an irrational number between 2 and 3.
Solution:    If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then $$\sqrt{ab}$$ is an irrational number lying between a and b. Also, if a,b are rational numbers, then $$\frac { a+b }{ 2 }$$ is a rational number between them.
∴ A rational number between 2 and 3 is
$$\frac { 2+3 }{ 2 }$$ = 2.5
An irrational number between 2 and 3 is
$$\sqrt{2\times 3}=\sqrt{6}$$

Example 11:     Find two irrational numbers between 2 and 2.5.
Solution:    If a and b are two distinct positive rational numbers such that ab is not a perfect square of a rational number, then $$\sqrt{ab}$$ is an irrational number lying between a and b.
∴ Irrational number between 2 and 2.5 is
$$\sqrt{2\times 2.5}=\sqrt{5}$$
Similarly, irrational number between 2 and $$\sqrt{5}$$ is $$\sqrt{2\times \sqrt{5}}$$
So, required numbers are $$\sqrt{5}$$ and $$\sqrt{2\times \sqrt{5}}$$.

Example 12:    Find two irrational numbers lying between √2 and √3.
Solution:    We know that, if a and b are two distinct positive irrational numbers, then is an irrational number lying between a and b.
∴ Irrational number between √2 and √3 is $$\sqrt{\sqrt{2}\times \sqrt{3}}=\sqrt{\sqrt{6}}$$ = 61/4
Irrational number between √2 and 61/4 is $$\sqrt{\sqrt{2}\times {{6}^{1/4}}}$$ = 21/4 × 61/8.
Hence required irrational number are 61/4 and 21/4 × 61/8

Example 13:    Find two irrational numbers between 0.12 and 0.13.
Solution:    Let a = 0.12 and b = 0.13. Clearly, a and b are rational numbers such that a < b.
We observe that the number a and b have a 1 in the first place of decimal. But in the second place of decimal a has a 2 and b has 3. So, we consider the numbers
c = 0.1201001000100001 ……
and, d = 0.12101001000100001…….
Clearly, c and d are irrational numbers such that a < c < d < b.

Example 14:    Find two rational numbers between 0.232332333233332…. and 0.252552555255552……
Solution:    Let a = 0.232332333233332….  and b = 0.252552555255552…..
The numbers c = 0.25 and d = 0.2525
Clearly, c and d both are rational numbers such that a < c < d < b.

Example 15:    Find a rational number and also an irrational number between the numbers a and b given below:
a = 0.101001000100001….,  b = 0.1001000100001…
Solution:   Since the decimal representations of a and b are non-terminating and non-repeating. So,
a and b are irrational numbers.
We observed that in the first two places of decimal a and b have the same digits. But in the third place of decimal a has a 1 whereas b has zero.
∴ a > b
Construction of a rational number between a and b : As mentioned above, first two digits after the decimal point of a and b are the same. But in the third place a has a 1 and b has a zero. So, if we consider the number c given by
c = 0.101
Then, c is a rational number as it has a terminating decimal representation.
Since b has a zero in the third place of decimal and c has a 1.
∴ b < c
We also observe that c < a, because c has zeros in all the places after the third place of decimal whereas the decimal representation of a has a 1 in the sixth place.
Thus, c is a rational number such that
b < c < a.
Hence , c is the required rational number between a and b.
Construction of an irrational number between a and b : Consider the number d given by
d = 0.1002000100001……
Clearly, d is an irrational number as its decimal representation is non-terminating and non-repeating.
We observe that in the first three places of their decimal representation b and d have the same digits but in the fourth place d and a 2 whereas b has only a 1.
∴ d > b
Also, comparing a and d, we obtain a > d
Thus, d is an irrational number such that
b < d < a.

Example 16:    Find one irrational number between the number a and b given below :
a = 0.1111….. = $$0.\bar{1}$$ and b = 0.1101
Solution:    Clearly, a and b are rational numbers, since a has a repeating decimal and b has a terminating decimal. We observe that in the third place of decimal a has a 1, while b has a zero.
∴ a > b
Consider the number c given by
c = 0.111101001000100001…..
Clearly, c is an irrational number as it has non-repeating and non-terminating decimal representation.
We observe that in the first two places of their decimal representations b and c have the same digits. But in the third place b has a zero whereas c has a 1.
∴ b < c
Also, c and a have the same digits in the first four places of their decimal representations but in the fifth place c has a zero and a has a 1.
∴ c < a
Hence, b < c < a
Thus, c is the required irrational number between a and b.