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CBSE Class 8 Rational Numbers

RATIONAL NUMBERS

1.1 Definition of Rational Numbers:

Question:

What are rational numbers?

Answer:

A number that can be written as \frac{p}{q}, where p and q are integers and q \neq  0, is known as Rational Number.

Example: \frac{2}{3}, \frac{4}{5}, -\frac{9}{11}, 12, -18 etc.

If the signs of numerator and denominator are either both positive or both negative, the rational number is known as Positive Rational Number.

Example:  \frac{-2}{-7}, \frac{14}{25} etc. 

If the signs of numerator and denominator are opposite to each other, the rational number is known as Negative Rational Number.

Example: \frac{-2}{9}, \frac{4}{-17} etc.

1.2 Types of Rational Numbers: 

Question:

How many types of rational numbers?

Answer:

There are following types of rational numbers.

Natural Numbers:

All the counting numbers are called NATURAL NUMBERS. Its set is represented by N. Therefore, N = {1, 2, 3, 4, 5, 6, 7, 8, 9, …} is the set of natural numbers. The number of natural number is infinite. 

Whole Numbers:

All counting numbers including 0 (zero) form the set of WHOLE NUMBERS. Its set is represented by W. Therefore, W = {0, 1, 2, 3, 4, 5, 6…} is the set of the whole numbers. Now, on comparing the set of natural numbers N = {1, 2, 3, 4, 5, 6, 7, 8, 9, …} is the subset of the whole numbers W = {0, 1, 2, 3, 4, 5, 6…}. Thus, N \subseteq  W.

Integers:

All the natural numbers, 0 (zero) and the negative of all the natural numbers from the set of INTEGERS. Its set is represented by are denoted by Z or I. Therefore, Z = {-3, -2, -1, 0,1,2,3,…} is the set of integers. Now, we observe that both the set of natural numbers N = {1, 2, 3, 4, 5, 6, 7, 8, 9, …} and the whole numbers W = {0, 1, 2, 3, 4, 5, 6…} are the subset of integers Z = {-3, -2, -1, 0,1,2,3,…}. Thus, N \subseteq  Z and W \subseteq  Z \Rightarrow N \subseteq  W \subseteq  Z. 

Rational Numbers:

We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq  0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq  W \subseteq  Z  \subseteq  Q.  

1.3 Decimal Representation of Rational Numbers:

Question:

How do we represent the rational number in decimal form? 

Answer:

We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq  0, is known as rational numbers. Now, to express the rational number \frac{p}{q} in decimal representation, we divide the numerator p by the denominator q. After dividing p by q, we get either the remainder zero or not.

Question:

What are the terminating, non-terminating and recurring decimal?

Answer:

In the division of a rational number \frac{p}{q}, when the remainder is zero, then that decimal representation is known as TERMINATING DECIMAL and when the remainder is not zero, then that decimal representation is known as NON-TERMINATING DECIMAL. Now, in the non-terminating decimal, either the decimal part is repeating, recurring or no repeat, non-recurring. When the decimal part is recurring, then that decimal number is known as RECURRING DECIMAL and when the decimal part is non-recurring, then that decimal representation is known as NON-RECURRING DECIMAL.

Example: \frac{1}{8} = 0.125, \frac{1}{2} = 0.5, \frac{1}{25} = 0.04 etc are known as terminating decimal. In the division of these rational numbers, we get the remainder zero after a finite steps.

Whereas, \frac{1}{3} = 0.33333…, \frac{9}{11} = 0.81818181…. etc are known as recurring decimal. In the division of these rational numbers, we don’t get the remainder zero after any finite steps and its decimal part is recurring. The recurring decimal numbers are also represented as 

 \frac{1}{3} = 0.33333… = 0. 3(Read as 0.3 bar) \frac{9}{11} = 0.81818181… = 0.81– (Read as 0.81 bar) etc.

Question:

How do we identify that the fraction is terminating or non-terminating decimal numbers without performing devision?

Answer:

In the rational number or fraction \frac{p}{q}, if the prime factorization of the denominator q is in the form of 2m \times 5n, it means, the prime factors of the denominator q are 2 or 5 or both. Otherwise, the fraction is non-terminating decimal.

Example: 

In \frac{1}{8}, we have q =8 = 23 \times 50. Thus, \frac{1}{8} is terminating decimal and \frac{1}{8} = 0.125.

In \frac{1}{25}, we have q = 25 = 20 \times 52. Thus, \frac{1}{25} is terminating decimal and \frac{1}{25} = 0.04.

In \frac{8}{15} , we have q = 15 = 31 \times 51. Thus, \frac{8}{15} is non-terminating decimal.

Question:

How do we express the recurring decimal as fraction?

Answer:

To express the recurring decimal number in a fraction, we use a formula as below.

Example: 12.253 = \frac{12253-122}{990} = \frac{11263}{990}

1.4 Comparison of Rational Numbers:

Question:

How do we compare two or more rational numbers? 

Answer:

To compare two or more rational numbers, we do the following steps:

Step 1: Express each of the rational numbers with the positive denominator.

Step 2: Find the LCM of these positive denominators.

Step 3: Express each of the rational number with this LCM as the common denominator. 

Step 4: The numerator having the greater numerator is greater.

Question:

Arrange the numbers \frac{-3}{5}\frac{7}{-10} and \frac{-5}{8}

Answer:

We are given \frac{-3}{5}\frac{7}{-10} = \frac{-7}{10} and \frac{-5}{8}. LCM of (5, 10 and 8) = 40

\frac{-3}{5} = \frac{-3 \times 8}{5 \times 8} = -\frac{24}{40}

\frac{-7}{10} = \frac{-7 \times 4}{10 \times 4} = -\frac{28}{40}

\frac{-5}{8} = \frac{-5 \times 5}{8 \times 5} = -\frac{25}{40}

Thus, we get  \frac{-28}{40} <\frac{-25}{40} <\frac{-24}{40} \Rightarrow\frac{-7}{10} <\frac{-5}{8} <\frac{-3}{5}.

1.5 Representation of Rational Numbers on Number Lines:

Question:

How do we represent the rational numbers on number line?

Answer:

To represent a rational number, we draw a line. Take a point O on it. Mark it as zero (0). Set off equal distances on the right of O and the left of O. These distances is known as a unit length. Clearly, the points A, B, C, D, E represent the integers 1, 2, 3, 4 and 5 respectively and the points A’, B’ C’, D’ and E’ represent the integers -1, -2, -3, -4 and -5 respectively.

Number Line

Thus, we represent any integer by a point on the number line. Clearly, every positive integer lies to the right of O and every negative integers lies to the left of O. Similarly, we represent other rational numbers.

Question:

Represent \frac{2}{3} and -\frac{2}{3} on number line.

Answer:

Number Line

1.6 Addition of Rational Numbers

Question:

How to add two rational numbers if the given rational numbers have same denominator?

Answer:

When the given rational numbers have same denominator, then we use the method as \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}.

Question:

Add the rational numbers, \frac{-7}{9} + \frac{11}{9} .

Answer:

\frac{-7}{9} + \frac{11}{9} = \frac{-7 + 11}{9}=\frac{4}{9}

Question:

How to add two rational numbers if the given rational numbers have different denominator?

Answer:

When the given rational numbers have same denominator, then we take the LCM of their denominators and express each of the given numbers with this LCM as the common denominator. Now, we add these numbers as \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}.

Question:

Find the sum \frac{-5}{6} + \frac{4}{9} .

Answer:

The denominator of the given numbers are 6 and 9. LCM of (6, 9) = 18. Thus, \frac{-5}{6}=\frac{-5\times3}{6\times3}=\frac{-15}{18} and \frac{4}{9}=\frac{4\times2}{9\times2}=\frac{8}{18}.

Now, \frac{-5}{6}+\frac{4}{9}=\frac{-15}{18}+\frac{8}{18}=\frac{-15+8}{18}=\frac{-7}{18}.

1.7 Properties of Addition of Rational Numbers

Question:

What are the properties of addition of rational numbers?

Answer:

There are following properties of addition of rational numbers:

Property 1: The sum of two rational numbers is always a rational number. It means, if \frac{a}{b} and \frac{c}{d} are numbers, then \left ( \frac{a}{b} + \frac{c}{d} \right ) are rational number. This property is known as CLOSURE PROPERTY.

Property 2: Two rational numbers can be added in any order. It means, if \frac{a}{b} and \frac{c}{d} are numbers, then \left ( \frac{a}{b} + \frac{c}{d} \right )  = \left ( \frac{c}{d} + \frac{a}{b} \right ) . This property is known as COMMUTATIVE PROPERTY.

Property 3: While adding three rational numbers, they can be grouped in any order. It means, if \frac{a}{b} , \frac{c}{d} and \frac{e}{f} are three numbers, then (\frac{a}{b} + \frac{c}{d}) + \frac{e}{f} = \frac{a}{b} + (\frac{c}{d} + \frac{e}{f}) . This property is known as ASSOCIATIVE PROPERTY.

Property 4: 0 (zero) is a rational number such that the sum of any rational number and 0 (zero) is the rational number itself. Thus, (\frac{a}{b} + 0) =( 0 + \frac{a}{b} )= \frac{a}{b}. This property is known as EXISTENCE OF ADDITIVE IDENTITY and 0 (zero) is known as ADDITIVE IDENTITY for rational number.

Property 5: For any rational number \frac{a}{b}, there exist a rational number \frac{-a}{b} such that (\frac{a}{b} + \frac{-a}{b}) = 0. This property is known as EXISTENCE OF ADDITIVE INVERSE and \frac{-a}{b} is known as ADDITIVE INVERSE of \frac{a}{b}.

1.8 Subtraction of Rational Numbers

Question:

How to subtract two rational numbers?

Answer:

For rational number \frac{a}{b} and \frac{c}{d}, we subtract as  (\frac{a}{b} – \frac{c}{d}) = (\frac{a}{b} + \frac{-c}{d}) = (\frac{a}{b} + Additive Inverse of \frac{c}{d}).

Question:

Subtract \frac{3}{4} from \frac{2}{3}

Answer:

\frac{2}{3} - \frac{3}{4} = \frac{2}{3} + \frac{-3}{4} = \frac{8+(-9)}{12} = \frac{-1}{12}.

1.9 Properties of Subtraction of Rational Numbers

Question:

What are the properties of subtraction of rational numbers?

Answer:

There are following properties of subtraction of rational number:

Property 1: The subtraction of two rational numbers is always a rational number. It means, if \frac{a}{b} and \frac{c}{d} are rational numbers, then (\frac{a}{b} – \frac{c}{d}) is rational number. This property is known as CLOSURE PROPERTY.

Property 2: Two rational numbers cannot be subtracted in any order. It means, if \frac{a}{b} and \frac{c}{d} are numbers, then (\frac{a}{b} – \frac{c}{d}\neq (\frac{c}{d} – \frac{a}{b})  . It means, the subtraction does not follow the rule COMMUTATIVE PROPERTY as in addition. Similarly, the subtraction does not follow  ASSOCIATIVE PROPERTY.

1.10 Multiplication of Rational Numbers

Question:

How do we multiply two rational numbers?

Answer:

For two rational numbers  \frac{a}{b} and \frac{c}{d} , we get the multiplication as \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.

Question:

Multiply \frac{-3}{7} \times \frac{14}{5}

Answer:

We multiply as \frac{-3}{7} \times \frac{14}{5} = \frac{-6 }{5}.

1.11 Properties of Multiplication of Rational Numbers

Question:

What are the properties of multiplication of rational numbers?

Answer:

There are following properties of multiplication of rational numbers:

Property 1: The multiplication or the product of two rational numbers is always a rational number. It means, if \frac{a}{b} and \frac{c}{d} are numbers, then \frac{a}{b} \times \frac{c}{d}  is rational number. This property is known as CLOSURE PROPERTY.

Property 2: Two rational numbers can be multiplied in any order. It means, if \frac{a}{b} and \frac{c}{d} are numbers, then \frac{a}{b} \times \frac{c}{d}   = \frac{c}{d} \times \frac{a}{b} . This property is known as COMMUTATIVE PROPERTY.

Property 3: While multiplying three rational numbers, they can be grouped in any order. It means, if \frac{a}{b} , \frac{c}{d} and \frac{e}{f} are three numbers, then (\frac{a}{b}  \times  \frac{c}{d})  \times  \frac{e}{f} = \frac{a}{b}  \times (\frac{c}{d}  \times  \frac{e}{f}) . This property is known as ASSOCIATIVE PROPERTY.

Property 4: 1 (one) is a rational number such that the product of any rational number and 1 is the rational number itself. Thus, (\frac{a}{b}  \times  1) =( 1  \times  \frac{a}{b} )= \frac{a}{b}. This property is known as EXISTENCE OF MULTIPLICATIVE IDENTITY and 1 is known as MULTIPLICATIVE IDENTITY for rational number.

Property 5: For any rational number \frac{a}{b}, there exist a rational number \frac{b}{a}  such that (\frac{a}{b}  \times  \frac{b}{a} ) = 1. This property is known as EXISTENCE OF MULTIPLICATIVE INVERSE and \frac{b}{a}  is known as MULTIPLICATIVE INVERSE or RECIPROCAL of \frac{a}{b}. Note that 0 (zero) has no reciprocal, because \frac{1}{0}  is not defined. The reciprocal of 1 is 1 and the reciprocal of (-1) is (-1). 

Property 6: For any three numbers  \frac{a}{b} , \frac{c}{d} and \frac{e}{f}, we have  \frac{a}{b}  \times  (\frac{c}{d} + \frac{e}{f} )= (\frac{a}{b}  \times  \frac{c}{d}) + (\frac{a}{b}  \times  \frac{e}{f}) . This property is known as DISTRIBUTIVE LAW OF MULTIPLICATION OVER ADDITION PROPERTY.

Property 7: If we multiply any rational number with 0 (zero), then the result is always 0 (zero). It means,   (\frac{a}{b}  \times  0)= (0  \times  \frac{a}{b}) = 0 . This property is known as MULTIPLICATIVE PROPERTY OF 0 (ZERO).

1.12 Division of Rational Numbers

Question:

How do we divide two rational numbers?

Answer:

If \frac{a}{b} and \frac{c}{d} are two numbers such that \frac{c}{d} \neq  0, then the division of \frac{a}{b} by \frac{c}{d} is defined as (\frac{a}{b} \div \frac{c}{d}) = (\frac{a}{b}  \times  \frac{d}{c}). It means, when \frac{a}{b} is divided by \frac{c}{d}, then \frac{a}{b} is known as dividend and \frac{c}{d} is known as divisor and the product/result is known as the quotient of the division.

Question:

Divide \frac{9}{16} by \frac{5}{8}.

Answer:

\frac{9}{16} \div \frac{5}{8} = \frac{9}{16}  \times  \frac{8}{5} = \frac{9}{10}.

1.13 Properties of Division of Rational Numbers

Question:

What are the properties of division of rational numbers?

Answer:

There are following properties of division of rational numbers:

Property 1: If \frac{a}{b} and \frac{c}{d} are two numbers such that \frac{c}{d} \neq  0, then (\frac{a}{b} \div \frac{c}{d}) is also a rational number. This property is known as CLOSURE PROPERTY.

Property 2: For every rational number, we have then (\frac{a}{b} \div 1) = \frac{a}{b}. This property is known as PROPERTY OF 1.

Property 3: For every rational number, we have then (\frac{a}{b} \div \frac{b}{a}) = 1. Here, \frac{a}{b} and \frac{b}{a} are RECIPROCAL TO EACH OTHER.

1.14 Finding more than one Rational Number between two Rational Numbers

Question:

How do we find a rational number between two rational number?

Answer:

If x and y are two rational numbers such that x < y, then \frac{x + y}{2} is the rational number between x and y

Question:

Find a rational number between \frac{1}{3} and \frac{1}{2}.

Answer:

 Required Number = (\frac{1}{3}+\frac{1}{2})/2 = (\frac{5}{6})/2 = \frac{5}{12}.

Question:

Find three rational numbers between 3 and 5.

Answer:

First number between 3 and 5 = \frac{3+5}{2}=4

Second number between 3 and 4  = \frac{3+4}{2}= \frac{7}{2}

Third number between 4 and 5 = \frac{4+5}{2}= \frac{9}{2}

Thus, the three rational number between 3 and 5 are 4, 7/2, 9/2.

Question:

Find 9 rational number between 1 and 2.

Answer:

Since 1 = \frac{10}{10} and 2 = \frac{20}{10}, the nine rational numbers between 1 and 2 are

\frac{11}{10},\frac{12}{10},\frac{13}{10},\frac{14}{10},\frac{15}{10},\frac{16}{10},\frac{17}{10},\frac{18}{10},\frac{19}{10},

1.15 Rational Numbers Notes & Sample Questions in PDF

To download This Chapter Slide, Click at Download.

To Download Practice Question paper 1 For Rational Number, Click at Download.

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To Download Practice Question paper 8 For Rational Number, Click at Download.

To Download Practice Question paper 9 For Rational Number, Click at Download.

To Download CBSE NCERT BOOK Chapter 1- Rational Number, Click at Download.

Next Chapters:

2. Exponent

3. Square And Square Root

4. Cubes And Cube Root

5. Playing With Numbers

6. Operation on Algebraic Expression

7. Factorization

8. Linear Equation in One Variable

9. Percentage

10. Profit and Loss

11. Compound Interest

12. Direct And Inverse Proportion

13. Time And Work

14. Polygon

15. Quadrilateral

16. Parallelogram

17. Area of a Trapezium and a Polygon

18. Three-Dimensional Figures

19. Volumes and Surface Area of Solids

20. Data Handling

21. Construction and Interpreting Graph

22. Pie Chart

23. Probability

24. Graphs

 

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