Chapter-1 Rational numbers Class 8th

In this slide we are going to study about Rational number, which is the first chapter of NCERT Class 8th Mathematics.
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1. RATIONAL NUMBERS -BY ABHISHEK MISHRA

2. CONTENT PRIOR KNOWLEDGE RATIONAL NUMBER – DEFINITION PROPERITIES OF NUMBERS PROPERTIES OF RATIONAL NUMBERS REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS VARIOUS TYPES OF QUESTIONS

3. PRIOR - KNOWLEDGE Brief knowledge on fraction, i.e fraction addition subtraction, multiplication, and division. Also there should be sufficient knowledge on LCM and HCF. Some knowledge on properties of numbers that you learn in previous classes. Also some knowledge on number lines

4. RATIONAL NUMBERS DEFINATION - The number which can be written in the form of where p and q are integers and q≠0. All the integers are rational numbers Rational numbers are represented by ‘ Q’ . Examples of Rational numbers – , , 5 , -6 , , If q is equal to 0 then, becomes ‘not defined’ then q can’t be equal to 0 Positive rational number= , Negative rational number= ,

5. NUMBER SYSTEM

6. PROPERITIES OF NUMBERS Under the properties of numbers, we studies the following properties of number system- CLOSRE PROPERTY Closer under addition Closer under subtraction Closer under multiplication Closer under division COMMUTATIVE PROPERTY Commutative for addition Commutative for subtraction Commutative for multiplication Commutative for division

7. ASSOCIATIVE PROPERTY Associative for addition Associative for subtraction Associative for multiplication Associative for division DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION ADDITIVE INVERSE ADDITIVE IDENTITY MULTIPLICATIVE INVERSE MULTIPLICATIVE IDENTITY

8. CLOSER PROPERTY CLOSER UNDER ADDITION A number system is said to closed under addition if and only if the result of the sum of two numbers in a system also exist in that system. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The sum of two natural number is always a natural number. For example- 5+7=12 (12 is a natural number). WHOLE NUMBERS- The sum of two whole number is always a whole number. For example- 8+3=11 (11 is a whole number) 0+0=0 (0 is whole number) INTEGERS- The sum of two integers is always a integers . For example- (- 8)+4=(-4) [-4 is a integer] 15+(-8)=7 [7 is a integer]

9. CLOSER UNDER SUBTRACTION A number system is said to closed under subtraction if and only if the result of the difference of two numbers in that system also exist in that system. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The difference of two natural number is not always a natural number. For example- 5-7=(-2) [-2 is not a natural number]. WHOLE NUMBERS- The difference of two whole number is not always a whole number. For example- 0-5=(-5) [-5 is not a whole number] INTEGERS- The difference of two integers is always a integers . For example- (- 8)-4=(-12) [-12 is a integer] 15-(-8)=7 [7 is a integer]

10. CLOSER UNDER MULTIPLICATION A number system is said to closed under multiplication if and only if the result of the product of two numbers in that system also exist in that system. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The product of two natural number is always a natural number. For example- 9×6=54 (54 is a natural number). WHOLE NUMBERS- The product of two whole number is always a whole number. For example- × 5=0 (0 is a whole number). INTEGERS- The product of two integers is always a integers . For example- (- 8) × 4=(-32) [-32 is a integer] (-5)×(-6)=30 [30 is a integer]

11. CLOSER UNDER DIVISION- A number system is said to closed under division if and only if the result of the division of two numbers in that system also exist in that system. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The division of two natural number is not always a natural number. For example- 5 ÷ 7= [ is not a natural number]. WHOLE NUMBERS- The division of two whole number is not always a whole number. For example- 4 ÷ 0 = (not defined) [it is not a whole number] INTEGERS- The division of two integers is not always a integers . For example- (- 5) ÷ 4 = ( ) [ is a integer] 4 ÷ 0 = (not defined) [it is not a integer]

12. COMMUTATIVE PROPERTY COMMUTATIVE FOR ADDITION - A number system is said to commutative under addition if and only if the result of the sum of two numbers in a system doesn’t depends on the order of addition . Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The sum of two natural number will always remains same if also we change there position. For example- 5 + 7 = 7 + 5 = 12 WHOLE NUMBERS- The sum of two whole number will always remains same if also we change there position . For example- 1 5 + 0 = 0 + 15 = 15 INTEGERS- The sum of two integers will always remains same if also we change there position . For example- (- 8) + 4 = 4 + (-8) = (-4) (-15) + (-8) = (-8) + (-15) = (-23)

13. COMMUTATIVE FOR SUBTRACTION- A number system is said to commutative under subtraction if and only if the result of the difference of two numbers in a system doesn’t depends on the order of subtraction. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The difference of two natural number will not be same if we change there position. For example- 5 - 7 ≠ 7 - 5 ⇒ (-2) ≠ (2) WHOLE NUMBERS- The difference of two whole number will not be same if we change there position. For example- 1 5 - 0 ≠ 0 - 15 ⇒ (15) ≠ (-15) INTEGERS- The difference of two integers will not be same if we change there position. For example- (- 8) – (4) ≠ (4) - (-8) (-15) - (-8) ≠ (-8) - (-15) ⇒ (-12) ≠ (12) ⇒ (-15) + 8 ≠ (-8) + 15 ⇒ (-7) ≠ (7)

14. COMMUTATIVE FOR MULTIPLICATION- A number system is said to commutative under multiplication if and only if the result of the product of two numbers in a system doesn’t depends on the order of multiplication. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The product of two natural number will always remains same if also we change there position. For example- 5 × 7 = 7 × 5 = 35 WHOLE NUMBERS- The product of two whole number will always remains same if also we change there position. For example- 1 5 × 0 = 0 × 15 = 0 INTEGERS- The product of two integer will always remains same if also we change there position. For example- (- 8) × 4 = 4 × (-8) = (-32) (-15) × (-8) = (-8) × (-15) = (120)

15. COMMUTATIVE FOR DIVISION- A number system is said to commutative under subtraction if and only if the result of the difference of two numbers in a system doesn’t depends on the order of subtraction. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The division of two natural number will not be same if we change there position. For example- 5 ÷ 7 ≠ 7 ÷ 5 ⇒ ≠ WHOLE NUMBERS- The division of two whole number will not be same if we change there position. For example- 1 5 ÷ ≠ ÷ 15 ⇒ (not defined) ≠ 0 INTEGERS- The division of two integers will not be same if we change there position. For example- (- 8) ÷ (4) ≠ (4) ÷ (-8) (-15) ÷ (-8) ≠ (-8) ÷ (-15) ⇒ (-2) ≠ ( ) ⇒ ≠

16. ASSOCIATIVE PROPERTY ASSOCIATIVE FOR ADDITION - A number system is said to associative under addition if you can add regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other words, if you are adding it does not matter where you put the parenthesis . Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The sum of natural number will remains same regardless of how the numbers are grouped . For example- ( 5 + 7) + 3 = 5 + (7 + 3) ⇒ 12 + 3 = 5 + 10 ⇒ 15 = 15 WHOLE NUMBERS- The sum of whole number will remains same regardless of how the numbers are grouped . For example- ( 3 + 0) + 6 = 3 + (0 + 6) ⇒ 3 + 6 = 3 + 6 ⇒ 9 = 9 INTEGERS- The sum of integers will remains same regardless of how the numbers are grouped . For example- { (- 8) + 4 }+ 6 = (- 8) + {4 + 6} = 2 {(-15) + (-8)} + 23 = (-15) + {(-8) + 23} = 0

17. ASSOCIATIVE FOR SUBTRACTION- A number system is said to associative under subtraction if you can subtract regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other words, if you are subtracting it does not matter where you put the parenthesis. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The difference of natural numbers will change if the grouping changes . For example- ( 5 - 7) - 3 ≠ 5 - (7 - 3) ⇒ (-2) - 3 ≠ 5 – (4) ⇒ (-5) ≠ 1 WHOLE NUMBERS- The difference of whole numbers will change if the grouping changes. For example- (3 - 0) - 6 ≠ 3 - (0 - 6) ⇒ 3 - 6 ≠ 3 + 6 ⇒ (-3) ≠ (9) INTEGERS- The difference of integers will change if the grouping changes . For example- { (- 8) - 4 } - 6 ≠ (- 8) - {4 - 6} {(-15) - (-8)} - 23 ≠ (-15) - {(-8) - 23} ⇒ (-12) - 6 ≠ (-8) – (-2) ⇒ (-7) – 23 ≠ (-8) – (-31) ⇒ (-18) ≠ (-6) ⇒ (-30) ≠ (23)

18. ASSOCIATIVE FOR MULTIPLICATION - A number system is said to associative under multiplication if you can multiply regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other words, if you are multiplying it does not matter where you put the parenthesis . Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The product of natural number will remains same regardless of how the numbers are grouped . For example- ( 5 × 7) × 3 = 5 × (7 × 3) ⇒ 35 × 3 = 5 × 21 ⇒ 105 = 105 WHOLE NUMBERS- The sum of whole number will remains same regardless of how the numbers are grouped . For example- ( 3 × 0) × 6 = 3 × (0 × 6) ⇒ 0 × 6 = 3 × ⇒ 0 = 0 INTEGERS- The sum of integers will remains same regardless of how the numbers are grouped . For example- { (- 8) × 4 } × 6 = (- 8) × {4 × 6} {(-15) × (-8)} × 3 = (-8) × {(-15) × 3} ⇒ (-32) × 6 = (-8) × 24 ⇒ 120 × 3 = (-8) × ( -45) ⇒ (-192) = (-192) ⇒ 360 = 360

19. ASSOCIATIVE FOR DIVISION- A number system is said to associative under division if you can divide regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other words, if you are dividing it does not matter where you put the parenthesis. Lets understand with some number systems that we have studied in previous classes- NATURAL NUMBERS- The division of natural numbers will change if the grouping changes . For example- ( 150 ÷ 10) ÷ 5 ≠ 150 ÷ (10 ÷ 5) ⇒ (15) ÷ 5 ≠ 150 ÷ (2) ⇒ 3 ≠ 75 WHOLE NUMBERS- The division of whole numbers will change if the grouping changes. For example- (45 ÷ 9) ÷ 3 ≠ 45 ÷ (9 ÷ 3) ⇒ 5 ÷ 3 ≠ 45 ÷ 3 ⇒ ≠ 15 INTEGERS- The division of integers will change if the grouping changes . For example- { (- 150) ÷ 10 } ÷ (-5) ≠ (- 150) ÷ {10 ÷ (-5)} ⇒ (-15) ÷ (-5) ≠ (-150) ÷ (-2) ⇒ (3) ≠ (75)

20. DISTRIBUTIVE PROPERTY The distributive property of multiplication states that when multiplying a number by the sum/difference of two numbers, the final value is equal to the sum/difference of each addend multiplied by the third number . Lets understand with some number systems that we have studied in previous classes- DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION 5 × (6 + 8) 5 × (6 + 8) ⇒ (5 × 6) + (5 × 8) OR ⇒ 5 × 14 = 70 ⇒ 30 + 40 = 70 DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION 8 × (7 – 2) 8 × (7 – 2) ⇒ (8 × 7) – (8 × 2) OR ⇒ 8 × 5 = 40 ⇒ 56 – 16 = 40

21. ADDITIVE INVERSE In mathematics, the  additive inverse  of a number a is the number that, when added to a, yields zero . This number is also known as the opposite (number), sign change, and negation . For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. For example- Additive inverse of 7 is (-7) as 7 + (-7) = 0 ADDITIVE IDENTITY In mathematics , the additive identity of a number is a number that should added to the number so that we get the number itself. When we add zero to any number we get the number itself. For example- Additive identity of 7 is 0 as 7 + 0 = 7

22. MULTIPLICATIVE INVERSE   In mathematics, the  multiplicative inverse  of a number a is the number that, when multiplied to the number, yields one. This number is also known as the reciprocal (number) as when we multiply a number with its reciprocal we get 1. For example- Multiplicative inverse of 7 is as 7 × = 1 MULTIPLICATIVE IDENTITY In mathematics, the  multiplicative identity  of a number a is the number that, when multiplied to the number, gives the number itself as a result . When we multiply one to a number we get the number itself. For example – Multiplicative identity of 7 is 1 as 7 × 1 = 7

23. PROPERITIES OF RATIONAL NUMBERS Under the properties of numbers, we studies the following properties of number system- CLOSRE PROPERTY Closer under addition Closer under subtraction Closer under multiplication Closer under division COMMUTATIVE PROPERTY Commutative under addition Commutative under subtraction Commutative under multiplication Commutative under division

24. ASSOCIATIVE PROPERTY Associative under addition Associative under subtraction Associative under multiplication Associative under division DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION ADDITIVE INVERSE ADDITIVE IDENTITY MULTIPLICATIVE INVERSE MULTIPLICATIVE IDENTITY

25. Closer Property Of Rational Number CLOSER UNDER ADDITION- Sum of any two rational number is always a rational number. Example - + = = (it is a rational number) CLOSER UNDER SUBTRACTION- Difference of any two rational number is always a rational number. Example - - = = (it is a rational number) CLOSER UNDER MULTIPLICATION- Product of any two rational number is always a rational number. Example - × = (it is a rational number) CLOSER UNDER DIVISION- Division of two rational number is not always a rational number. When we divide any number with zero we get a non rational number ( not defined ).

26. Commutative Property Of Rational Number COMMUTATIVE FOR ADDITION- The sum of two rational number will always remains same if also we change there position . Example - + = + = = COMMUTATIVE FOR SUBTRACTION- The difference of two natural number will not be same if we change there position . Example - - ≠ - ⇒ ≠ ⇒ ≠ COMMUTATIVE FOR MULTIPLICATION- The sum of two rational number will always remains same if also we change there position. Example - × = × = COMMUTATIVE FOR DIVISION- The division of two rational number will not be same if we change there position.

27. Associative Property Of Rational Number ASSOCIATIVE FOR ADDITION- The sum of rational number will remains same regardless of how the numbers are grouped Example- + [ + ] = + = OR [ + ] + = + = ASSOCIATIVE FOR SUBTRACTION- The difference of rational numbers will change if the grouping changes . Example - - [ - ] ≠ [ - ] - ASSOCIATIVE FOR MULTIPLICATION- The product of rational number will remains same regardless of how the numbers are grouped Example - × [ × ] = × = OR [ × ] × = × = ASSOCIATIVE FOR DIVISION- The division of rational numbers will change if the grouping changes.

28. Distributive Property Of Rational Number DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION- Lets us consider three rational numbers , , × [ + ] × [ + ] ⇒ [ × ] + [ × ] OR ⇒ + ⇒ + = ⇒ + = = DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER SUBTRACTION- × [ - ] × [ - ] OR ⇒ [ × ] – [ × ] = ⇒ - =

29. ADDITIVE INVERSE In mathematics, the  additive inverse  of a number a is the number that, when added to a, yields zero . This number is also known as the opposite (number), sign change, and negation . For example- Additive inverse of is as + = 0 ADDITIVE IDENTITY In mathematics , the additive identity of a number is a number that should added to the number so that we get the number itself. When we add zero to any number we get the number itself. For example- Additive identity of is 0 as + 0 =

30. MULTIPLICATIVE INVERSE   In mathematics, the  multiplicative inverse  of a number a is the number that, when multiplied to the number, yields one. This number is also known as the reciprocal (number) as when we multiply a number with its reciprocal we get 1. For example- Multiplicative inverse of is as × = 1 MULTIPLICATIVE IDENTITY In mathematics, the  multiplicative identity  of a number a is the number that, when multiplied to the number, gives the number itself as a result . When we multiply one to a number we get the number itself. For example – Multiplicative identity of is 1 as × 1 =

31. SUMMARISATION OF PROPERTIES

32. CLOSER PROPERTIES NUMBERS CLOSED UNDER ADDITION a + b belongs to system SUBTRACTION a - b belongs to system MULTIPLICATION a × b belongs to system DIVISION a ÷ b belongs to system RATIONAL NUMBER YES YES YES NO INTEGERS YES YES YES NO WHOLE NUMBER YES NO YES NO NATURAL NUMBER YES NO YES NO

33. COMMUTATIVE PROPERTIES NUMBERS COMMUTATIVE FOR ADDITION a + b = b + a SUBTRACTION a – b = b - a MULTIPLICATION a × b = b × a DIVISION a ÷ b = b ÷ a RATIONAL NUMBER YES NO YES NO INTEGERS YES NO YES NO WHOLE NUMBER YES NO YES NO NATURAL NUMBER YES NO YES NO

34. ASSOCIATIVE PROPERTIES NUMBERS ASSOCIATIVE FOR ADDITION ( a + b )+ c = a + ( b + c ) SUBTRACTION ( a - b ) - c = a - ( b - c ) MULTIPLICATION ( a × b ) × c = a × ( b × c ) DIVISION ( a ÷ b ) ÷ c = a ÷ ( b ÷ c ) RATIONAL NUMBER YES NO YES NO INTEGERS YES NO YES NO WHOLE NUMBER YES NO YES NO NATURAL NUMBER YES NO YES NO

35. DISTRIBUTIVE PROPERTY DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION- a × ( b + c ) = (a × b ) + (a × c) DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER SUBTRACTION- a × ( b - c ) = (a × b ) - (a × c) ADDITIVE IDENTITY PROPERTY - a + 0 = a ADDITIVE INVERSE PROPERTY - a + (-a) = 1 MULTIPLICATIVE IDENTITY PROPERTY - a × = 1 ( ‘a’ can’t be 0 ) MULTIPLICATIVE INVERSE PROPERTY - a × 1 = a

36. REPRESENTATION OF RATIONAL NUMBER ON NUMBER LINE -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Let us understand how to represent a rational number on a number line. Let us understand that there are many numbers between any two integers. 1 2

37. RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBER There are many rational numbers between any two rational number Let the two rational numbers be You may thought there are only But if I multiply 10 in both numerator and denominator we get - So numbers between are We will find the rational numbers in two different ways LCM method Average method

38. LCM METHORD OF FINDING RATIONAL NUMBER When you are given with two rational numbers and asked to find some rational numbers between them then follow the steps- STEP 1 - Make sure that the denominator of the two fractions are same otherwise make the denominator same by taking the LCM. STEP 2 – If the desire number of rational numbers are present in between the resultant rational numbers the answer is over STEP 3 – If desire number of rational numbers are not obtained then multiply suitable numbers at numerator as well as denominator and get the resultant rational numbers

39. AVERAGE METHORD OF FINDING RATIONAL NUMBER Let us assume that ‘x’ and ‘y’ are two rational numbers 1 st rational number between x and y is 2 nd rational number between 3 rd rational number between y and is x y x y and so on.........

40. SOME IMPORTANT CONCEPTS Comparison of two rational numbers can be done by making there denominator same (taking LCM) and then comparing. Equality of two rational number can be checked by cross multiplication method- (numerator of 1 st ) × (denominator of 2 nd ) = (numerator of 2 nd ) × (denominator of 1 st ) Methods of finding prime numbers- (Sieve of Eratosthenes) Let us assume that we are given with a number ‘n’ and we have to check weather it is a prime number or not. First of all find and see all the prime numbers less than and check weather ‘n’ is divisible by these prime numbers or not. Example- Let the number be 223. So clearly . Prime numbers below 14 are { 2 , 3 , 5 , 7 , 11 , 13 } and 223 is not divisible by any of these numbers. So it is a prime number

41. VARIOUS TYPES OF QUESTIONS TYPE 1 – Application of Mathematical operations of rational numbers (BODMAS Formula) TYPE 2 – Proper use of all the properties of rational numbers TYPE 3 – Rational numbers between two rational numbers TYPE 4 – Representation of rational number on number line TYPE 5 – Comparison of two rational numbers TYPE 6 – Verification of properties of rational numbers

42. TYPE 1 – Application of Mathematical operations of rational numbers (BODMAS Formula) Add Sol- Subtract Sol- Multiply Sol-

43. TYPE 2 – Proper use of all the properties of rational numbers Solve Sol-

44. TYPE 3 – Rational numbers between two rational numbers Find 10 rational numbers between Sol- LCM of 6 and 8 is 24 ,so Thus we have and we can choose any 10 out of these.

45. TYPE 4 – Representation of rational number on number line Represent Sol- ( i ) Clearly Divide the line between 1 and 2 into 4 equal parts 1 2 -7 7

46. TYPE 5 – Comparison of two rational numbers Compare the rational numbers Lets make there LCM same and then compare them. So, and

47. TYPE 6 – Verification of properties of rational numbers Verify : for , , We have , LHS = RHS =

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