CLASS VII -operations on rational numbers(1).pptx

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1. OPERATIONS ON RATIONAL NUMBERS SL. NO SUB-TOPIC SLIDE NO 1 INTRODUCTION 3 2 ADDITION OF RATIONAL NUMBERS AND ITS PROPERTIES 5 3 SUBTRACTION OF RATIONAL NUMBERS AND ITS PROPERTIES 8 4 MULTIPLICATION OF RATIONAL NUMBERS AND ITS PROPERTIES 12 5 DIVISION OF RATIONAL NUMBERS AND ITS PROPERTIES 18 6 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS 24

2. INTRODUCTION O ne day two children of Real number, Whole number and Rational number have a talk with each other; Whole number: Do you know rational ,different operations like Addition,Subtraction ,MULTIPLICATION and DIVISION can act on me and they also satisfies different properties. Rational number : But brother I do not have any knowledge that how these operations act on me. Let’s ask our teacher about this… Teacher- I will definitely clear your doubts but before that answer my questions… What is the additive identity for whole numbers ? Does whole number obeys commutative law under Subtraction ? What is the multiplicative inverse of 5 ? Define distributive property of Multiplication over Addition ? Say two whole numbers between 5 and 12 ? Tell whether division obeys closure property or not ? After getting correct answers teacher proceed further ..

3. T he numbers of the form , Where Rational numbers include natural numbers, whole numbers, integer sand all positive and negative fractions. Example: , ,… etc   Rational Numbers Can you say why ‘0’ is a Rational number ? As that of Whole numbers for rational numbers,also there are four operations they are as following. Let’s discuss one after another… ADDITION SUBTRACTION MULTIPLICATION DIVISION Click here for DAV CAE Book..

4. PRACTICE 1. Add and 2. Add and   PRACTICE 1. Add and 2. Add and  

5. PROPERTIES OF ADDITION OF RATIONAL NUMBERS Closure property : Sum of any two rational number is always a rational number i.e if x and y are two rational numbers then x+y will also became a rational number. EXAMPLE : = Here and are both Rational number And sum is also a rational number. Hence Rational numbers are closed under Addition. Commutative property : The sum of two rational numbers does not depend on the order in which they are added.i.e where x and y are rational numbers. Example : Verify that sum of two rational number and remains same even if the order of addends are changed. LHS= = = RHS= = = LHS=RHS Hence Addition of rational number satisfy Commutative property Associative Property : The sum of three rational numbers remains same even after changing the grouping of addends. If EXAMPLE :Verify that for x = ,y = and z = i.e sum will remains same even after changing the grouping of numbers.   LHS= + = + = = =   RHS= = = = =   Hence Addition of rational number satisfy Associative property.

6. Additive identity property: W hen ‘o’ is added to any rational number the sum of the rational number is becomes itself, If x is a rational number then . EXAMPLE: let LHS = RHS= As LHS = RHS so this property valids . Zero is called the identity element in Addition . Additive inverse property: The negative of a rational number is it’s additive inverse. The sum of a rational number and its additive inverse is always equal to Zero. If x is a rational number then (-x) is the additive inverse of it and Let a rational number be Then = Negative of a rational number is called its additive inverse.   Try these — (a) Simplify (b) For x = and y= , verify that –( x+y ) = (-x) + (-y) (c) For x = and y= and z= , verify that ( x+y )+ z = x + ( y+z )  

7. PRACTICE 1. Add and 2. Add and   PRACTICE 1. Subtract and 2. Subtract and  

8. PROPERTIES OF SUBTRACTION OF RATIONAL NUMBERS EXAMPLE : Difference of any two rational number is always a rational number i.e if x and y are two rational numbers then x - y will also became a rational number. VERIFICATION : = Here and are both Rational numbers And their Difference is also a rational number. Hence Rational numbers are closed under Subtraction . PROPERTY-1 : The difference of two rational numbers does not remain same if their order is changed i.e where x and y are rational numbers. Example : Verify that difference of two rational number and does not remains same even if the order are changed. LHS= = = , RHS= = = LHS RHS Hence Subtraction of rational number does not satisfy Commutative property. PROPERTY-2: The Difference of three rational numbers does not remains same even after changing the grouping of numbers.i.e If EXAMPLE :Verify that for x = ,y = and z =   LHS= - = = = =   RHS= = = = =   Hence Subtraction of rational number does not satisfy Associative property.

9. EXAMPLE ; For all rational number we have Show that,If x is a rational number then . VERIFICATION: let LHS = But RHS= As LHS RHS so this property does not valid. I dentity element in Subtraction does not exist . EXAMPLE : Since identity of subtraction does not exist so inverse for Subtraction does not arise.   Try thesee — (a) Simplify (b) For x = and y= and z= , verify that (x-y)-z = x –(y-z)  

10. MULTIPLICATION OF RATIONAL NUMBERS Example: Find the product of and Solution: = =   If and are two rational numbers then =   Solve the followings… Multiply and Multiply 3 and 1  

11. PROPERTIES OF MULTIPLICATION OF RATIONAL NUMBERS CLOSURE PROPERTY : Product of any two rational number is always a rational number EXAMPLE: = Here and are both Rational number And product are rational number Hence Rational numbers are closed under multiplication. COMMUTATIVE PROPERTY : The product of two rational numbers remain same even if we change their order . If and are two rational number then Example : Verify that product of two rational number and remains same even if the order is changed . LHS= = Hence LHS=RHS   RHS= =   Commutative property holds TRUE in multiplication of rational number.

12. Associative Property: The product of three rational numbers remains same even after changing the groups. If Example : Verify that product of three rational numbers , and remains same even after changing the groupings.   LHS= = =   RHS= = =   Hence rational number obeys associative under multiplication. LHS=RH S

13. MULTIPLICATIVE IDENTITY: W hen 1 is multiplied to any rational number then the product of the rational number is equal to itself If x is a rational number then Example: let   LHS= RHS=   1 is the identity element under the multiplication PRODUCT OF A RATIONAL NUMBER AND ZERO IS ALWAYS ZERO If EXAMPLE : The product of = Again  

14. Distributive property : If ( ii) Example: For three rational number   LHS = = = = =   RHS= ( = = = =   LHS=RHS Hence verified… So Rational numbers obeys Distributive property of multiplication over Addition and Subtraction.

15. Reciprocal of a rational number(Multiplicative Inverse ) Reciprocal of rational number means interchanging of numerator and denominator in other words Rational number obtained after inverting the given rational number is called Reciprocal of rational number Example : The Reciprocal of is   Note: (1)Zero has no Reciprocal (ii)Reciprocal of 1 is 1 (iii) if x is any non zero rational number,then its reciprocal is denoted by which is equal to  

16. DIVISION OF RATIONAL NUMBERS EXAMPLE: Divide Solution :   Dividing one rational number by another except by zero, is the same as the multiplication of the first by the reciprocal of the second, i.e x÷y = x y -1   Divide followings… -10 by by -2  

17. PROPORTIES OF DIVISION OF RATIONAL NUMBER Property: 1 Division of a rational number by another rational number except zero is always rational number. Example: (i) (ii) (iii) Property-2 When a nonzero rational number divided by itself we get always 1 Example : Property-3 When non zero rational number divided by 1 the quotient is the same rational number Example =  

18. PROPERTY-4 The division of two rational number does not remain same if the order of the number are changed. If are two rational number then Example:Verify that if and then   LHS= = = =   RHS= = = =   LHS RHS    

19. PROPERTY-5 The division of three rational numbers does not remains same even after changing the groups. If Example: Verify that if , , then   LHS= = = = =   RHS = ) = = = =   LHS RHS   Associative property does not holds true for division

20. PROPERTY-6: If are rational numbers, then and ) Example : If   LHS = = = = =   RHS = ) = = = =   LHS = RHS

21. PROPERTY- 7 For three non-zero rational numbers Example :- If show that   LHS= = = = =   RHS= = = = =   LHS RHS   Distributive property does not hold true for division

22. RATIONALS BETWEEN TWO RATIONAL NUMBERS Example : Find two rational number between Solution: Step-1 Find a rational number between Step-2 Find a rational number between Hence , are two rational number between   If .  

23. Introduction of Rational number Operation of rational number Properties of addition of rational number Properties of subtraction of rational number Properties of Multiplication of rational number Properties of division of rational number Insert rational numbers between two given rational number KEY POINTS

24. TATA DAV PUBLIC SCHOOL,JODA KEONJHAR,ODISHA