Page 1 RATIONAL NUMBERS 1 1.1 Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + 2 = 13 (1) is solved when x = 11, because this value of x satisfies the given equation. The solution 11 is a natural number. On the other hand, for the equation x + 5 = 5 (2) the solution gives the whole number 0 (zero). If we consider only natural numbers, equation (2) cannot be solved. T o solve equations like (2), we added the number zero to the collection of natural numbers and obtained the whole numbers. Even whole numbers will not be sufficient to solve equations of type x + 18 = 5 (3) Do you see ‘why’? W e require the number –13 which is not a whole number. This led us to think of integers, (positive and negative). Note that the positive integers correspond to natural numbers. One may think that we have enough numbers to solve all simple equations with the available list of integers. Now consider the equations 2x = 3 (4) 5x + 7 = 0 (5) for which we cannot find a solution from the integers. (Check this) W e need the numbers 3 2 to solve equation (4) and 7 5  to solve equation (5). This leads us to the collection of rational numbers. We have already seen basic operations on rational numbers. W e now try to explore some properties of operations on the different types of numbers seen so far. Rational Numbers CHAPTER 1 201920 Page 2 RATIONAL NUMBERS 1 1.1 Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + 2 = 13 (1) is solved when x = 11, because this value of x satisfies the given equation. The solution 11 is a natural number. On the other hand, for the equation x + 5 = 5 (2) the solution gives the whole number 0 (zero). If we consider only natural numbers, equation (2) cannot be solved. T o solve equations like (2), we added the number zero to the collection of natural numbers and obtained the whole numbers. Even whole numbers will not be sufficient to solve equations of type x + 18 = 5 (3) Do you see ‘why’? W e require the number –13 which is not a whole number. This led us to think of integers, (positive and negative). Note that the positive integers correspond to natural numbers. One may think that we have enough numbers to solve all simple equations with the available list of integers. Now consider the equations 2x = 3 (4) 5x + 7 = 0 (5) for which we cannot find a solution from the integers. (Check this) W e need the numbers 3 2 to solve equation (4) and 7 5  to solve equation (5). This leads us to the collection of rational numbers. We have already seen basic operations on rational numbers. W e now try to explore some properties of operations on the different types of numbers seen so far. Rational Numbers CHAPTER 1 201920 2 MATHEMATICS 1.2 Properties of Rational Numbers 1.2.1 Closure (i) Whole numbers Let us revisit the closure property for all the operations on whole numbers in brief. Operation Numbers Remarks Addition 0 + 5 = 5, a whole number Whole numbers are closed 4 + 7 = ... . Is it a whole number? under addition. In general, a + b is a whole number for any two whole numbers a and b. Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed whole number . under subtraction. Multiplication 0 × 3 = 0, a whole number Whole numbers are closed 3 × 7 = ... . Is it a whole number? under multiplication. In general, if a and b are any two whole numbers, their product ab is a whole number. Division 5 ÷ 8 = 5 8 , which is not a whole number . Check for closure property under all the four operations for natural numbers. (ii) Integers Let us now recall the operations under which integers are closed. Operation Numbers Remarks Addition – 6 + 5 = – 1, an integer Integers are closed under Is – 7 + (–5) an integer? addition. Is 8 + 5 an integer? In general, a + b is an integer for any two integers a and b. Subtraction 7 – 5 = 2, an integer Integers are closed under Is 5 – 7 an integer? subtraction. – 6 – 8 = – 14, an integer Whole numbers are not closed under division. 201920 Page 3 RATIONAL NUMBERS 1 1.1 Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + 2 = 13 (1) is solved when x = 11, because this value of x satisfies the given equation. The solution 11 is a natural number. On the other hand, for the equation x + 5 = 5 (2) the solution gives the whole number 0 (zero). If we consider only natural numbers, equation (2) cannot be solved. T o solve equations like (2), we added the number zero to the collection of natural numbers and obtained the whole numbers. Even whole numbers will not be sufficient to solve equations of type x + 18 = 5 (3) Do you see ‘why’? W e require the number –13 which is not a whole number. This led us to think of integers, (positive and negative). Note that the positive integers correspond to natural numbers. One may think that we have enough numbers to solve all simple equations with the available list of integers. Now consider the equations 2x = 3 (4) 5x + 7 = 0 (5) for which we cannot find a solution from the integers. (Check this) W e need the numbers 3 2 to solve equation (4) and 7 5  to solve equation (5). This leads us to the collection of rational numbers. We have already seen basic operations on rational numbers. W e now try to explore some properties of operations on the different types of numbers seen so far. Rational Numbers CHAPTER 1 201920 2 MATHEMATICS 1.2 Properties of Rational Numbers 1.2.1 Closure (i) Whole numbers Let us revisit the closure property for all the operations on whole numbers in brief. Operation Numbers Remarks Addition 0 + 5 = 5, a whole number Whole numbers are closed 4 + 7 = ... . Is it a whole number? under addition. In general, a + b is a whole number for any two whole numbers a and b. Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed whole number . under subtraction. Multiplication 0 × 3 = 0, a whole number Whole numbers are closed 3 × 7 = ... . Is it a whole number? under multiplication. In general, if a and b are any two whole numbers, their product ab is a whole number. Division 5 ÷ 8 = 5 8 , which is not a whole number . Check for closure property under all the four operations for natural numbers. (ii) Integers Let us now recall the operations under which integers are closed. Operation Numbers Remarks Addition – 6 + 5 = – 1, an integer Integers are closed under Is – 7 + (–5) an integer? addition. Is 8 + 5 an integer? In general, a + b is an integer for any two integers a and b. Subtraction 7 – 5 = 2, an integer Integers are closed under Is 5 – 7 an integer? subtraction. – 6 – 8 = – 14, an integer Whole numbers are not closed under division. 201920 RATIONAL NUMBERS 3 – 6 – (– 8) = 2, an integer Is 8 – (– 6) an integer? In general, for any two integers a and b, a – b is again an integer. Check if b – a is also an integer. Multiplication 5 × 8 = 40, an integer Integers are closed under Is – 5 × 8 an integer? multiplication. – 5 × (– 8) = 40, an integer In general, for any two integers a and b, a × b is also an integer. Division 5 ÷ 8 = 5 8 , which is not Integers are not closed an integer. under division. Y ou have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However , integers are closed under addition, subtraction and multiplication but not under division. (iii) Rational numbers Recall that a number which can be written in the form p q , where p and q are integers and q ? 0 is called a rational number. For example, 2 3  , 6 7 , 9 5  are all rational numbers. Since the numbers 0, –2, 4 can be written in the form p q , they are also rational numbers. (Check it!) (a) Y ou know how to add two rational numbers. Let us add a few pairs. 3 ( 5) 8 7  + = 21 ( 40) 19 56 56 +   = (a rational number) 3 ( 4) 8 5   + = 15 ( 32) ... 40  +  = Is it a rational number? 4 6 7 11 + = ... Is it a rational number? W e find that sum of two rational numbers is again a rational number. Check it for a few more pairs of rational numbers. We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number. (b) Will the difference of two rational numbers be again a rational number? We have, 5 2 7 3   = 5 3 – 2 7 29 21 21  × ×  = (a rational number) 201920 Page 4 RATIONAL NUMBERS 1 1.1 Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + 2 = 13 (1) is solved when x = 11, because this value of x satisfies the given equation. The solution 11 is a natural number. On the other hand, for the equation x + 5 = 5 (2) the solution gives the whole number 0 (zero). If we consider only natural numbers, equation (2) cannot be solved. T o solve equations like (2), we added the number zero to the collection of natural numbers and obtained the whole numbers. Even whole numbers will not be sufficient to solve equations of type x + 18 = 5 (3) Do you see ‘why’? W e require the number –13 which is not a whole number. This led us to think of integers, (positive and negative). Note that the positive integers correspond to natural numbers. One may think that we have enough numbers to solve all simple equations with the available list of integers. Now consider the equations 2x = 3 (4) 5x + 7 = 0 (5) for which we cannot find a solution from the integers. (Check this) W e need the numbers 3 2 to solve equation (4) and 7 5  to solve equation (5). This leads us to the collection of rational numbers. We have already seen basic operations on rational numbers. W e now try to explore some properties of operations on the different types of numbers seen so far. Rational Numbers CHAPTER 1 201920 2 MATHEMATICS 1.2 Properties of Rational Numbers 1.2.1 Closure (i) Whole numbers Let us revisit the closure property for all the operations on whole numbers in brief. Operation Numbers Remarks Addition 0 + 5 = 5, a whole number Whole numbers are closed 4 + 7 = ... . Is it a whole number? under addition. In general, a + b is a whole number for any two whole numbers a and b. Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed whole number . under subtraction. Multiplication 0 × 3 = 0, a whole number Whole numbers are closed 3 × 7 = ... . Is it a whole number? under multiplication. In general, if a and b are any two whole numbers, their product ab is a whole number. Division 5 ÷ 8 = 5 8 , which is not a whole number . Check for closure property under all the four operations for natural numbers. (ii) Integers Let us now recall the operations under which integers are closed. Operation Numbers Remarks Addition – 6 + 5 = – 1, an integer Integers are closed under Is – 7 + (–5) an integer? addition. Is 8 + 5 an integer? In general, a + b is an integer for any two integers a and b. Subtraction 7 – 5 = 2, an integer Integers are closed under Is 5 – 7 an integer? subtraction. – 6 – 8 = – 14, an integer Whole numbers are not closed under division. 201920 RATIONAL NUMBERS 3 – 6 – (– 8) = 2, an integer Is 8 – (– 6) an integer? In general, for any two integers a and b, a – b is again an integer. Check if b – a is also an integer. Multiplication 5 × 8 = 40, an integer Integers are closed under Is – 5 × 8 an integer? multiplication. – 5 × (– 8) = 40, an integer In general, for any two integers a and b, a × b is also an integer. Division 5 ÷ 8 = 5 8 , which is not Integers are not closed an integer. under division. Y ou have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However , integers are closed under addition, subtraction and multiplication but not under division. (iii) Rational numbers Recall that a number which can be written in the form p q , where p and q are integers and q ? 0 is called a rational number. For example, 2 3  , 6 7 , 9 5  are all rational numbers. Since the numbers 0, –2, 4 can be written in the form p q , they are also rational numbers. (Check it!) (a) Y ou know how to add two rational numbers. Let us add a few pairs. 3 ( 5) 8 7  + = 21 ( 40) 19 56 56 +   = (a rational number) 3 ( 4) 8 5   + = 15 ( 32) ... 40  +  = Is it a rational number? 4 6 7 11 + = ... Is it a rational number? W e find that sum of two rational numbers is again a rational number. Check it for a few more pairs of rational numbers. We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number. (b) Will the difference of two rational numbers be again a rational number? We have, 5 2 7 3   = 5 3 – 2 7 29 21 21  × ×  = (a rational number) 201920 4 MATHEMATICS TRY THESE 5 4 8 5  = 25 32 40  = ... Is it a rational number? 3 7 8 5   ? ? ? ? ? ? = ... Is it a rational number? Try this for some more pairs of rational numbers. W e find that rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number. (c) Let us now see the product of two rational numbers. 2 4 3 5  × = 8 3 2 6 ; 15 7 5 35  × = (both the products are rational numbers) 4 6 5 11   × = ... Is it a rational number? T ake some more pairs of rational numbers and check that their product is again a rational number. We say that rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a × b is also a rational number. (d) We note that 5 2 25 3 5 6   ÷ = (a rational number) 2 5 ... 7 3 ÷ = . Is it a rational number? 3 2 ... 8 9   ÷ = . Is it a rational number? Can you say that rational numbers are closed under division? We find that for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. However , if we exclude zero then the collection of, all other rational numbers is closed under division. Fill in the blanks in the following table. Numbers Closed under addition subtraction multiplication division Rational numbers Y es Y es ... No Integers ... Y es ... No Whole numbers ... ... Y es ... Natural numbers ... No ... ... 201920 Page 5 RATIONAL NUMBERS 1 1.1 Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + 2 = 13 (1) is solved when x = 11, because this value of x satisfies the given equation. The solution 11 is a natural number. On the other hand, for the equation x + 5 = 5 (2) the solution gives the whole number 0 (zero). If we consider only natural numbers, equation (2) cannot be solved. T o solve equations like (2), we added the number zero to the collection of natural numbers and obtained the whole numbers. Even whole numbers will not be sufficient to solve equations of type x + 18 = 5 (3) Do you see ‘why’? W e require the number –13 which is not a whole number. This led us to think of integers, (positive and negative). Note that the positive integers correspond to natural numbers. One may think that we have enough numbers to solve all simple equations with the available list of integers. Now consider the equations 2x = 3 (4) 5x + 7 = 0 (5) for which we cannot find a solution from the integers. (Check this) W e need the numbers 3 2 to solve equation (4) and 7 5  to solve equation (5). This leads us to the collection of rational numbers. We have already seen basic operations on rational numbers. W e now try to explore some properties of operations on the different types of numbers seen so far. Rational Numbers CHAPTER 1 201920 2 MATHEMATICS 1.2 Properties of Rational Numbers 1.2.1 Closure (i) Whole numbers Let us revisit the closure property for all the operations on whole numbers in brief. Operation Numbers Remarks Addition 0 + 5 = 5, a whole number Whole numbers are closed 4 + 7 = ... . Is it a whole number? under addition. In general, a + b is a whole number for any two whole numbers a and b. Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed whole number . under subtraction. Multiplication 0 × 3 = 0, a whole number Whole numbers are closed 3 × 7 = ... . Is it a whole number? under multiplication. In general, if a and b are any two whole numbers, their product ab is a whole number. Division 5 ÷ 8 = 5 8 , which is not a whole number . Check for closure property under all the four operations for natural numbers. (ii) Integers Let us now recall the operations under which integers are closed. Operation Numbers Remarks Addition – 6 + 5 = – 1, an integer Integers are closed under Is – 7 + (–5) an integer? addition. Is 8 + 5 an integer? In general, a + b is an integer for any two integers a and b. Subtraction 7 – 5 = 2, an integer Integers are closed under Is 5 – 7 an integer? subtraction. – 6 – 8 = – 14, an integer Whole numbers are not closed under division. 201920 RATIONAL NUMBERS 3 – 6 – (– 8) = 2, an integer Is 8 – (– 6) an integer? In general, for any two integers a and b, a – b is again an integer. Check if b – a is also an integer. Multiplication 5 × 8 = 40, an integer Integers are closed under Is – 5 × 8 an integer? multiplication. – 5 × (– 8) = 40, an integer In general, for any two integers a and b, a × b is also an integer. Division 5 ÷ 8 = 5 8 , which is not Integers are not closed an integer. under division. Y ou have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However , integers are closed under addition, subtraction and multiplication but not under division. (iii) Rational numbers Recall that a number which can be written in the form p q , where p and q are integers and q ? 0 is called a rational number. For example, 2 3  , 6 7 , 9 5  are all rational numbers. Since the numbers 0, –2, 4 can be written in the form p q , they are also rational numbers. (Check it!) (a) Y ou know how to add two rational numbers. Let us add a few pairs. 3 ( 5) 8 7  + = 21 ( 40) 19 56 56 +   = (a rational number) 3 ( 4) 8 5   + = 15 ( 32) ... 40  +  = Is it a rational number? 4 6 7 11 + = ... Is it a rational number? W e find that sum of two rational numbers is again a rational number. Check it for a few more pairs of rational numbers. We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number. (b) Will the difference of two rational numbers be again a rational number? We have, 5 2 7 3   = 5 3 – 2 7 29 21 21  × ×  = (a rational number) 201920 4 MATHEMATICS TRY THESE 5 4 8 5  = 25 32 40  = ... Is it a rational number? 3 7 8 5   ? ? ? ? ? ? = ... Is it a rational number? Try this for some more pairs of rational numbers. W e find that rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number. (c) Let us now see the product of two rational numbers. 2 4 3 5  × = 8 3 2 6 ; 15 7 5 35  × = (both the products are rational numbers) 4 6 5 11   × = ... Is it a rational number? T ake some more pairs of rational numbers and check that their product is again a rational number. We say that rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a × b is also a rational number. (d) We note that 5 2 25 3 5 6   ÷ = (a rational number) 2 5 ... 7 3 ÷ = . Is it a rational number? 3 2 ... 8 9   ÷ = . Is it a rational number? Can you say that rational numbers are closed under division? We find that for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. However , if we exclude zero then the collection of, all other rational numbers is closed under division. Fill in the blanks in the following table. Numbers Closed under addition subtraction multiplication division Rational numbers Y es Y es ... No Integers ... Y es ... No Whole numbers ... ... Y es ... Natural numbers ... No ... ... 201920 RATIONAL NUMBERS 5 1.2.2 Commutativity (i) Whole numbers Recall the commutativity of different operations for whole numbers by filling the following table. Operation Numbers Remarks Addition 0 + 7 = 7 + 0 = 7 Addition is commutative. 2 + 3 = ... + ... = .... For any two whole numbers a and b, a + b = b + a Subtraction ......... Subtraction is not commutative. Multiplication ......... Multiplication is commutative. Division ......... Division is not commutative. Check whether the commutativity of the operations hold for natural numbers also. (ii) Integers Fill in the following table and check the commutativity of different operations for integers: Operation Numbers Remarks Addition ......... Addition is commutative. Subtraction Is 5 – (–3) = – 3 – 5? Subtraction is not commutative. Multiplication ......... Multiplication is commutative. Division ......... Division is not commutative. (iii) Rational numbers (a) Addition Y ou know how to add two rational numbers. Let us add a few pairs here. 2 5 1 5 2 1 and 3 7 21 7 3 21   ? ? + = + = ? ? ? ? So, 2 5 5 2 3 7 7 3   ? ? + = + ? ? ? ? Also,  +  ? ? ? ? ? ? 6 5 8 3 = ... and Is  +  ? ? ? ? ? ? =  ? ? ? ? ? ? +  ? ? ? ? ? ? 6 5 8 3 6 5 8 3 ? 201920Read More
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