Welcome dear students! Today we are going to learn about Rational Numbers from Class 8 Maths.
In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + 2 = 13 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 the solution gives the whole number 0. If we consider only natural numbers, this second equation cannot be solved. To solve equations like this, 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. Do you see why? We require the number -13 which is not a whole number. This led us to think of integers, which include positive and negative numbers. 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 and 5x + 7 = 0 for which we cannot find a solution from the integers. Check this yourself. We need the numbers 3/2 to solve the first equation and -7/5 to solve the second equation. This leads us to the collection of rational numbers. We have already seen basic operations on rational numbers. We now try to explore some properties of operations on the different types of numbers seen so far.
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Let us revisit the closure property for all the operations on whole numbers in brief. For addition, 0 + 5 = 5, a whole number. 4 + 7 = 11, which is also a whole number. In general, a + b is a whole number for any two whole numbers a and b. So whole numbers are closed under addition. For subtraction, 5 - 7 = -2, which is not a whole number. So whole numbers are not closed under subtraction. For multiplication, 0 × 3 = 0, a whole number. 3 × 7 = 21, a whole number. In general, if a and b are any two whole numbers, their product ab is a whole number. Whole numbers are closed under multiplication. For division, 5 ÷ 8 = 5/8, which is not a whole number. Whole numbers are not closed under division. You should check for closure property under all four operations for natural numbers yourself. Now let us recall the operations under which integers are closed. For addition, -6 + 5 = -1, an integer. Is -7 + (-5) an integer? Yes, it is -12. Is 8 + 5 an integer? Yes, it is 13. In general, a + b is an integer for any two integers a and b. Integers are closed under addition. For subtraction, 7 - 5 = 2, an integer. Is 5 - 7 an integer? Yes, -2. -6 - 8 = -14, an integer. -6 - (-8) = 2, an integer. Is 8 - (-6) an integer? Yes, 14. In general, for any two integers a and b, a - b is again an integer. Check if b - a is also an integer. Integers are closed under subtraction. For multiplication, 5 × 8 = 40, an integer. Is -5 × 8 an integer? Yes, -40. -5 × (-8) = 40, an integer. In general, for any two integers a and b, a × b is also an integer. Integers are closed under multiplication. For division, 5 ÷ 8 = 5/8, which is not an integer. Integers are not closed under division. You 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.
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Now let us look at 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 yourself. Let us add a few pairs of rational numbers. 3/8 + (-5)/7 = (21+(-40))/56 = -19/56, a rational number. (-3)/8 + (-4)/5 = (-15+(-32))/40. The textbook leaves this as a blank for you to solve. The result is -47/40. Is it a rational number? Yes. 4/7 + 6/11. The textbook leaves this blank. The sum is 86/77. Is it a rational number? Yes. We 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. Will the difference of two rational numbers be again a rational number? We have (-5)/7 - 2/3 = (-5×3-2×7)/21 = -29/21, a rational number. 5/8 - 4/5 = (25-32)/40. The textbook leaves this blank. The difference is -7/40. Is it a rational number? Yes. 3/7 - (-8/5). The textbook leaves this blank. The difference is 71/35. Is it a rational number? Yes. Try this for some more pairs of rational numbers. We 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. Let us now see the product of two rational numbers. (-2)/3 × 4/5 = -8/15. 3/7 × 2/5 = 6/35. Both products are rational numbers. -4/5 × (-6)/11. The textbook leaves this blank. The product is 24/55. Is it a rational number? Yes. Take 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. We note that (-5)/3 ÷ 2/5 = -25/6, a rational number. 2/7 ÷ 5/3. The textbook leaves this blank. The quotient is 6/35. Is it a rational number? Yes. (-3)/8 ÷ (-2)/9. The textbook leaves this blank. The quotient is 27/16. Is it a rational number? Yes. 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.
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Now let us complete the closure table for different number systems. For rational numbers, they are closed under addition, closed under subtraction, closed under multiplication, and not closed under division. For integers, they are closed under addition, closed under subtraction, closed under multiplication, and not closed under division. For whole numbers, they are closed under addition, not closed under subtraction, closed under multiplication, and not closed under division. For natural numbers, they are closed under addition, not closed under subtraction, closed under multiplication, and not closed under division.
Let us recall the commutativity of different operations for whole numbers. For addition, 0 + 7 = 7 + 0 = 7. 2 + 3 = 3 + 2 = 5. For any two whole numbers a and b, a + b = b + a. Addition is commutative. For subtraction, it is not commutative. For multiplication, it is commutative. For division, it is not commutative. Check whether the commutativity of the operations hold for natural numbers also. Now let us check commutativity for integers. Addition is commutative. For subtraction, is 5 - (-3) = -3 - 5? No, 8 does not equal -8. Subtraction is not commutative. Multiplication is commutative. Division is not commutative.
Now for rational numbers. Let us add a few pairs. (-2)/3 + 5/7 = 1/21 and 5/7 + ((-2)/3) = 1/21. So (-2)/3 + 5/7 = 5/7 + ((-2)/3). Also (-6)/5 + ((-8)/3) = -58/15 and (-8)/3 + ((-6)/5) = -58/15. Is (-6)/5 + ((-8)/3) = ((-8)/3) + ((-6)/5)? Yes. Is (-3)/8 + 1/7 = 1/7 + ((-3)/8)? Yes. You find that two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, a + b = b + a. Is 2/3 - 5/4 = 5/4 - 2/3? No. Is 1/2 - 3/5 = 3/5 - 1/2? No. You will find that subtraction is not commutative for rational numbers. Note that subtraction is not commutative for integers and integers are also rational numbers. So subtraction will not be commutative for rational numbers too.
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For multiplication, we have (-7)/3 × 6/5 = (-42)/15 = 6/5 × ((-7)/3). Is (-8)/9 × ((-4)/7) = (-4)/7 × ((-8)/9)? Yes. Check for some more such products. You will find that multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b. For division, is (-5)/4 ÷ 3/7 = 3/7 ÷ ((-5)/4)? No. You will find that expressions on both sides are not equal. So division is not commutative for rational numbers.
Let us complete the commutativity table. Rational numbers are commutative for addition, not for subtraction, commutative for multiplication, and not for division. Integers are commutative for addition, not for subtraction, commutative for multiplication, and not for division. Whole numbers are commutative for addition, not for subtraction, commutative for multiplication, and not for division. Natural numbers are commutative for addition, not for subtraction, commutative for multiplication, and not for division.
Now let us look at associativity. For whole numbers, addition is associative. Subtraction is not associative. Multiplication is associative. Division is not associative. Check for yourself the associativity of different operations for natural numbers. For integers, addition is associative. Subtraction is not associative. Multiplication is associative. Division is not associative.
Now for rational numbers. We have (-2)/3 + [3/5 + ((-5)/6)] = (-2)/3 + ((-7)/30) = (-27)/30 = (-9)/10. [(-2)/3 + 3/5] + ((-5)/6) = (-1)/15 + ((-5)/6) = (-27)/30 = (-9)/10. So (-2)/3 + [3/5 + ((-5)/6)] = [(-2)/3 + 3/5] + ((-5)/6). Find (-1)/2 + [3/7 + ((-4)/3)] and [(-1)/2 + 3/7] + ((-4)/3). Are the two sums equal? Yes. Take some more rational numbers, add them as above and see if the two sums are equal. We find that addition is associative for rational numbers. That is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
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You already know that subtraction is not associative for integers, then what about rational numbers. Is (-2)/3 - [(-4)/5 - 1/2] = [(-2)/3 - ((-4)/5)] - 1/2? Check for yourself. Subtraction is not associative for rational numbers. Let us check the associativity for multiplication. (-7)/3 × (5/4 × 2/9) = (-7)/3 × 10/36 = (-70)/108 = (-35)/54. ((-7)/3 × 5/4) × 2/9 = (-35)/12 × 2/9 = (-70)/108 = (-35)/54. We find that (-7)/3 × (5/4 × 2/9) = ((-7)/3 × 5/4) × 2/9. Is 2/3 × ((-6)/7 × 4/5) = (2/3 × ((-6)/7)) × 4/5? Yes. Take some more rational numbers and check for yourself. We observe that multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c. Recall that division is not associative for integers, then what about rational numbers? Let us see if 1/2 ÷ [(-1)/3 ÷ 2/5] = [1/2 ÷ ((-1)/3)] ÷ 2/5. We have LHS = 1/2 ÷ ((-1)/3 ÷ 2/5) = 1/2 ÷ ((-1)/3 × 5/2) = 1/2 ÷ (-5/6) = -3/5. RHS = [1/2 ÷ ((-1)/3)] ÷ 2/5 = (1/2 × (-3)/1) ÷ 2/5 = (-3)/2 ÷ 2/5 = -15/4. Is LHS = RHS? No. Check for yourself. You will find that division is not associative for rational numbers.
Let us complete the associativity table. Rational numbers are associative for addition, not for subtraction, associative for multiplication, and not for division. Integers are associative for addition, not for subtraction, associative for multiplication, and not for division. Whole numbers are associative for addition, not for subtraction, associative for multiplication, and not for division. Natural numbers are associative for addition, not for subtraction, associative for multiplication, and not for division.
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Now let us solve Example 1. Find 3/7 + ((-6)/11) + ((-8)/21) + (5/22). Solution: 3/7 + ((-6)/11) + ((-8)/21) + (5/22) = 198/462 + ((-252)/462) + ((-176)/462) + (105/462). Note that 462 is the LCM of 7, 11, 21 and 22. This equals (198 - 252 - 176 + 105)/462 = (-125)/462. We can also solve it as 3/7 + (-6/11) + (-8/21) + 5/22 = [3/7 + (-8/21)] + [-6/11 + 5/22] by using commutativity and associativity. This equals [(9+(-8))/21] + [(-12+5)/22]. LCM of 7 and 21 is 21; LCM of 11 and 22 is 22. This equals 1/21 + (-7/22) = (22-147)/462 = -125/462. Do you think the properties of commutativity and associativity made the calculations easier? Yes, they did.
Now let us solve Example 2. Find (-4/5) × (3/7) × (15/16) × (-14/9). Solution: We have (-4/5) × (3/7) × (15/16) × (-14/9) = (-(4×3)/(5×7)) × ((15×(-14))/(16×9)) = (-12/35) × (-35/24) = (-12×(-35))/(35×24) = 1/2. We can also do it as (-4/5) × (3/7) × (15/16) × (-14/9) = ((-4/5) × (15/16)) × [(3/7) × (-14/9)] using commutativity and associativity. This equals (-3/4) × (-2/3) = 1/2.
Now let us look at section 1.2.4, The role of zero. Look at the following. 2 + 0 = 0 + 2 = 2. This is addition of 0 to a whole number. -5 + 0 = 0 + (-5) = -5. This is addition of 0 to an integer. (-2/7) + 0 = 0 + (-2/7) = -2/7. This is addition of 0 to a rational number. You have done such additions earlier also. Do a few more such additions. What do you observe? You will find that when you add 0 to a whole number, the sum is again that whole number. This happens for integers and rational numbers also. In general, a + 0 = 0 + a = a, where a is a whole number. b + 0 = 0 + b = b, where b is an integer. c + 0 = 0 + c = c, where c is a rational number. Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
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Now section 1.2.5, The role of 1. We have 5 × 1 = 5 = 1 × 5. This is multiplication of 1 with a whole number. -2/7 × 1 = 1 × (-2/7) = -2/7. 3/8 × 1 = 1 × 3/8 = 3/8. What do you find? You will find that when you multiply any rational number with 1, you get back the same rational number as the product. Check this for a few more rational numbers. You will find that a × 1 = 1 × a = a for any rational number a. We say that 1 is the multiplicative identity for rational numbers. Is 1 the multiplicative identity for integers? Yes. For whole numbers? Yes.
Think, Discuss and Write. If a property holds for rational numbers, will it also hold for integers? For whole numbers? Which will? Which will not? The properties of closure, commutativity, associativity, and distributivity that hold for rational numbers generally hold for integers and whole numbers for addition and multiplication. However, subtraction and division are not commutative or associative for integers and whole numbers either. The additive and multiplicative identities also hold for integers and whole numbers.
Now let us look at section 1.2.6, Distributivity of multiplication over addition for rational numbers. To understand this, consider the rational numbers -3/4, 2/3 and -5/6. -3/4 × {2/3 + (-5/6)} = -3/4 × (4/6 + (-5/6)) = -3/4 × (-1/6) = 3/24 = 1/8. Also -3/4 × 2/3 = (-3 × 2)/(4 × 3) = -6/12 = -1/2. And (-3/4) × (-5/6) = 5/8. Therefore ((-3/4) × (2/3)) + ((-3/4) × (-5/6)) = (-1/2) + (5/8) = 1/8. Thus, (-3/4) × {2/3 + (-5/6)} = ((-3/4) × (2/3)) + ((-3/4) × (-5/6)). Distributivity of Multiplication over Addition and Subtraction. For all rational numbers a, b and c, a(b + c) = ab + ac and a(b - c) = ab - ac.
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Now let us solve the Try These distributivity problems. First, find {7/5 × (-3/12)} + {7/5 × 5/12} using distributivity. This equals 7/5 × {(-3/12) + 5/12}. This equals 7/5 × (2/12), which simplifies to 7/5 × 1/6, giving 7/30. Second, find {9/16 × 4/12} + {9/16 × (-3/9)} using distributivity. This equals 9/16 × {4/12 + (-3/9)}. This equals 9/16 × {1/3 + (-1/3)}. This equals 9/16 × 0, which equals 0.
Now let us solve Example 3. Find (2/5) × (-3/7) - (1/14) - (3/7) × (3/5). Solution: (2/5) × (-3/7) - (1/14) - (3/7) × (3/5) = (2/5) × (-3/7) - (3/7) × (3/5) - (1/14) by commutativity. This equals (2/5) × (-3/7) + ((-3/7)) × (3/5) - (1/14). This equals (-3/7)(2/5 + 3/5) - (1/14) by distributivity. This equals (-3/7) × 1 - (1/14) = (-3/7) - (1/14). Converting to common denominator fourteen, this equals (-6-1)/14 = (-1/2).
Now let us solve Exercise 1.1 completely. Question one asks to name the property under multiplication used in each of the following. Part one is (-4/5) × 1 = 1 × (-4/5) = -4/5. The property used is the Multiplicative Identity property, and also Commutativity. Part two is (-13/17) × (-2/7) = (-2/7) × (-13/17). The property used is Commutativity of multiplication. Part three is (-19/29) × (29/-19) = 1. The property used is the Multiplicative Inverse property.
Question two asks what property allows you to compute (1/3) × (6 × 4/3) as ((1/3) × 6) × 4/3. The property that allows this regrouping is the Associative property of multiplication.
Question three states the product of two rational numbers is always a rational number. This is due to the Closure property of rational numbers under multiplication.
Let us review what we have discussed. Rational numbers are closed under the operations of addition, subtraction and multiplication. The operations addition and multiplication are commutative for rational numbers. They are also associative for rational numbers. The rational number 0 is the additive identity for rational numbers. The rational number 1 is the multiplicative identity for rational numbers. For distributivity of rational numbers, for all rational numbers a, b and c, a(b + c) = ab + ac and a(b - c) = ab - ac. Between any two given rational numbers there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]