Integers Addition and Subtraction PDF

## Addition of Integers * Addition of two or more integers with like signs * To add two or more integers with like signs, we add their absolute values and affix the common sign. * **Illustration 3:** * (a) 4 + 5 = 9 * (b) -3 + (-12) = -( |-3| + |-12| ) = - (3 + 12) = -15 * A...

## Addition of Integers * Addition of two or more integers with like signs * To add two or more integers with like signs, we add their absolute values and affix the common sign. * **Illustration 3:** * (a) 4 + 5 = 9 * (b) -3 + (-12) = -( |-3| + |-12| ) = - (3 + 12) = -15 * Addition of two or more integers with unlike signs * To add two integers with unlike signs, we subtract their absolute values (the smaller from the larger) and append the sign of the integer with the larger absolute value. * **Illustration 4:** * (-23) + 15 = - ( |-23| - |15| ) = - (23 - 15) = -8 * Similarly, 4 + (-8) = - ( |-8| - |4| ) = -( 8 - 4 ) = - 4 * and 10 + (-3) = + ( |10| - |-3| ) = + (10 - 3) = 7. ## Additive Inverse * For any integer *a*, there exists its opposite (-*a*) such that *a* + (-*a*) = (-*a*) + *a* = 0. * Integers *a* and -*a* are opposites or additive inverse of each other. * For example, 4 + (-4) = (-4) + 4 = 0. So, the additive inverse of 4 is -4 and the additive inverse of -4 is 4. ## Subtraction of Integers * Subtraction is the inverse operation of addition. * So, to subtract an integer from another integer, we add the additive inverse of the integer to be subtracted to the other integer, i.e., if *a* and *b* are any integers, then * *a* - *b* = *a* + additive inverse of (*b*) = *a* + (-*b*), * and *a* - (-*b*) = *a* + [additive inverse of (-*b*)] = *a* + (+ *b*) * **Illustration 5:** * (a) -15 - 4 = -15 + (-4) = -19 * (b) 7 - (-8) = 7 + (8) = 15 ## Exercise 1.1 1. Fill in the blanks. * (a) 1 + (-1) + (-1) + 1 + 1 + (-1) = * (b) (-60) + (-11) + (-9) + 100 = * (c) The additive inverse of (3 - 8) - (-4) is * (d) The sum of an integer and its additive inverse is always 2. Compare the following integers using >, < or =. * (a) 7 -6 * (b) -2 0 * (c) 1 - 8 -31 -4 * (d) -19 |-20 -11| 3. Arrange the following integers in ascending order. * (a) -8, 7, -5, 13, 0, 4 * (b) -13, 25, 7, 15, 1, -7 4. Simplify the following * (a) -15 - (-40) * (b) 136 + (-127) * (c) 19 - (-5) + 17 + 0 * (d) 57 + (-11) - (-10) + 7 * (e) (-50) + (-150) + 200 * (f) 9 - (-5) - 110 + 71 5. Verify *a* - (-*b*) = *a* + *b* for the following values of *a* and *b*. * (a) *a* = 3, *b* = -5 * (b) *a* = 18, *b* = 25 6. Subtract the sum of -110 and 235 from the sum of 103 and -117. 7. Write True (T) or False (F) for the following statements. * (a) Additive inverse of a positive integer is always negative. * (b) The sum of two integers is always positive. * (c) 0 is a positive integer. * (d) The sum of four different integers can never be zero. * (e) -1 is the greatest negative integer. 8. Two cyclists start from the same point on a mountain. One travels 1200 m downward the mountain in 20 minutes, and the other travels 350 m upward the mountain in the same time. How far are they from each other after 20 minutes? 9. The sum of two integers is -51. If one of the integers is 14, find the other integer. 10. A submarine was floating at a depth of 640 m below the sea level. If it ascends 230 metres, what will be its new position? Express the position in terms of depth. 11. The temperature of a city drops by 15 °C in one month. If the present day temperature is -4 °C, what was the temperature one month ago? ## Properties of Addition and Subtraction of Integers * **Closure Property of Addition** * Observe the following additions of two integers. * (a) 17 + 32 = 49, 49 is an integer. * (b) (-10) + 12 = 2, 2 is an integer. * (c) (-12) + 5 = -7, -7 is an integer. * (d) (25) + (-25) = 0, 0 is an integer. * We observe that, when we add two integers, the sum obtained is again an integer. * So, we say that integers are closed under addition. This property is called the closure property of addition of integers. * **In general, it can be stated as follows:** * For any two integers *a* and *b*, *a* + *b* is an integer. * **Closure Property of Subtraction** * Recall that if *a* and *b* are two whole numbers, then *a* - *b* may not be a whole number. * For example, 7 and 9 are whole numbers but 7 - 9 = -2 is not a whole number. So, whole numbers are not closed under subtraction. * Let us see what this property holds for integers. Observe the following subtractions. * (a) 7 - 9 = - 2, -2 is an integer. * (b) 12 - (-10) = 22, 22 is an integer. * (c) (-15) - (-15) = 0, 0 is an integer. * (d) 27 - 0 = 27, 27 is an integer. * We observe that when we subtract two integers, the result obtained is again an integer. * So, we say that integers are closed under subtraction. This property is called the closure property of subtraction of integers. * **In general, it can be stated as follows:** * For any two integers *a* and *b*, *a* - *b* is an integer. * **Commutative Property of Addition** * We already know that whole numbers can be added in any order, i.e., 3 + 8 = 8 + 3 = 11. In other words, addition is commutative for whole numbers. Let us check if this property holds for integers also. * Consider the integers 8 and -3 and observe the addition of the two integers. * 8 + (-3) * Start from 8 and move 3 units to the left, we end up at 5. * Start from -3 and move 8 units to the right, we end up at 5. * So, 8 + (-3) = 5 and (-3) + 8 = 5 * 8 + (-3) = (-3) + 8. * Similarly, (-8) + (-3) = (-3) + (-8) = -11. * We observe that, addition is commutative for integers. This property is called the commutative property of addition of integers. In general, it can be stated as follows: * **For any two integers *a* and *b*, *a* + *b* = *b* + *a*.** * **Note:** Subtraction is not commutative for integers. For example, 7 - (-3) = 7 + 3 = 10 but (-3) - 7 = -3 - 7 = -10 i.e., *a* - *b* ≠ *b* - *a*. * **Associative Property of Addition** * Recall that if *a*, *b* and *c* are any three whole numbers, then (*a* + *b*) + *c* = *a* + (*b* + *c*). We say that addition is associative for whole numbers. * Consider the integers, -2, 5 and -8 and observe the addition of these integers. * We have, [ (-2) + 5] + (-8) = 3 + (-8) = - (8 - 3) = - 5 * and (-2) + [ 5 + (-8) ] = (-2) + (-3) = -( 2 + 3 ) = - 5 * We find that [ (-2) + 5 ] + (-8) = (-2) + [5 + (-8) ] = -5. * We observe that addition is associative for integers. This property is called the associative property of addition of integers. In general, it can be stated as follows: * **For any three integers *a*, *b* and *c*, *a* + (*b* + *c*) = (*a* + *b*) + *c*.** * **Note:** * If we first add -2 and -8 and, then add 5 to the sum obtained, we will get the same answer. [ (-2) + (-8) ] + 5 = ( - 2 - 8) + 5 = -10 + 5 = -5. * We conclude that to find the sum of three (or more) integers, we can group them in any manner. * **Remember** * Subtraction is not associative for integers. For example, (-2) - [ (-3) - 5 ] = -2 - (-8) = -2 + 8 = 6 but [ (-2) - (-3) ] - 5 = 1 - 5 = -4 i.e., for three integers *a*, *b*, and *c*, *a* - (*b* - *c*) ≠ (*a* - *b*) - *c*. * **Additive Identity** * Observe the following: * (a) 3 + 0 = 0 + 3 = 3; * (b) 0 + (-2) = (-2) + 0 = -2; * (c) 0 + (+9) = (+9) + 0 = +9 * We observe that if zero is added to any integer or any integer is added to zero, the value of the integer remains the same. So, 0 is the additive identity for integers. In general, it can be stated as follows: * **For any integer *a*, *a* + 0 = 0 + *a* = *a*.** * **Additive Inverse** * We have, 3 + (-3) = 0 and (-3) + 3 = 0. * We say that 3 is the additive inverse of -3 or -3 is the additive inverse of 3. * Since, 3 and -3 are opposites of each other, we find that the additive inverse of an integer is the same as its opposite. * **In general, it can be stated as follows:** * **For any integer *a*, -*a* is its additive inverse as *a* + (-*a*) = (-*a*) + *a* = 0.** ## Exercise 1.2 1. Identify the property demonstrated * (a) (-7) + 0 = -7 * (c) [ 6 + (-3) ] + (-9) = 6 + [ (-3) + (-9) ] 2. Write down a pair of integers whose: * (a) sum is -7. * (b) difference is 10. * (c) sum is 0. 3. Write a pair of integers whose sum gives * (a) a negative integer. * (b) an integer smaller than both the integers. * (c) an integer greater than both the integers. * (d) an integer smaller than only one of the integers. 4. Fill in the blanks to make the following statements true. * (a) (-9) + (-11) = (-11) + ( ) * (b) -17 + ( ) = - 17 * (c) [(-15) + (-10)] + ( ) = -15 + [ (-10) + (-9)] * (d) 10 + ( ) = 0 * (e) 246 + (-132) = ( ) + 246 * (f) 65 + {(-85 + 7) } = {( ) + (-85) } + 7 5. In a quiz competition, Radhika scored -10, 20, 0, -20 and Seema scored 0, -30, 10, 20 in four successive rounds. Who scored more? Can we say that integers can be added in any order? 6. If a = 8 and b = -7, show that *a* - *b* = *b* - *a*. 7. In a competitive exam of 40 questions, +2 is awarded for every correct answer and -1 for every wrong answer. Find the total score if a student had 12 wrong answers in his attempt of all the questions. ## Multiplication of Integers * **Multiplication of Two Positive integers** * We know that positive integers are whole numbers. * We have learnt multiplication of whole numbers in earlier classes. * For example, 2 x 3 = 6, 11 x 8 = 88, etc. * **Multiplication of One Positive Integer and One Negative Integer** * We know that multiplication of whole numbers is a repeated addition. * For example, 3 + 3 + 3 + 3 = 4 x 3 = 12. * What is (-3) + (-3) + (-3) + (-3)? * From the number line, we find that (-3) + (-3) + (-3) + (-3) = -12 * We can write it as (-3) + (-3) + (-3) + (-3) = 4 x (-3) = (-12) * Similarly, (-4) + (-4) + (-4) + (-4) = 4 x (-4) = -16. * We also have, 4 x (-3) = -12. * So, -3 x 4 = 4 x (-3) = -12. * Similarly, using such pattern, (-4) x 4 = -16. * Thus, 4 x 4 = 4 x (-4) = -16. * From the above discussion, we observe that when multiplying a positive integer and a negative integer, we multiply them as whole numbers (ignoring their signs) or multiply their absolute values and put a minus sign (-) before the product. We conclude that the product of a negative and a positive integer is a negative integer. * **Multiplication of Two Negative Integers ** * We already know how to multiply two positive integers and also a positive integer and a negative integer. We use the product of a positive integer and a negative integer to develop a procedure for multiplying two negative integers. * Consider two negative integers, say -4 and -3. Let us find (-4) x (-3). * Observe the following pattern: * -4 x (5) = -20 * -4 x (4) = -16 -16 - 20 + 4 * -4 x (3) = -12 -12 - 16 + 4 * -4 x (2) = -8 -8 - 12 + 4 * -4 x (1) = -4 -4 - 8 + 4 * -4 x (0) = 0 0 - 4 + 4 * We observe that each product is 4 more than the one above it. Continuing this pattern, we have: * -4 x (-1) = 4 (positive integer) 0 + 4 = 4 * -4 x (-2) = 8 (positive integer) 4 + 4 = 8 * -4 x (-3) = 12 (positive integer) 8 + 4 = 12 * Also, observe: * -4 x (-2) = 8 = 4 x 2 = |-4| x |-2| * -4 x (-3) = 12 = 4 x 3 = |-4| x |-3| * We conclude that the product of two negative integers is always positive. * From the above discussion, we observe that to find the product of two negative integers, first multiply the two integers as whole numbers (ignoring their signs) and put a positive sign before the product or multiply the absolute values of the integers and then put a positive sign (+) before the product. * **Remember** * The product of two integers with like signs (that is, either both positive or both negative) is positive and is equal to the product of their absolute values. * Thus, it can be stated as follows: * The product of two integers with like signs is positive, i.e., (+) (+) = (+) and (-) (-) = (+). * **Multiplication of More than Two Integers ** * We can multiply more than two integers by following the procedure given below: * **To find the product of two (or more) integers,** * **Step 1:** Find the product of their absolute values. * **Step 2:** * (a) Keep the product positive, if there is an even number of negative factors (integers) or no negative factor. * (b) Keep the product negative, if there is an odd number of negative factors. ## Exercise 1.3 1. Answer the following questions. * (a) Find the integer whose product with -1 is 72. * (b) If x x (-1) = -27, then what is the value of integer x? * (c) Find a pair of integers whose product is -136. 2. What will be the sign of the product, if * (a) 11 negative integers and 6 positive integers are multiplied together? * (b) (-2) is multiplied with itself 300 times? 3. Find the product of the following. * (a) 3 x (-8) * (b) (-15) x (-4) * (c) (-11) x 0 * (d) (-19) x (4) 4. Simplify the following: * (a) (-1) x (-2) x (-3) x (-4) * (b) -6 + 2 x (5 - 8) * (c) 7 - 5 x (9 - 10) * (d) (-110) x (27) x (13) x 0 * (e) (-8) x 7 + (-8) x 3 * (f) 9 x (-12) + 9 x (-8) * (g) 125 x (-55) + (-125) x 45 * (h) 25 x (-32) x (-4) 5. Write the correct integers in the blanks. * (a) (-4) x ( ) = 100 * (b) (-5) x (-5) x ( ) = 625 * (c) ( 4 x 5 ) x 9 = ( 4 x 9 ) x ( ) * (d) -6 x ( 3 + 4 ) = ( ) x 3 + ( ) x 4 6. Fill in the blanks to make the statements true. * (a) If we multiply five negative integers and five positive integers, then the resulting integer is * (b) If we multiply ( ) number of negative integers, then the resulting integer is positive. * (c) Multiplicative identity for integers is ( ) * (d) Additive inverse of an integer *a* is obtained by multiplying it by ( ) * (e) The opposite of (-3) x 2 x (-5) is ( ) * (f) For three integers *a*, *b* and *c*, (*a* x *b*) x *c* = *a* x (*b* x *c*). This property is called the ( ) of multiplication. * (g) The product of two integers with like signs is ( ) * (h) The integer whose product with '-1' is -40, is ( ) 7. State True (T) or False (F) for the following statements. * (a) The product of five negative integers is a negative integer. * (b) Of the two integers, if one is negative, then their product must be negative. * (c) For all non-zero integers *a* and *b*, *a* × *b* is always greater than either *a* or *b*. * (d) The product of a positive and a negative integer may be zero. * (e) Multiplication is commutative for integers. * (f) (-1) is the multiplicative identity of integers. 8. A certain freezing process requires that room temperature be lowered from 40 °C at the rate of 4°C per hour. What will be the room temperature after 12 hours of the process begins. 9. Raghav earns a profit of 8 per bag of white cement sold and a loss of 5 per bag of grey cement sold. If he sells 3000 bags of white cement and 5000 bags of grey cement in a month, what is his profit or loss? ## Division of Integers * In earlier classes, we have learnt division of whole numbers. Recall that division operation is the inverse of the multiplication operation. * In this section, we shall learn division of integers. In the previous section, we have also learnt multiplication of integers. The rules for division of integers may be obtained from the rules of multiplication of integers. * We already know for a multiplication fact, there are two corresponding division facts. * Let us observe few multiplication and their corresponding division facts. | Multiplication fact | Corresponding division facts | |:---|:---| | 3 x 5 = 15 | 15 ÷ 3 = 5; 15 ÷ 5 = 3 | | (-2) x (8) = -16| -16 ÷ (-2) = 8; -16 ÷ 8 = -2 | | 9 x (-4) = -36 | -36 ÷ 9 = -4; -36 ÷ (-4) = 9 | | (-8) x (-7) = 56 | 56 ÷ (-8) = -7; 56 ÷ (-7) = -8 | * **Division of Two Integers with Like Signs** * Observe the Table 1.2 given above. You will find that -16 ÷ (-2) = 8 (positive); -36 ÷ (- 4) = 9; (positive); 15 ÷ 3 = 5 (positive); 15 ÷ 5 = 3 (positive) * We observe that the quotient in each case is positive. * From the above examples, we conclude that **If both the integers are of the same sign (like signs), i.e., either both are positive or both are negative, the quotient is always positive.** * **In general, it can be stated as follows:** * For any two positive integers *a* and *b*, (-*a*) ÷ (-*b*) = *a* + *b*. * **Division of Two Integers with Unlike Signs** * Observe the table 1.2 given, we will find that -16 ÷ 8 = -2 (negative); -36 ÷ 9 = - 4; (negative); 56 ÷ (-7) = -8 (negative); 56 ÷ (-8) = -7 (negative) * We observe that the quotient in each case is negative. From the above examples we conclude that if both the integers (dividend and divisor) are of opposite signs (unlike signs), i.e., one is positive and the other is negative, then the quotient is always negative. * **In general, it can be stated as follows:** * For two positive integers a and b, -*a* ÷ *b* = (-*a* + *b*); *a* ÷ (-*b*) = -(*a* + *b)*. * **Remember** | Integer | Integer | Quotient | | -------- | ---------- | ----------- | | (+ve) | (+ve) | (+ve) | | (+ve) | (-ve) | (-ve) | | (-ve) | (+ve) | (-ve) | | (-ve) | (-ve) | (+ve) | * **Division by Zero** * Like whole numbers, any integer divided by 0 is meaningless. If integer 0 is divided by any integer other than 0, the quotient is 0. * **In general, it can be stated as follows:** * For any integer *a* ≠ 0, *a* ÷ 0 is not defined and 0 ÷ *a* = 0. * **Division by 1** * Observe the following divisions. * -10 ÷ 1 = -10; 15 ÷ 1 = 15 * We observe that, any integer divided by 1 gives the same integer. * **In general, it can be stated as follows:** * For any integer *a*, *a* ÷ 1 = *a*. ## Exercise 1.4 1. Find the quotient. * (a) (-96) ÷ (-8) * (b) 64 ÷ (-8) * (c) (-65) ÷ 5 * (d) (-480) ÷ (-48) * (e) 0 ÷ (-1000) 2. Fill in the blanks. * (a) (-76) ÷ ( ) = 1 * (b) 314 ÷ ( ) = (-314) * (c) ( ) ÷ (-2) = 75 * (d) (-98) ÷ 98 = ( ) 3. Write True (T) or False (F) for the following statements. * (a) (-1) ÷ (-1) = -1 * (b) (-9) ÷ (-1) = 9 * (c) (-10) ÷ 0 = 0 * (d) 0 ÷ (-100) = 0 * (e) - 11 x 11 ÷ (-11) = -1 4. Find the value of the following expressions. * (a) 90 ÷ [(-5) + (-4)] * (b) (-64) ÷ [6 - (-2)] * (c) [ {40 ÷ (-4) } ÷ (-2) ] * (d) [ 57 - (-3) ] + [ (-60) ÷ 10] * (e) (12 - 48) ÷ (-6) * (f) (-63) ÷ [ (-20) + (-1) ] * (g) [(-25) - (-7) ] ÷ [ (-2) - (-1) ] 5. Find the sum of (-36) ÷ 4 and 48 ÷ (-12). 6. For what value of x, (-288) ÷ x = -144? 7. What is the additive inverse of [ (-4) x (-9) x (-25) ] + [ (-2) x (-3) x (-5) ]? 8. Which of the following does not represent an integer? * (a) 0 ÷ (-5) * (b) 25 ÷ (-5) * (c) (-9) ÷ 3 * (d) (-12) ÷ 7 9. A city recorded the following temperature on 5 days. * Day 1: -4°C, Day 2: -1°C, Day 3: 3°C, Day 4: 2°C, Day 5: -5°C. * Find the average temperature for these 5 days. 10. The product of two numbers is 315. One of the numbers is -15, what is the other number? 11. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach -350 m deep under the ground? 12. In a mathematics paper there are 10 problems. Each correct answer carries 5 marks whereas for each incorrect answer 2 marks are deducted from the marks scored. Abdul answered all the questions but got only 29 marks. Find the number of correct answers given by Abdul. ## Now You Know! * Integers are closed under addition, subtraction and multiplication. * Addition and multiplication are commutative and associative for integers. * 0 is the additive identity for integers. * 1 is the multiplicative identity for integers. * For any three integers *a*, *b* and *c*, *a* x (*b* x *c*) = (*a* x *b*) x *c* and (*b* + *c*) x *a* = *b* x *a* + *c* x *a* * The product and quotient of two integers with unlike signs is always negative. * The product and quotient of two integers with like signs is always positive. * For any integer *a*, *a* x 0 = 0; 0 x *a* = 0, i.e., the product of an integer and 0 is always 0. This property is known as zero factor property. * For any integer *a*, *a* ÷ 0 is not defined and *a* ÷ 1 = *a*. ## Let Us Assess 1. Tick (✔) the correct answer. * (a) The product of 6 and -12 and the quotient of -45 and -5 add upto * (i) -9 * (ii) -27 * (iii) -63 * (iv) -81 * (b) The integer which should be multiplied by -5 to get 60 is * (i) -300 * (ii) -12 * (iii) 12 * (iv) 300 * (c) The integer which is used as a divisor for the dividend 45 gives the quotient as -9, is * (i) -5 * (ii) 5 * (iii) 9 * (iv) 45 * (d) If [ 39 ÷ (-3) ] is divided by [ -104 ÷ (-8) ], then the quotient obtained is * (i) -13 * (ii) -1 * (iii) 1 * (iv) 13 * (e) If an operation * is defined on integers *a* and *b* such that *a* * *b* = (*a* + *b*) - (*a* × *b*), then the value of [(2 * 3) *( 6 ÷ 5)] is equal to * (i) -19 * (ii) -1 * (iii) 18 * (iv) 19 * (f) Fill in the correct operations in the box to make the mathematical statement true. * 8 ( ) 8 ( ) 8 ( ) 8 = 15 * (g) -45 x 107 is not same as * (i) -45 × (100 + 7) * (ii) (-45) x 7 + (-45) x 100 * (iii) -45 x *7* + 100 * (iv) (-40 - 5) x 107 * (h) Find the number of zeros used in writing non-negative integers up to 100. * (i) 9 * (ii) 10 * (iii) 11 * (iv) 12 2. Fill in the blanks. * (a) ( ) ÷ (-2) = -15 * (b) (-35) x (-45) x (100) x 0 = ( ) * (c) (-59) ÷ (59) = ( ) * (d) [(-4) x (-5)] x (-6) = 4 x 6 x ( ) * (e) ( ) ÷ 6 = -5 * (f) (-1) multiplied 8 times with 1 multiplied 101 times gives ( ) 3. State True (T) or False (F) for the following statements. * (a) (-9) ÷ (-11) is greater than (-50) ÷ 10. * (b) Product of seven negative integers is positive. * (c) (-1729) ÷ (1729) = -1 * (d) (-67) ÷ (+1) = -67 4.