Page 1 : PRT Bey yaa aie, , , , In this chapter you will learn :, 1. To define integers., , 2. Addition, subtraction, multiplication and division of integers., , 3. To understand and investigate the properties related to various operations on integers., 4. To realize the importance and use of integers in your day to day life., , 5. To use number lines to represent integers., , , , , , OUR NATION’S PRIDE, , Brahmagupta : Brahmagupta was the Indian, Mathematician who defined zero and set the, tules for its computation, which further led to, make the mathematical problems real and easily, solvable. Not only this, there are numerous, areas in the field of mathematics where the, contribution of Indian mathematician has been, immense. It includes the discovery of zero, the, tules of working with negative numbers and, most importantly a system of expressing all the, numbers using only ten symbols., , , , , , As we know that the integers are the numbers used for counting forward as well as backward., The use of integers in our real life makes them important in mathematics too. They help in, computing the efficiency in positions, to calculate how more or less measures to be taken for, achieving better results and many more., , We have studied numbers used for counting called Natural numbers (N = 1, 2, 2, All natural numbers along with 0 (Zero) are called whole numbers (W = 0, 1, 2, 3, ss, these numbers cannot help us solve all our daily life problems. Therefore we shall study integers,, which is a collection of whole numbers and the negatives of natural numbers., , seseeeeeees 4, -3, -2,-1,0, 1, 2, 3,4...

Page 2 : 2 Mathematics - VII, ° 1, 2, 3, 4..., ° —1,-2,-3, 4, , ° 0 (Zero) is an integer which is neither positive nor negative, , are positive integers., , , , , , are negative integers, , REPRESENTATION OF INTEGERS ON A NUMBER LINE, , Draw a line. Mark a point on the line. Label it as O. Now mark some more points at equal, distances on the right as well as the left of 0. Label the point on the right side of 0 as 1, 2, 3, 4, ARR and label the points on the left side as —1, —2, -3, -4............... as shown below :, , Oo, , eS,, +—+—_+—_+—_ ++, Sih S59 =) = 3 SSE), The arrowheads on both the sides of the number line indicate the continuation of integers, infinitely on each side., ABSOLUTE VALUE OF AN INTEGER, , The absolute value of an integer ‘a’ is the number value of ‘a’ regardless of its sign. It is, denoted by | a |, called the modulus of a, , for example (i) 151=5 and|—5 1=5 (i) 1-31 =3 and [31 =3, “EXample-1 : Write all integers (i) between -5 and 5 (ii) between —20 and -13, , Sol: (i All integers between —5 and 5 are, 4, -3,-2,-1,0, 1,2, 3,4, (i) All integers between —20 and —13 are, —19, -18, -17, -16, -15, -14, , “Example-2 : Compare the integers, , ()-7and0 = (ii) -Sand-13 (iii) -193 and -128 (iv) -26 and 23, Sol: (i) We know that every negative integer is less than 0, -7<0, (ii) —S lies to the right of -13 on number line., 5 >-13, (iii) —193 lies to the left of -128 on a number line, —193 <-128, (iv) We know that every negative integer is less than every positive integer, -26 < 23, , “Example-3 : Evaluate (i)17 - |-12| (a) |-21I-191 (aii) 127-18! + +-91, Sol : We have, , () 17-1-12|=17-12=5 [eo |-121= 12], , (i) 12119) = 21-9 =12 [. 1-211=21 and191=9], , (ii) 127- 181+ -91=94+9=18 = [- 127-181=191=9 and|-91=9

Page 3 : Intergers 3, , “EXmple-4 : Arrange the following integers in ascending order., , 135, -87, -9, 87, -23, 263, -172, 18, Sol : Given positive integers are 135, 87, 263, 18, , In ascending order 18 < 87 < 135 < 263, Given negative integers are -87, —9, -23, -172, In ascending order —-172 <-87 <-23 <-9, Hence all given integers in ascending order are, 172 <-87 <-23 <-9 < 18 < 87 < 135 < 263, , ie. —172, -87,-23, -9, 18, 87, 135, 263, , a,, = d EXERCISE - 1.1 |, , 1. Use the appropriate symbol >, < = to fill in the blanks, (i) 3 i -5 @ 2g s4, ai) s4[ ]3 (wv) 6 50, , (v) s[ |383 w of ]3, , 2. Arrange the following integers in ascending order., (i) —2, 12, 43, 31, 7, -35, -10, (ii) -20, 13, 4,0,-5,5, , 3. Arrange the following integers in descending order., (@) 0,-7, 19, -23, -3, 8, 46, (ii) 30, -2, 0, -6, -20, 8, , 4. Evalute :@ 30-121) (ii) L25|H-18) (iii) 6 - 41 (iv) 1-125] + 11101, , 5. Fill in the blanks :(i) Ois greater than every, (i) Modulus of a negative integer is always ...........0.00., (iii) The smallest postive integer is ., (iv) The largest negative integer is .. a, (v) Every negative integer is less than every ................. integer., , , , , , , , , , ... integer., , , , , , , , FOUR FUNDAMENTAL OPERATIONS, (i) Addition of Integers, , , , Addition of Integers, , J, J d, , Addition of two Integers Addition of two integers, having same signs having different signs

Page 4 : 4, , Addition of two integers having same signs :, , Step 1 : Add the values regardless of their sign., , Step 2 : Write the sum with sign of both the integers., Example 1. Solve 10 + 23, , Solution, , Example 2. Solve 70 + 18, , Solution, , 10 + 23, =33, , 70 +18, = 88, , Example 3. Solve (-50) + (—32), , Solution, , Example 4. Solve (42) + (60), , Solution, , (50) + (-32), =- 82, , (42) + 60), =- 102, , Addition of two integers having different signs :, , Mathematics - VII, , , , , , Both the numbers, have same sign, 1 (6349), , , , Step 1 : Find the difference between the values regardless of their sign., Step 2 : Write the difference with the sign of the integer having greater value., Example 5. Solve (-17) + 35, , Solution, , Example 6. Solve (-63) + 27, , Solution, , Positive, Integer, , , , , , , (-17) +35, =18, , (63) +27, =-36, , , , , , , , Positive, Integer, , , , , Negative, Integer, , , , , , The numbers are of, opposite signs, ie. (+, -) or -, +), , Negative, Integer, , , , , , , , Add their values, regardless of their, signs and write the, sum with positive, sign., , , , , , Find the difference between their, numerical values regardless of their, signs and write the difference with, the sign of greater value., , , , , , Add their values, regardless of their, signs and write the, sum with negative, sign.

Page 5 : Intergers 5, PROPORTIES OF ADDITION OF INTEGERS, , 1. Closure property : The sum of two integers is also an integer. i.e if a and b are integers, then a + b is also an integer. For example 2 + (4) = —2, (-3) + (7) =4,8+5=13, 2. Commutative property : For all integers a and b, , a+b = bta, For example 5+8 = 8+5=13, 3. Associative property : For all integers a, b and c, at(b+c) = (at+b)+c, For example (-2)+(5+9) | [(2)+5]+9, = (2)+(14) | =(@)4+9, = 12 | =12, (-2)+(5+9) = [(-2)+5]+9, , 4, Additive identity : The integer ‘0’ is such thata +0 =0 + a=a ‘0’ is called the additive, identity of integers., 5. Additive Inverse : For any integer a, we have (-a) + a=0 =a + (-a), The negative of an integer a is (a) and the sum of an integer and its negative is ‘0’, Additive inverse of a is (—a), Similarly additive inverse of (a) is—(- a) =a, , “EXmple-7 : In a quiz, Manjeet Singh scored 65, -30, 25, where as Ramandeep scored, , —30, 65, 25 in three rounds. Find who scored better ? What conclusion do you draw ?, Sol. Manjeet Singh’s score = [(65 + (-30)] +25, = 354+25, 60, Ramandeep’s score = (—30)+(65 +25), , = -30+90, , = 60, We see that both Manjeet and Ramandeep scored same. We conclude that addition of, integers is associative., Subtraction of Integers : Subtracting an integer from another integer is same as adding, first integer with additive inverse of second integer. In other words, if a and b are two, integers then a — (+b) =a + (0)., , “Exmple-8 : Solve 15 — (-8), , Sol. We have, 15-(-8) = 15+8, & 23, a 15-(-8) = 23, “Exmple-9 : Solve (-3) - (+ 21), Sol. We have, @3)-G21) = ©@3)+€C21), , = —24