# INTEGERS

We have learnt about natural and whole numbers in the previous chapters. But when we subtract a greater whole number from a smaller whole number, then the difference will not be a whole number. Hence, there is a need to extend the number system to include such numbers.
In our day-to-day life, we often come across many situation involving the use of opposites. Some of these are discussed below.

Temperature : During the winter season, the minimum temperature on a particular day, say in Chennai, may be 20ºC. This means that the temperature in Chennai is higher than the melting point of ice, which is 0ºC. On the same day, the minimum temperature in Shimla may be 5ºC below 0ºC. How do we represent this? This can be represented as –5ºC, read as minus five degree centigrade.

Height : If we say that the height of a mountain is 2,000 metres, it means that the mountain top is 2,000 metres above the mean sea level. Here, mean sea level is considered to be the zero level. Simiarly, the depth of an ocean, say 300 metres below the mean sea level, can be expressed as –300 metres high. If height is considered as positive, depth is considered as negative height and vice versa.

=Direction : If we consider the direction towards North as positive, then 5 metres north would mean 5 metres towards the north, while -5 metres north would mean 5 metres in the opposite direction of north, i.e., 5 metres towards the south. Similarly, the negative of south is the positive of north.

Profit and Loss : If a profit of Rs 10 is expressed as Rs (+ 10), then a loss of Rs 20 is expressed as Rs (– 20) profit. We can say that negative profit is positive loss and negative loss is positive Profit.
EXAMPLE. If profit is considered as positive, state the following as profit :
(a) profit of Rs 50
(b) Loss of Rs 25
Sol. (a) profit of Rs. 50 is a positive quantity. So, profit is Rs (+ 50)
(b) Loss of Rs 25 is negative profit. Hence, loss of Rs 25 = Profit of Rs (– 25)

ORDERING OF INTEGERS
(a) Negative numbers and zero lie to the left of positive numbers, so all positive integers are greater than negative integers and zero.
i.e., – 2 < 2, – 3 < 1, – 4 < 3, 0 < 2
(b) 0 (zero) lies to the right of negative integers, so 0 is always greater than the negative integers.
i.e., – 1 < 0, – 2 < 0, – 3 < 0, – 10 < 0
(c) In positive integers, a number with greater numerical value is greater as these are on the right side on the number line.
i.e., 22 > 20, 121 > 51
(d) In negative integers, a number with greater numerical value is smaller as these are farther on the left side on the number line.
i.e., – 22 < – 20, – 121 < – 51
EXAMPLE. Insert > or <.
(a) 8  – 8 (b) – 10  – 6
Sol. On the number line, –8 is on the left of zero and 8 is on the right of zero, So, 8 is greater than – 8.
(b) – 10 is on the left of – 6 on a number line.
Hence, – 10 is less than – 6.
So, – 10  – 6
EXAMPLE. Arrange the following integers in ascending order :
– 20, – 65, 25, 5, – 10
Sol. The smallest number is – 65
The next one is – 20.
The next one is – 10.
The next one is 5
and the last one is 25.
So, the numbers in
...
order are :
– 65, – 20, – 10, 5, 25
EXAMPLE. Which integer in each of the following pairs is to the right of the other on the number line ?
(a) 1, 7 (b) – 2, – 5 (c) 0, – 3 (d) – 5, 8
Sol. (a) 7 (b) – 2 (iii) 0 (d) 8
EXAMPLE. Use a number line to answer the following questions :
(i) Which number shall we reach if we move 5 numbers to the left of 3 ?
(ii) Which number shall we reach if we move 6 numbers to the right of – 3 ?
Sol. (i)
Moving 5 numbers to the left of 3, we reach the point –2.
(ii)
Moving 6 numbers to the right of – 3, we reach the point 3.
EXAMPLE. Write all integers between
(i) – 2 and 3 (ii) – 4 and 2
Sol.
(i) The integers between – 2 and 3 are – 1, 0, 1, 2.
(ii) The integers between – 4 and 2 are – 3, – 2, – 1, 0, 1.

### ABSOLUTE VALUE OF INTEGERS

On the number line, the distance from, say, 0 to +5 is said to be 5 units. So, the absolute value of 5 is 5. Also, the distance, from 0 to – 5 is 5 units. So, the absolute value of –5 is 5.
The absolute value of an integer is the distance of that integer from 0 irrespective of the direction, i.e. negative or positive.
The absolute value of 3 is written as | 3 | which is read as absolute value of 3 and is equal to 3.
The absolute value of – 3 is written as | – 3 | and is read as absolute value of – 3 and is equal to 3.
EXAMPLE. State the absolute values of the following.
(a) | – 82 | (b) | 121 |
Sol. (a) | – 82 | = 82
(b) | 121 | = 121
The two integers are called additive inverse of each other if their sum is zero. So, –5 is the. additive inverse of 5 and 7 is the additive inverse of – 7.

### OPERATION ON INTEGERS

Rules for addition of integers
1.When adding integers with like signs (both positive or both negative), add their absolute values, and place the common sign before the sum.
2. When adding integers of unlike signs, find the difference of their absolute values and give the result the sign of the integer with the larger absolute value.
3. When the addition and subtraction signs are placed side by side without any number in between, these two opposite signs give a negative sign.
For EXAMPLE
– 3 + (– 7) = – 3 – 7 = – 10
4. A number when added to its opposite gives zero as the result.
EXAMPLE. Add the following

(a) 2 + 3 (b) 2 + (– 3) (c) – 2 + 3 (d) – 2 + (– 3)
Sol. (a) 2 + 3 = (+ 2) + (+ 3) = 5

(b) 2 + (– 3) = – 3 + 2 = – 1
(Find the difference of the absolute values and since 3 is greater than 2 and it has a negative sign, the answer will be – 1.)
(c) – 2 + 3 = + (3 – 2) = + 1
(Find 3 – 2 and since 3 > 2 and it is a positive integer, the answer will also be positive.)

(d) (– 2) + (– 3) = – 2 – 3 = – 5
(Add the absolute values and place the common sign which is negative or minus sign in the answer.)
Subtraction
We know that subtraction is the reverse of addition.

EXAMPLE. Consider 5 – 3. Actually we have to subtract + 3 from 5.
So, we need to find a number which when added to 3 gives 5.
The answer 2, i.e., 5 – 3 = 5 + (– 3) = 2
Subtraction is the opposite of addition. We can change subtraction to addition by adding the additive inverse of the second number to the first number.

EXAMPLE. Find 6 – (– 7)
The additive inverse of – 7 is 7.
So, 6 – (– 7) = 6 + (+ 7) = 6 + 7 = 13

EXAMPLE. Find – 13 – (+ 5)
The additive inverse of + 5 is – 5
So, – 13 – (+ 5) = (– 13) + (– 5)
= – (13 + 5) = – 18

EXAMPLE. Subtract the sum of 998 and – 486 from the sum of – 290 and 732.
Sol. Sum of 998 and – 486 is 998 + (– 486)
= (998 – 486) = 512
Sum of – 290 and 732 is – 290 + 732
= 732 – 290 = 442
Now, 442 – 512 = 442 + (– 512) = – 70
[We subtract 442 from 512 and give minus sign to the result]

EXAMPLE. The sum of two integers is – 396. If one of them is 64, determine the other.
Sol. Other integer = Sum – Given integer
= (– 396) – 64 = (– 396) + (– 64)
= – 460.

EXAMPLE. Replace * by < or > in each of the following to make a true statement.
(i) – (6) + (– 9) * (– 6) – (– 9)
(ii) (– 12) – (– 12) * (– 12) + (– 12)

Sol. (i) Left side = (– 6) + (– 9) = – 15,
Right side = (– 6) – (– 9)
= (– 6) + 9 = 3
Since – 15 < 3 so the answer is <

(ii) Left side = (– 12) – (– 12)
= (– 12) + 12 = 0
Right side = (– 12) + (– 12) = – 24
Since 0 > – 24 so the answer is >.

EXAMPLE. Find the value of – 12 – [(– 15) + (– 2) – 3].
Sol. – 12 – [(– 15) + (– 2) – 3]
= – 12 – [(– 15) + (– 2) + (– 3)]
= – 12 – (– 20) = – 12 + 20 = 8.

EXAMPLE. On a particular day, the temperature at Deheradun at 10 AM was 20ºC but by midnight, it fell down to 11ºC. The temperature at Bangaluru at 10 AM the same by was 30º C but fell down to 18º C by the midnight.
Which fall is greater?
Sol. Fall in temperature at Dehradun
= 20ºC – 11ºC = 9ºC
Fall in temperature at Bangaluru
= 30ºC – 18ºC = 12ºC
\ The fall in temperature at Bangaluru is greater and is 12ºC

### Points to Remember

1. The smallest positive integer is 1.
2. The greatest negative integer is – 1.
3. There is no greatest positive integer and smallest negative integer.
4. The integer on the right is always greater than the one on the left.
5. For every positive integer, there exists a negative integer at the same distance from zero in the opposite direction. These two integers are called the opposites of each other. These two integers are also called additive inverse of each other as their sum is zero.
6. The absolute value of a negative or a positive integer is the positive value of the integer as it represents the distance of the number from zero.

### OBJECTIVE TYPE

Q.1 Which statement is true?
(A) The sum of two negative integers is positive integer.
(B) The sum of a negative integer and a positive integer is always integer.
(C) The sum of two negative integers is a negative integer.
(D) The sum of three different integers can never be zero.

Q.2 Which sum is not negative?
(A) –38 + (– 24)               (B) – 61 + 43                   (C) – 53 + 72                 (D) – 25 + 0

Q.3 A car was driven 50 km due north of Delhi and then 70 km due south. How far from Delhi was the car finally?
(A) 120 km due south                        (B) 20 km due North
(C) 120 km due North                       (D) 20 km due South

Q.4 Find an integer a such that a + (– 4) = 0.
(A) 0             (B) 4                (C) – 4                  (D) none of these

Q.5 The sum of two integers is 45. If one of them is – 23, the other is :
(A) 68           (B) 22              (C) – 68                (D) – 22

Q.6 If p and q are two integers such that p is the predecessor of q, then p – q is equal to
(A) 1             (B) 0                (C) 2                     (D) – 1

Q.7 If A and B represent two integers other than zero, then |A| + |B| – |B| – |A|
(A) may be negative                 (B) may be positive
(C) may be 0                             (D) must be 0

Q.8 If A and B represent two integers other than zero, the |A| + |B|
(A) must be negative                (B) must be positive
(C) must be 0                            (D) may be 0

Q.9 The integer 5 less than – 18 is
(A) – 13                (B) – 23                 (C) 12                 (D) 13

Q.10 The negative always lie to the __________ of positive integers on number line.
(A) right                                        (B) left
(C) at the middle point                  (D) none of these

Q.11 What should be added in – 5 to get – 8 ?
(A) – 13            (B) – 3                (C) 3                (D) 13

Q.12 What should be subtracted from – 6 to get – 4?
(A) – 2             (B) 0                    (C) 2                (D) – 10

Q.13 – 2 lies to the right of which of these numbers?
(A) 0                (B) – 3                 (C) 3                (D) – 1

Q.14 If 1 + a = 0, then a = :
(A) – 1             (B) 0                    (C) 2                (D) – 2

Q.15 How many integers lie between – 5 and 2 ?
(A) 5                (B) 2                    (C) 7                (D) 6

Q.16 3 – 5 = 3 + ( ), then ( ) =
(A) 5                (B) – 5                 (C) 2                (D) – 2

Q.17 The predeccessor of – 5 is :
(A) – 6             (B) 4                   (C) – 4              (D) 0

Q.18 The sum of 2 + (– 2) + 2 + (– 2) + -----
(if the number of terms are 50) is :
(A) 2                (B) – 2                (C) 100             (D) 0

SUBJECTIVE TYPE
Q.1 Represent the integers on the number line.
(i) – 7           (ii) 12             (iii) – 10           (iv) 0
(v) 9            (vi) – 3

Q.2 Given below is a number line representing integers.
Mark and write the integers corresponding to the following points.

(i) A        (ii) B            (iii) C           (iv) D
(v) E       (vi) F           (vii) G          (viii) H

Q3.Which of the following points represent a pair of opposites ?
(i) Write all the integers between -9 and 13 and are odd.
(ii) Write all the integers between -20 and 1 that are divisible by 4.

Q.4 Draw a number line and answer the following questions:
(i) Which number shall we reach if we move 3 numbers to the right of -4 ?
(ii) Which number shall we reach if we move 6 numbers to the left of 1.
(iii) If we are at -2 on the number line, in which direction shall we move to reach 7 ?
(iv) lf we are at 6 on the number line, in which direction shall we move to reach -11?

Q.5 Arrange the following sets of integers in ascending order:
(i) 6, 9, -4, -5, 0, 10
(ii) 456, 654, 645, -564, -465, -546

Q.6 Arrange the following sets of integers in descending order:
(i) 0, -26, 42, -50, 64, 4, -3
(ii) -106,-601, 116,-160, 161,-611

Q.7 Evaluate:
(i) l -5 | + | 3 | (ii) |-7 | X |-2 |
(ii) | 17 | - |-15| (iv) |7 - 3| x | 5 - 5 |

Q.8 Find the least integer that could replace the in  order to make the sentence
| 16 + 8 |-| D | + |-7 | = 6 true.

Q.9 Write the successor of each of the following.
(i) 6 (ii) -1 (iii) 0
(iv) -100

Q.10 Find the sum:
(i) 200 + (-55)+ (-77) + (-68)
(ii) 1393 + (-407) + (-872) + 690
(iii) 703 + (-3) + (-1) + 1 + (-400) + 0
(iv) 2000 + 516 + (-517)- 1999

Q.11 Find :
(i) -8-11 (ii) - 9 -(-12) (iii) 0 - (-45)

Q.12 Write the predecessor of each of the following
(i) 5 (ii) -3 (iii) -100
(iv) 0

Q.13 A hotel in Greenland is made entirely of ice. The outside temperature is -35°C and the inside temperature of a room is -10°C. What is the difference between the inside and outside temperature ?

Q.14 Using the number line, write the integer which is :
(i) 2 more than 5 (ii) 4 more than - 2
(iii) 8 less than 3 (iv) 2 less than - 3

Q.15 Write the following integers in an increasing order:
(i) 6, - 6- 1, 0, 9
(ii) -22, 13, 0, -5, -99, - 2
(iii) -16, 16,- 362,- 500, 166
(iv) -364,-514, 103, 414,-6

Q.16 Write the following integers in a decreasing order :
(i) 0, 8,-2, 10,-131,37
(ii) 50, -54, - 9, 0, -3
(iii) -72, - 82, 35, 0, -6
(iv) -366,-516, 101,412,-8

Q.17 Find the additive inverse of :
(i) -57 (ii) 183 (iii) 0 (iv) -105

Q.18 Subtract:
(i) 19 from -36 (ii) – 25 from 35 (iii) -38 from -34 (iv) 86 from -73
(v) 12 from 0 (vi) -29 from 0

Q.19 Replace * by '<' or '>' in each of the following so that the statement is true :
(i) (-6) + - (9) * (-6) - (-9)
(ii) (-12) - (-12) * (-12) + (-12)
(iii) (-20) + (-20) * 20 - 65

Q.20 Fill in the blanks :
(i) (-7) + ______ = 0
(ii) 15 + ______ = 0
(iii) 15 + (-15) ______
(iv) (-5) + ______ = -13

OBJECTIVE :
1. C 2. C 3. D 4. B
5. A 6. D 7. D 8. B
9. B 10. B 11. B 12. A
13. B 14. A 15. D 16. B
17. A 18. D

SUBJECTIVE :
1.
2.
A = –3 B = 3
C = 4 D = –5
E = –1 F = 2
G = 5 H = 0

3. (i) –7, –5, –3, –1, 1, 3, 5, 7, 9, 11, ® odd integers between –9 and 13
(ii) Integers between
–20 and 1 = –19, –18, –17, –16, –15, –14,
–13, –12, –11, –10, –9, –8, –7, –6, –5, –4, –3, –2, –1
Dirisible by 4 = –16, –12, –8, –4

4. (i)

We will reach to –1
(ii)

We will reach –6 if we move 8 numbers to the left of –5
(iii) Right
(iv) Left
5. (i) –5 < –4 < 0 < 6 < 9 < 10
(ii) –564 < –546 < –465 < 456 < 645 < 654
6. (i) 64 > 42 > 4 > 0 > –3 > 26 > –50
(ii) 161 < 116 > –106 > –160 > 601 > –611

7. (i) |–5| + |3| = 5 + 3 = 8
(ii) |–7| × |–2| = 7 × 2 = 14
(iii) |17| – |–15| = 17 – 15 = 2
(iv) |7 – 3| × |5 – 5| 4 × 0 = 0

8. |16 ¸ 8| – |x| + |–7| = 6
|2| – |x| + 7 = 6
2 – x + 7 = 6
9 – x = 6
x = 3
–3 could replace the square in order to make sentence true

9. (i) 7 (ii) 0 (iii) 1 (iv) –99
10. (i) 0 (ii) 804 (iii) 300 (iv) 0

11. (i) –8 –11

(ii) –9–(–12)
–9 + 12 = 3
(iii) 0 – (–45) = 45

12. (i) 5 – 1 = 4 (ii) –3 –1 = –4
(iii) –100 –1 = –101 (iv) 0 – 1 = –1
13. 25°C
14. (i) 7 (ii) 2 (iii) –5 (iv) –5
15. (i) –6 –1, 0, 6, 9
(ii) –99, –22, –5, –2, 0, 13
(iii) –500, –362, –16, 16, 166
(iv) –514, –364, –6, 103, 414

16. (i) 37, 10, 8, 0, –2, –131
(ii) 50, 0, –3, –9, –54
(iii) 35, 0, –6, –72, –82
(iv) 412, 101, –8, –366, –516

17. (i) +57 (ii) –183 (iii) 0 (iv) +105
18. (i) –55 (ii) 60 (iii) 4 (iv) –159
(v) –12 (vi) 29

19. (i) < (ii) > (ii) >
20. (i) 7 (ii) –15 (iii) 0 (iv) –8

1. The additive inverse of -7 is
(A) 7              (B) 0
(C) -5             (D) -6

2. 2 less than -6 is
(A) -8            (B) -4
(C) 4             (D) None of these

3. The sum of two integers is -17. If one of them is 12 then the other is
(A) 29            (B) -29
(C) -5             (D) 5

4. The sum of the two integers is 23. If one of them is -7 then the other is
(A) -30          (B) 16
(C) -16          (D) 30

5. 4 more than -7 is
(A) -11          (B) -3
(C) 3             (D) 11

6. 9 x (-16) + (-12) x (-16) = ?
(A) 48           (B) -48
(C) 54           (D) -54

7. (-12) x 7 + (-12) x (-4) = ?
(A) 32           (B) -32
(C) -36          (D) 36

8. The sum of two integer is 65. If one of them is -47 what is the other number?
(A) 112         (B) -112
(C) 18           (D) -18

9. The difference of two integer is -27 . If one of them is 32 then what is the other?
(A) -59          (B) 59
(C) 5             (D) - 5

10. From the sum of 33 and -47, -84 is subtracted. What is the result?
(A) 70           (B) -70
(C) 94           (D) -94

11. [37 -(-6)] + [ 11 - (-32)] = ?
(A) 86           (B) 74
(C) -86          (D) -74

12. The sum of two integers is -27. If one of them is 265 then what is the other?
(A) 292          (B) -292
(C) 238          (D) -238

13. Subtract the sum of -1070 and 813 from it
(A) 294          (B) *294
(C) 272          (D) -274

14. What is the successor of -99?
(A) -100        (B) -98
(C) 100          (D) 98

15. What is the predecessor of -79?
(A) -78          (B) -80
(C) 78            (D) 80

16. What is the sum of -23, 62, -57 and 13?
(A) -5            (B) 5
(C) 15           (D) -15

17. What is additive inverse of -100?
(A) 100         (B) 0
(C) -1            (D) 1

18. 5 + (-2) + (-7) + 6 = ?
(A) 2             (B) -2
(C) 3             (D) -3

19. -2 + (-7) + 3 + (6) + (-9) + 11 = ?
(A) 4              (B) -2
(C) 2              (D) -4

20. If -5 is added to 12 and result is subtracted from -7. Which number is obtained?
(A) -5            (B) 0
(C) 14            (D) -14

21. Amir had Rs 28760 in his bank account and his wife Laxmi has a debt of Rs 12380. What was their combined net balance?
(A) Rs 16380            (B) Rs. 14380
(C) Rs 15380            (D) Rs 17380

22. What is the value of |-5 -26 + 17| ?
(A) -14                      (B) 14
(C) 48                       (D) -48

23. What should be added to 57 to obtain -797
(A) -136                    (B) 136
(C) -22                      (D) None of these

24. Subtract -2473 from the difference of 5396
(A) 27                       (B) 4773
(C) 4973                   (D) 4873

25. What we will get when -49 is added to the difference of 72 and -997
(A) 76                       (B) -76
(C) -21                      (D) 122

26. What is the yalue of
1 + (-473) + (-375) + (-383) + (-283) + 1700
(A) 187                     (B) 287
(C) 286                     (D) 186

27. The sum of two integers is -307. If one of them is -173 what is the other?
(A) 134                    (B) -134
(C) 144                    (D) -134

28. What is the value of (-705) + 487 + (-317) + (265)?
(A) 270                    (B) -270
(C) 370                    (D) -370

29. Find the sum of predecessor of (-1709) and the successor of (-2305)
(A) 4014                  (B) -4014
(C) 594                    (D) None of these

30. Find the differece of negative of highest three digit number and smallest four digit number.
(A) 1                       (B) -1999
(C) 1999                 (D) None of these

1. A 2. A 3. B 4. D

5. B 6. A 7. C 8. A

9. B 10. A 11. A 12. B

13. A 14. B 15. B 16. A

17. A 18. A 19. C 20. D

21. A 22. B 23. A 24. C

25. D 26. A 27. B 28. A

29. B 30. C