Rd Sharma 2020 2021 for Class 6 Maths Chapter 5 - Negative Numbers And Integers

Answer:

$$\begin{gathered} \left(\mathrm{i}\right) \mathrm{Here}, \mathrm{we} \mathrm{have} \mathrm{to} \mathrm{add} \mathrm{integers} \mathrm{of} \mathrm{unlike} \mathrm{sign} ,\mathrm{therefore} \mathrm{we} \mathrm{find} \mathrm{the} \mathrm{difference} \mathrm{of} \mathrm{their} \mathrm{absolute} \mathrm{values} \& \mathrm{assign} \mathrm{the} \mathrm{sign} \mathrm{of} \mathrm{the} \mathrm{addend} \mathrm{having} \mathrm{greater} \mathrm{absolute} \mathrm{value} \\ (-557)+488 \\ =-\left[\right|-557|-|488\left|\right] ( \mathrm{As}, |-557|=557,|488|=488) \\ =-[557-488] \\ =-69 \\ \left(\mathrm{ii}\right) \mathrm{Here}, \mathrm{we} \mathrm{have} \mathrm{to} \mathrm{add} \mathrm{integers} \mathrm{that} \mathrm{are} \mathrm{both} \mathrm{negative}. \\ (-552)+(-160) \\ =-\left[\right|-552|+|-160\left|\right] (\mathrm{As},|-552|=552,|-160|=160) \\ =-[552+160] \\ =-682 \\ \left(\mathrm{iii}\right) \mathrm{Here},\mathrm{we} \mathrm{have} \mathrm{to} \mathrm{add} \mathrm{integers} \mathrm{of} \mathrm{unlike} \mathrm{signs} ,\mathrm{therefore} \mathrm{we} \mathrm{find} \mathrm{the} \mathrm{difference} \mathrm{of} \mathrm{their} \mathrm{absolute} \mathrm{values} \& \mathrm{assign} \mathrm{sign} \mathrm{of} \mathrm{the} \mathrm{addend} \mathrm{having} \mathrm{greater} \mathrm{absolute} \mathrm{value} . \\ \left(2567\right)+(-325) \\ =\left[\right|2567|-|-325\left|\right] (\mathrm{As},|2567|=2567,|-325|=325) \\ =[2567-325] \\ =2242 \\ \left(\mathrm{iv}\right) \mathrm{Here}, \mathrm{we} \mathrm{have} \mathrm{to} \mathrm{add} \mathrm{integers} \mathrm{of} \mathrm{unlike} \mathrm{sign} ,\mathrm{therefore} \mathrm{we} \mathrm{find} \mathrm{the} \mathrm{difference} \mathrm{of} \mathrm{their} \mathrm{absolute} \mathrm{values} \& \mathrm{assign} \mathrm{the} \mathrm{sign} \mathrm{of} \mathrm{the} \mathrm{addend} \mathrm{having} \mathrm{greater} \mathrm{absolute} \mathrm{value} . \\ (-10025)+139 \\ =-\left[\right|-10025|-|139\left|\right] (As,|-10025|=10025,|139|=139) \\ =-[10025-139] \\ = -9886 \\ \left(v\right) \mathrm{Here},\mathrm{we} \mathrm{have} \mathrm{to} \mathrm{add} \mathrm{integers} \mathrm{of} \mathrm{unlike} \mathrm{signs} ,\mathrm{therefore} \mathrm{we} \mathrm{find} \mathrm{the} \mathrm{difference} \mathrm{of} \mathrm{their} \mathrm{absolute} \mathrm{values} \& \mathrm{assign} \mathrm{sign} \mathrm{of} \mathrm{the} \mathrm{addend} \mathrm{having} \mathrm{greater} \mathrm{absolute} \mathrm{value} . \\ \left(2547\right)+(-2548) \\ =-\left[\right|2548|-|-2547\left|\right] (\mathrm{As},|2548|=2548,|-2547|=2547) \\ =-[2548-2547] \\ =-1 \\ \left(\mathrm{vi}\right) \mathrm{Here},\mathrm{we} \mathrm{have} \mathrm{to} \mathrm{add} \mathrm{integers} \mathrm{of} \mathrm{unlike} \mathrm{signs} ,\mathrm{therefore} \mathrm{we} \mathrm{find} \mathrm{the} \mathrm{difference} \mathrm{of} \mathrm{their} \mathrm{absolute} \mathrm{values} \& \mathrm{assign} \mathrm{sign} \mathrm{of} \mathrm{the} \mathrm{addend} \mathrm{having} \mathrm{greater} \mathrm{absolute} \mathrm{value} . \\ \left(2884\right)+(-2884) \\ =\left[\right|2884|-|-2884\left|\right] (\mathrm{As},|2884|=2884,|-2884|=2884) \\ =[2884-2884] \\ =0 \end{gathered}$$

Answer:

The additive inverse of the number a  is the number added to a,  yields zero.  So, we find the following:

Answer:

For the integer a, a + 1 will be its successor.
(i) −41 is the successor of −42.
(ii) 0 is the successor of −1.
(iii) 1 is the successor of 0.
(iv) −199 is the successor of 200.
(v) −98 is the successor of −99.

Answer:

For the integer a, the predecessor is (a − 1).
(i)  −1 is the predecessor of 0.
(ii) 0 is the predecessor of  1.
(iii) −2 is the predecessor of −1.
(iv) −126 is the predecessor of −125.
(v) 999 is the predecessor of 1000.

Answer:

(i) True − It is the definition of additive inverse; for example, 5 + (−5) = 0.
(ii) False − For example, −2 − 3 = −5; it is a negative integer.
(iii) False − −3 +  5 = 2; it is a positive integer.
(iv) False − 0 is the successor of −1.
(v) False − It can be zero like (−2) + (−1) + (3).

Answer:

Let x be an integer such that |x| < 5. 
∴ -5 < x < 5 
=> x = ] -4, 4 [
These are the nine integers whose absolute values are less than 5, namely, -4, -3, -2, -1, 0, 1, 2, 3 and 4.

Answer:

(i) |4|+ |2| = 6 = |4| + |2|; True
(ii) |2 − 4| = |−2| = 2 ≠ 2 + 4 = 6; False
(iii) |4 − 2| = 2 = |4| − |2|; True
(iv) |(−2) + (−4)| = |−6| = 6 = |−2| + |(−4)|; True

Answer:

(i) (6, −6), (4, −4), (2, −2), (0, 0), (−2, 2), (−4, 4) and (−6,6).
(ii) (−4) + (−2) = −6 = (−2) + (−4); Yes
(iii) 0 + (−6) = −6; Yes
 

Answer:

(i) x +  1 = 0 ⇒ x = −1
(ii) x + 5 = 0 ⇒ x = −5
(iii) −3 + x = 0 ⇒ x = 3
(iv) x + (−8) = 0 ⇒ x = 8
(v) 7 + x = 0 ⇒ x = −7
(vi) x + 0 = 0  ⇒ x = 0

Answer:

(i) (−5) − 12 = −17
(ii) 8 − (−12) = 8 + 12 = 20
(iii) −135 − (−225) =  −135 + 225 = 90
(iv) 101 − 1001 = −900
(v) 3126 − (−812) = 3126 + 812 = 3938
(vi) −8 − 7560 = −7568
(vii) −4109 − (−3978) =  −4109 + 3978 = −131
(viii) −1005 − 0 = −1005

Answer:

(i) −27 − (−23)
= −27 + 23
= −4

(ii) −17 −18 − (−35)
= −17 − 18 + 35
= −35 + 35
= 0 

(iii) −12 − (−5) − (−125) + 270
= −12 + (5 + 125 + 270)
= −12 + 400
= 388

(iv) 373 + (−245) + (−373) + 145 + 3000
= 373 − 245 − 373 + (145 + 3000)
= 128 − 373 + 3145
= −245 + 3145
= 2900

(v) 1 + (−475) + (−475) + (−475) + (−475) + 1900
= 1 (−475 − 475 − 475 − 475 ) + 1900
= 1 − 1900 + 1900
= 1

(vi) (−1) + (−304) + 304 + 304 + (−304) + 1
= −1 + (−304 + 304) + ( 304 − 304) + 1
= −1 + 0 + 0 + 1
= 0

Answer:

We have to subtract the sum of −5020 and 2320 from −709.
Sum:
−5020 + 2320 = −2700
Now,
−709 − (−2700) = −709 + 2700 = 1991

Answer:

Sum of −1250 and 1138 = (−1250) + 1138 = −112
Sum of 1136 and −1272 = 1136 + (−1272) = −136
Now,
−136 − (−112) = −136 + 112 = −24

Answer:

We have to subtract −284 from the sum of 233 and −147.
Sum of 233 and (−147) = 233 + (−147) = 233 − 147 = 86
Now, we will subtract −284 from 86.
86 − (−284) = 86 + 284 = 370

Answer:

Let x and y be two integers such that x + y =  238.
Given: x = −122 
Now,
x + y = 238
⇒ −122 + y = 238
⇒  y = 238 + 122 = 360
So, the other integer is 360.

Answer:

Let  x and y be two integers such that x + y = −223.
Given: x = 172
Now,
x + y = −223
⇒ 172 + y = −223
⇒ y = −223 − 172
⇒ y = −395

Answer:

(i) −8 − 24 + 31 − 26 − 28 + 7 + 19 − 18 − 8 + 33
  = (−8 − 24) + (31 − 26) + (−28 + 7) + (19 − 18) + (−8 + 33)
  = (−32 + 5 − 21) + (1 + 25)
  = −48 + 26
  = −22

(ii) −26 − 20 + 33 − (−33) + 21 + 24 − (−25) − 26 − 14 − 34
    = (−26 − 20) + (33 + 33) + (21 + 24) + (25 − 26) + (−14 − 34)
    = (−46 + 66) + (45 − 1 − 48)
    = 20 − 4
    = 16

Answer:

1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 10 + 11 − 12 + 13  − 14 + 15 − 16
= (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15) − (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16)
= 64 − 72
= −8

Answer:

(i) If the number of terms is 10, then 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) = 0.
(ii) If the number of terms is 11, then 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 = 5.

 

Answer:

(i) (−6) + (−9) = −15 < (−6) − (−9) = −6 + 9 = 3
(ii) (−12) − (−12) = −12 + 12 = 0 > (−12) + (−12) = −12 − 12 = −24
(iii) (−20) − (−20) = −20 + 20 = 0 > 20 − 65 = −45
(iv) 28 − (−10) = 28 + 10 = 38 < −16 − (−76) = −16 + 76 = 60

Answer:

(i) −4 + 3 − (−2)
  = −4 + (3 + 2)
  = −4 + 5
  = 1

(ii) −(−2) + (−3) − (−2)
  = (2 − 3) + 2
  = −1 + 2
  = 1

(iii) −6 + (−5) − (−2)
  = −6 + (−5 + 2)
  = −6 − 3
  = −9

(iv) −(−5) + 6 − (−2)
  = 5 + (6 + 2)
  = 5 + 8
  = 13

Answer:

a and b are integers such that a is the predecessor of b, that is, a = b − 1.
∴ (a − b)
= (b − 1) − b
=  b − 1 − b
= −1 

Answer:

a and b are two integers such that a is the successor of b, that is, a = b + 1.
∴​ a − b 
= b + 1 − b
= 1

Answer:

(i) False; It should be −13 < −8 + 2 = −6. 
(ii) True; −4 − 2 = −6 < 2 
(iii) True; For example: −(−2) = 2
(iv) True; a > b 
(v) True; For example: 3 − 2 = 1, which is a integer.
(vi) False; For example: −2 + 2 = 0. Here, 2 is the additive inverse of −2; it is positive.
(vii) True   
(viii) True

Answer:

(i) −7 + 7 = 0  (−a and a are the negative and additive inverses of each other.)
(ii) 29 + (−29) = 0  (−a and a are the negative and additive inverses of each other.)
(iii) 132 + (−132) = 0  (−a and a are the negative and additive inverses of each other.)
(iv) −14 + 36 = 22
(v) −1256 + 514 = −742
(vi) −3305 − 1234 = −4539

Answer:

(a)
On the number line, − 7 is to the left of − 5, so − 7 < − 5.
(c)
Here, (− 7) + (− 5) = − ( 7 + 5) = − 12.
On the number line, − 12 is to the left of 0, so (− 7) + (− 5) < 0.
(d)
Here, (− 7) − (− 5) = (− 7) + (additive inverse of − 5) = (− 7) + (5) = − (7 − 5) = − 2
On the number line, − 2 is to the left of 0, so (− 7) − (− 5) < 0.
Hence, the correct option is (b).

Answer:

Here, 5 less than − 2 = (− 2) − (5) = − 2 − 5 = − 7.
Hence, the correct option is (c).

Answer:

6 more than − 7 = (− 7) + 6
                          = − ( 7 − 6)
                          = − 1
Hence, the correct option is (b).

Answer:

If x is positive integer, then |x| = x. So, x + |x| = x + x = 2x and x − |x| = x − x = 0.                    
Hence, the correct option is (b).

Answer:

If x is negative integer, then |x| = −x. So
x + |x| = x − x = 0
x − |x| = x − (− x) = x + x = 2x
Hence, the correct option is (a).

Answer:

If a is negative integer, then |a| = − a.
Here, x is greater than 2. So, 2 − x is negative.
Therefore, |2 − x| = − (2 − x) = − 2 + x = x − 2.
Hence, the correct option is (b).

Answer:

Here, |− 4| = 4. Therefore
9 + |− 4| = 9 + 4 = 13
Hence, the correct option is (c).

Answer:

(− 35) + (− 32) = − (35 + 32) = − 67
Hence, the correct option is (b).

Answer:

(− 29) + 5 = − (29 − 5) = − 24
Thus, the correct option is (d).

Answer:

|− |− 7| − 3| = |− 7 − 3|                                 (∵ |− 7| = 7)
                   = |− 10|                                
                   = 10                                          (∵ |− 10| = 10)
Hence, the correct option is (c).

Answer:

If a is an integer, then its successor is a + 1. So
Successor of − 22 = − 22 + 1 = − (22 − 1) = − 21
Hence, the correct option is (b).

Answer:

If a is an integer, then its predecessor is a − 1. So
Predecessor − 14 = − 14 − 1 = − 15
Hence, the correct option is (a).

Answer:

Sum of two integers = − 26
One of the two numbers = 14
∴ second number = − 26 − 14
                             = − (26 + 14)
                             = − 40
Hence, the correct option is (c).

Answer:

(a) 10 − 5 = 5
(b) (− 5) − (− 10) = − 5 + 10 = 5
(c) 15 − (− 20) = 15 + 20 = 35
Hence, the correct option is (d).

Answer:

Product of integers = 72
One of the two integers = − 9
So, the other integer is 72 $\text{÷}$ (− 9) = − 8.
Hence, the option is (a).

Answer:

Required number = − 14 − (− 7)
                             = − 14 + 7
                             = − (14 − 7)
                             = − 7
Hence, the correct option is (b).

Answer:

Subtract 12 and 7 from 64 and 72 respectively.
64 − 12 = 52
72 − 7 = 65
The required number is the HCF of 52 and 65.
Now
52 = 4 $\text{×}$ 13
65 = 5 $\text{×}$ 13
∴ HCF 52 and 65 = 13
Therefore, the largest number that divides 64 and 72 and leave the remainder 12 and 7 respectively, is 13.
Hence, the correct option is (b).

Answer:

Sum of integers = − 23
One of the numbers = 18
Other number = (− 23) − (18)
                       = − 23 − 18
                       = − (23 + 18)
                       = − 41
Therefore, the other number is − 41.
Hence, the correct option is (d).

Answer:

Sum of integers = − 35
One of the numbers = 40
Other number = (− 35) − (40)
                       = − 35 − 40
                       = − (35 + 40)
                       = − 75
Therefore, the other number is − 75.
Hence, the correct option is (b).

Answer:

Here, 0 − (− 5) = 0 + 5 = 5.
Therefore, on subtracting − 5 from 0, we get 5.
Hence, the correct option is (b).


 

Answer:

(− 16) + 14 − (− 13) = (− 16) + 14 + 13           (Additive inverse of − 13 is 13)
                                 = (− 16) + 27
                                 = 27 − 16
                                 = 11
Hence, the correct option is (c).

Answer:

(− 2) $\text{×}$ (− 3) $\text{×}$ 6 $\text{×}$ (− 1) = (2 $\text{×}$ 3) $\text{×}$ 6 $\text{×}$ (− 1)
                                          = 6 $\text{×}$ 6 $\text{×}$ (− 1)
                                          = 36 $\text{×}$ (− 1)
                                          = − (36 $\text{×}$ 1)
                                          = − 36
Hence, the correct option is (b).

Answer:

86 + (−28) + 12 + (−34) = 86 + (−28) − (34 − 12)
                                        = 86 + (−28) − 22
                                        = (86 − 28) − (34 − 12)
                                        = 58 − 22
                                        = 36
Hence, the correct option is (c).

Answer:

(−12) $\text{×}$ (−9) − 6 $\text{×}$ (−8) = (12 $\text{×}$ 9) − 6 $\text{×}$ (−8)
                                        = 108 − 6 $\text{×}$ (−8) 
                                        = 108 + 6 $\text{×}$ 8
                                        = 108 + 48
                                        = 156
Hence, the correct option is (a).

Answer:

If a is an integer, then its successor is a + 1. So
Successor of − 79 = − 79 + 1
                             = − (79 − 1)
                             = − 78
Hence, the correct option is (b).

Answer:

If a is an integer, then its predecessor is a − 1. So
Predecessor of − 99 = − 99 − 1
                                 = − (99 + 1)
                                 = − 100
Hence, the correct option is (b).

Answer:

The integer 8 more than − 12 is (− 12) + 8.
Now
(− 12) + 8 = − (12 − 8) = − 4
Hence, the correct option is (b).

Answer:

The required number is (− 34) − 18.
Now
(− 34) − 18 = − (34 + 18) = − 52
Hence, the correct option is (b).

Answer:

If a is an integer, then its additive inverse is − a.
So
Additive inverse of 17 = − 17
Hence, the correct option is (a).

Answer:

If x is negative integer, then |x| = − x.
Here, a is greater than 7, so 7 − a is negative.
Therefore
|7 − a| = − (7 − a) = − 7 + a = a − 7
Hence, the correct option is (b).

Answer:

If a is an integer, then a + 0 = 0 + a = a. Here, 0 is the additive identity element in the set of integers.
Hence, the correct option is (c).

Answer:

(a) 19 − 10 = 9
(b) − 10 − (− 19) = − 10 + 19 = 19 − 10 = 9
(c) 19 − (− 10) = 19 + 10 = 29
Thus, (a) and (b) both are correct.
Hence, the correct option is (d).

Answer:

(− 23) − (47) = − 23 − 47
                      = − (23 + 47)
                      = − 70
Thus, when 47 is subtracted from − 23, we get − 70.
Hence, the correct option is (d).

Answer:

Here, the operation ∆ is defined as a ∆ b = a − b − 2. So
7 ∆ (− 4) = 7 − (− 4) − 2
               = 7 + 4 − 2
               = 11 − 2
               = 9
Hence, the correct option is (c).

Answer:

(− 145) + 97 + (− 365) + (− 71) + 8 = (− 145) + 97 + (− 365) − (71 − 8)
                                                          = (− 145) + 97 + (− 365) − 63
                                                          = (− 145) + 97 − (365 + 63)
                                                          = (− 145) + 97 − 428
                                                          = (− 145) − (428 − 97)
                                                          = (− 145) − 331
                                                          = − (145 + 331)
                                                          = − 476
Hence, (− 145) + 97 + (− 365) + (− 71) + 8 = − 476.

Answer:

Sum of two integers = 84
One of the integers = 44
Other integer = Sum of two integers − One of the integers
                      = 84 − 44
                      = 40
Hence, the other integer is 40.

Answer:

9 $\text{×}$ (− 16) + (− 17) $\text{×}$ (− 16) = − (9 $\text{×}$ 16) + (17 $\text{×}$ 16)
                                               = − 144 + 272
                                               = 272 − 144
                                               = 128
Hence, 9 $\text{×}$ (− 16) + (− 17) $\text{×}$ (− 16) = 128.

Answer:

x = (− 23) + 22 + (− 23) + 22 + ... + (40 terms)
   = (− 23) + (− 23) + ... + (20 terms) + 22 + 22 + ... + (20 terms)
   = 20 $\text{×}$ (− 23) + 20 $\text{×}$ 22
   = 20 $\text{×}$ (− 23 + 22)
   = 20 $\text{×}$ (− 1)
   = − 20
Now
y = 11 + (− 10) + 11 + (− 10) + ... + (20 terms)
   = 11 + 11 + ... + (10 terms) + (− 10) + (− 10) + ... + (10 terms)
   = 11 $\text{×}$ 10 + (− 10) $\text{×}$ 10
   = 110 − 100
   = 10
Therefore, y − x = 10 − (− 20) = 10 + 20 = 30.

Answer:

1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ... + 49 − 50
= (1 − 2) + (3 − 4) + (5 − 6) + (7 − 8) + ... + (49 − 50)
= (− 1) + (− 1) + (− 1) + ... + 25 terms
= (− 1) $\text{×}$ 25
= − 25
Hence, 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ... + 49 − 50 = − 25.

Answer:

∵ |− 15| = 15 and |− 9| = 9
∴ 7 $\text{×}$ |− 15| − |− 9| $\text{×}$ 8 = 7 $\text{×}$ 15 − 9 $\text{×}$ 8
                                       = 105 − 72
                                       = 33
Hence, 7 $\text{×}$ |− 15| − |− 9| $\text{×}$ 8 = 33.

Answer:

38 − (− 25) − 58 + (− 15) + 23 − (− 8)
= 38 − (− 25) − 58 + (− 15) + 23 +8
= 38 − (− 25) − 58 + (− 15) + 31
= 38 − (− 25) − 58 + (31 − 15)
= 38 − (− 25) − 58 + 16
= 38 − (− 25) − (58 − 16)
= 38 − (− 25) − 42
= 38 + 25 − 42
= 38 − (42 − 25)
= 38 − 17
= 21
Hence, 38 − (− 25) − 58 + (− 15) + 23 − (− 8) = 21.

Answer:

(i) When the number of terms is 20             
5 + (− 5) + 5 + (− 5) + ... = (5 + 5 + 5 + ... + 10 terms) + (− 5) + 5 + (− 5) + ... + 10 terms
                                         = 10 $\text{×}$ 5 + 10 $\text{×}$ (− 5)
                                         = 50 − 50
                                         = 0

(ii) When the number of terms is 25
5 + (− 5) + 5 + (− 5) + ... = (5 + 5 + 5 + ... + 12 terms) + (− 5) + 5 + (− 5) + ... + 13 terms
                                         = 12 $\text{×}$ 5 + 13 $\text{×}$ (− 5)
                                         = 60 − 65
                                         = − (65 − 60)
                                         = − 5

Answer:

(i)
∵ a ∆ b = a − b − (− 5)
∴ (− 7) ∆ 3 = (− 7) − 3 − (− 5)
                   = − 7 − 3 + 5
                   = − (7 + 3) + 5
                   = − 10 + 5
                   = − (10 − 5)
                   = − 5
(ii)
 (− 9) ∆ (− 4) = (− 9) − (− 4) − (− 5)
                      = − 9 + 4 + 5
                      = − 9  + 9
                      = 0

(iii)
 2 ∆ 5 = 2 − 5 − (− 5)
           = 2 − 5 + 5
           = 2
(iv)
4 ∆ (− 5) = 4 − (− 5) − (− 5)
               = 4 + 5 + 5
               = 14

Answer:

− 36 − 40 + 43 − (− 29) + 18 − (− 74)
= − 36 − 40 + 43 − (− 29) + 18 + 74
= − 36 − 40 + 43 − (− 29) + 92
= − 36 − 40 + 43 + 29 + 92
= − 36 − 40 + 43 + 121
= − 36 − 40 + 164
= − (36 + 40) + 164
= − 76 + 164
= 164 − 76
= 88

Answer:

The numbers ... − 5, − 4, − 3, − 2, − 1, 0, 1, 2, 3, 4, 5, ... form the set of integers.
Thus, the largest negative integer is − 1.

Answer:

The collection ... − 4, − 3, − 2, − 1, 0, 1, 2, 3, 4, ... forms the set of integers.
Thus, the smallest positive integer is 1.

Answer:

∵ (− 22) + 21 = − (22 − 21) = − 1
∴  (− 22) + 21 + (− 22) + 21 + ... + 20 terms = (− 1) + (− 1) + (− 1) + ... + 10 terms
                                                                       = 10 $\text{×}$ (− 1)
                                                                       = − 10

Answer:

(− 3) (− 4) (12) (− 1) = (− 3) (− 4) (− 12)                       [ ∵ (12) (− 1) = (− 12)]
                                  = (− 3) (48)                                    [ ∵ (− 4) (− 12) = 48]
                                  = − (3 $\text{×}$ 48)
                                  = − 144
Hence, (− 3) (− 4) (12) (− 1) = − 144.

Answer:

Here, − 1 is multiplied 5 times and since, 5 is an odd integer, therefore
(− 1) (− 1) (− 1) (− 1) (− 1) = − 1

Answer:

(i) Decrease in population
(ii)Withdrawing money from a bank
(iii) Spending money
(iv) Going South
(v) Losing weight of 4 kg
(vi) A gain of Rs 1,000
(vii) −25
(viii) 15

Answer:

(i) If temperature is above zero, then it should be positive, i.e., +25°.
(ii) If temperature is below zero, then it should be negative, i.e., 5°.
(iii) If there is a profit of Rs 800, then it should be +800.
(iv) Deposition of Rs 2,500 indicates that money in the account is increased; therefore, it should be +2500 .
(v) Here, the distance is above the sea level; therefore, it  should be +3.
(vi) Here, the distance is below the sea level; therefore, it should be −2.

Answer:

Answer:

(i) 0 is greater than every negative integer, so −4 < 0.
(ii) Every positive integer is greater than every negative integer; therefore, −3 < 12.
(iii) Because 8 is to the left of 13 on a number line, 8 < 13.
(iv) Because −27 is to the left of −15 on a number line, −27 < −15.

Answer:

(i) Every positive integer is greater than every negative integer; therefore, 3 > −4.
(ii) Because −12 is to the left of −8 on a number line, −8 > −12. 
(iii) Every positive integer is greater than zero; therefore, 7 > 0.
(iv) Every positive integer is greater than every negative integer; therefore, 12 > −18.

Answer:

(i) There are nine integers in between −7 and 3, namely, −6, −5, −4, −3, −2, −1, 0, 1 and 2.
(ii) There are three integers in between −2 and 2, namely, −1, 0 and 1.
(iii) There are three integers in between −4 and 0, namely, −3, −2 and −1.
(iv) There are two integers in between 0 and 3, namely, 1 and 2.

Answer:

(i) There are six integers in between −4 and 3, namely, −3, −2, −1, 0, 1 and 2.
(ii) There are six integers in between 5 and 12, namely, 6, 7, 8, 9, 10 and 11.
(iii) There are six integers in between −9 and −2, namely, −8, −7, −6, −5, −4 and −3.
(iv) There are four integers in between 0 and 5, namely, 1, 2, 3 and 4.

Answer:

In the given pairs of numbers, the numbers that are to the left of the other numbers on a number line are smaller.

(i) 2 < 5
(ii) 0 < 3
(iii) 0 > −7
(iv) −18 < 15
(v) −235 > −532
(vi) −20 < 20

Answer:

(i)  −12 < −9 < −8 < 0 < 1 < 5 < 15 
(ii) −320 < −106 < −7 < 107 < 185

Answer:

(i) 8 > 7 > 6 > 0 > −2 > −5 > −9 > −15
(ii) 123 > −74 > −89 > −154 > −205

Answer:

(i) We want an integer two more than 3. So, on the number line, we will start from 3 and move 2 units to the right to obtain 5, as shown on the number line.

(ii) We want an integer five less than 3. So, on the number line, we will start from 3 and move 5 units to the left to obtain −2, as shown on the number line.

(iii) We want an integer four more than −9. So, on the number line, we will start from −9 and move 4 units to the right to obtain −5, as shown on the number line.


(i)



(ii)



(iii)

Answer:

(i) Absolute value of 14 is 14.
(ii) Absolute value of −25 is 25.
(iii) Absolute value of 0 is 0.
(iv) Absolute value of −125 is 125.
(v) Absolute value of −248 is 248.
(vi) Absolute value of (a − 7) is (a − 7) if a is greater than 7, that is, a − 7 > 0.
(vii) if a − 2 is less than 7, that is, a − 2 < 7 ⇒ a < 9 or a − 7 < 2
  So absolute value of a−7 = a−7 if  7 < a < 9 that is a−2 is less than 7 but a−2 is greater than 5.
and absolute value will be −(a−7) if  a < 7 if a−7 is less than 5.
(viii) Absolute value of (a + 4) is (a + 4) if a is greater than −4, that is, a > −4 ⇒ a + 4 > 0.
(ix) Absolute value of (a + 4) is −(a + 4) if a is less than −4, that is, a < −4 ⇒ a + 4 < 0.
(x) Absolute value of −3 is 3.
(xi) −|−5| is −5 and its absolute value is 5.
(xii) |12 − 5 | = | 7| and its absolute value is 7.

Answer:

(i) −9, −8, −7 and −6 are the four negative integers less than −10.
(ii) −11, −10, −9, −8, −7 and −6 are the six negative integers just greater than −12.

Answer:

(i) False
    Integers are negative also.
(ii) True
      0 is neither positive nor negative.
(iii) False
      0 is simply an integer that is neither positive nor negative.
(iv) True
       Every negative integer is to the left of 0 on a number line.
(v) False
       The absolute value of positive integer is integer itself. So both are equal.
(vi) True
        Its opposite will be a negative integer and positive integer is always greater than negative integer.
(vii) True
         Natural numbers start from 0, and 0 is greater than every negative integer.
(viii) False 
         0 is neither positive nor negative.

Answer:

(i) If we start from 5 and move 2 units to the left of 5, we will obtain 3, as shown on the number line.


(ii) If we start from -9 and move 4 units to the right of -9, we will obtain -5, as shown on the number line.

(iii) If we start from -3 and move 5 units to the left of -3, we will obtain -8, as shown on the number line.

(iv) If we start from 6 and move 6 units to the left of 6, we will obtain 0, as shown on the number line.

(v) If we start from -1 and move 2 units to the left of -1, we will obtain -3; and then if we start from -3 and move 2 units to the left, we will obtain -1, as shown on the number line.

(vi) If we start from -2 and move 7 units to the right, we will obtain 5; and then if we start from 5 and move 9 units to the left, we will obtain -4, as shown on the number line.