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(i) This representation is wrong. Because the numbers are not continuously marked as -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6,

(ii) This is the correct representation of Integers

(iii) The representation is wrong. – 2 is misplaced in the number line.

(iv) This representation is correct as the numbers are marked at equal intervals in the correct order.

(v) Here negative integers are marked wrongly to the left of 0

On observing the given expression, 1 + 2 – 3 + 4 + 5 – 6 – 7 + 8 – 9 = -5, we noticed that (-7) should be replaced by (+7) to get a result of 9.

Thus, 1 + 2 – 3 + 4 + 5 – 6 + 7 + 8 – 9

= (1 + 2 + 4 + 5 + 7 + 8) – (3 + 6 + 9)

= 27 – 18 

= 9

(- 1) × (- 1) = + 1 

(- 1) × (- 1) × (- 1) = – 1

 (- 1) × (- 1) × (- 1) × (- 1) = + 1 

(- 1) × (- 1) × (- 1) × (- 1) × (- 1) = – 1 

If negative integer multiplied even number of times, the product is positive integer. If negative integer multiplied odd number of times, the product is negative integer. 

(i) (-a) × (-a) × ………….. 24 times 

= + a (Positive integer) 

(ii) (-a) × (-a) × ……………. 35 times 

= – a (Negative integer)

(i) -11, 12, -13, 14, -15, 16, -17, 18, -19, -20

• First separating the positive integers 12, 14, 16, 18 and the negative integers -11,-13,-15,-17,-19,-20.
• Then arranging the positive integers in ascending order we get 12, 14, 16, 18 and negative integers in ascending order as -20, -19, -17, -15, -13, -11 4
• Now the ascending order : -20, -19, -17, -15, -13, -11, 12, 14, 16, 18.

(ii) -28, 6, -5, -40, 8, 0, 12, -1, 4, 22

• Positive integers are 6, 8, 12, 4, 22 Negative integers are -28, -5, -40, -1
• Arranging the positive integers in ascending order we get 4, 6, 8, 12, 22 and the negative integers in ascending order -40, -28, -5, -1
• The number 0 is neither positive nor negative and stand in the middle.
• In ascending order : -40, -28, -5, -1, 0, 4, 6, 8,12, 22

(iii) -100, 10, -1000, 100, 0, -1, 1000, 1, -10

• Separating positive integers 10, 100, 1000, 1 and negative integers -100, -1000, -1, -10.
• Now the positive integers in ascending order 1,10,100,1000 and the negative integers in ascending order. -1000, -100, -10, -1
• Also ‘0’ stand in the middle as its is neither positive nor negative.
• ∴ The numbers in ascending order: -1000, -100, -10, -1, 0, 1, 10, 100, 1000.

Ascending order of given integers is, -5, -3, -2, 0, 1, 4

1 + 2 – 3 + 4 +5 – 6 + 7 + 8 – 9 = 9

The numbers marked = -3 and 

2 biggest number = 2 

smallest number = -3

We have, 6 + 0 = 6,

6 – 0 = 6

∴ The required two integers are 6 and 0.

The two integers whose sum is 6 and difference is also 6 is 0, 6.

7 ×  x   = – 42 x 

= (- 42) ÷ 7 

We know, (- a) ÷ b = – (a ÷ b) 

x = – (42 ÷ 7) 

∴ x = – 6

(-5) × (-9) = 45 

(-5) × (-9) = 45

5 × 9 = 45 

45 = 45

-7 × 5 = – 35 

(-7) × 5 = – 35 

– (7 × 5) = – 35 

– 35 = – 35

-6) × (-8) = 48 

(-6) × (-8) = 48 

6 × 8 = 48 

48 = 48

– (6) × 7) = – 42 

– (6) × 7) = – 42 

– (6 × 7) = – 42 

– 42 = – 42

When we divide a negative integer by a positive integer, we dividethem as whole numbers and put a Minus sign before quotient.

fill in the blanks to make the statements true.When –16 is divided by Nigative integer the quotient is 4.

(- 4) × 5 = – 20 

(- 4) × 5 = – 20 

– (4 × 5) = – 20 

– 20 = – 20

negative

When we divide a negative integer by a positive integer, we divide them as whole numbers and put a negative sign before quotient.

-4:

Let -16 be divided by x and the quotient is 4

So, -16/x = 4

x = -4

Multiplication

Division is the inverse operation of Multiplication.

10:

( – 100) ÷ ( – 10) = ( – 100)/ ( – 10) = 10

-5:

65 ÷ (– 13) = 65/ (-13) = -5

(a) (-7) -8 – (-25)

(-7) – 8 – (-25) = (-7) – 8 + 25 

= -15 + 25 = 10

(b) (-13) + 32 – 8 – 1

(-13) + 32 – 8 – 1 = -13 + 32 – 8 – 1 

= 32 – 22 = 10

(c) (-7) + (-8) +(-90)

(-7) + (-8) + (-90) = -7 – 8 – 90 

= -105

(d) 50 – (-40) – (-2)

50 – (-40) – (-2) = 50 + 40 + 2 = 92

(a) (-8) + 8 = 0

(b) 13 + (-13) = 0

(c) 12 + (- 12) = 0

(d) (-4) + (-8) =-12

(e) 5 – 15 = -10

(a) (-3) + (-6) = -9

(-3) – (-6) = -3 + 6 = 3 – 9 < 3

Hence (3) +(-6) < (-3)-(-6) 

(b) -31 + (-11) = -42 – 11 > -42

(9 – 21) – (- 10) ≥ (-31) +(-11)

(c) 45 – (-11) ...... 57 + (-4)

57 + (-4) = 57 – 4 = 53

56 > 53

(d) -42 - (-25) = -42 + 25 = -17

117 > -17

(-25) - (-42) ≥ (-42) - (-25)

On the number line, an integer on the right of a given integer is always larger than the integer.

True

(a) 137 and-354

137 + (- 354) = -217

(b) -52 and 52

-52 + 52 = 0

(c) -312, 39 and 192

-312 + 39 + 192 = -312 + 231 = -81

(d) -50, -200 and 300

-50 + (-200) + 300 = -250 + 300 = 50

(a) 35 – (20)

35 – (20) = 15

(b) 72 – (90)

72 – (90) = -18

(c) (-15) – (-18)

(-15) – (-18) = -15 + 18 = 3

(d) (-20) – (-13)

(-20) – (-13) = -33

(e) 23 – (-12)

23 – (-12) = 23 + 12 = 35

(f) (-32) – (-40)

(-32) – (-40) = -32 + 40 = 8

True.

Zero is neither positive nor negative.

(a) ( – 4) ÷ 16

(– 16) ÷ 4 = -4

But, ( – 4) ÷ 16 = -1/4

Zero is neither positive nor negative.

True

(d) (– 12) ÷ 5

0 ÷ (– 7) = 0, an integer

20 ÷ (– 4) = -5, an integer

(– 9) ÷ 3 = -3, an integer

But, (– 12) ÷ 5 = -2.4, which is a decimal and not an integer

(a) (-7) + (-9) + 4 + 16

(-7) + (-9) + 4 + 16 = -16 + 20 = 4

(b) (37) + (-2) + (-65) + (-8)

37 + (-2) + (-65) + (-8) = 37 + (-75) = -38

False.

Zero is an integer even though it is neither positive nor negative.

(c) (– 5) × (–1)

As all the remaining options give a value of -5

20 + (–25) = (– 37) – (– 32) = ( 45 ) ÷ (– 9) = -5

But, (– 5) × (–1) = 5

(b) – 10° to + 1°

As the difference in the temperature = 1° – (10°) = 11° (maximum)

Whereas,

23° to 32° = 32° – 23° = 9°

– 18° to – 11° = -11° – (-18)° = 7°

– 5° to 5° = 5° – (-5)° = 10°

(d) a ÷ b

If a and b are two integers, then

a + b will always be an integer

a – b will always be an integer

a × b will always be an integer

(a) a ÷ 0

a ÷ 0 = a/0 is undefined

(c) (–6, 1)

–3 + 3 = 0

–5 + 5 = 0

–8 + 8 = 0

–6 + 1 = -5

Hence, (–6, 1) is the odd one.

(c) (–9) × (–5) × (–6) × 3

(–9) × 5 × 6 × (–3) = 810

9 × (–5) × 6 × (–3) = 810

(–9) × (–5) × (–6) × 3 = -810

9 × (–5) × (–6) × 3 = 810

(d) (–32) ÷ 9

Since, only (–32) ÷ 9 doesn’t give an integer i.e. -32/9 = -3.5555555556

(d) (–9, +7)

–1 + (–2) = -3

–5 + 2 = -3

–4 + 1 = -3

–9 + 7 = -2

Hence, (–9, +7) is the odd one.