Integers

We use the fact that integers are used to represent statements which are opposite to each other in real life.

(i) The opposite of "Loss" is "Gain" or "Profit".

Hence, the opposite of Loss of ₹5000 is Gain (Profit) of ₹5000.

(ii) The opposite of "East" is "West".

Hence, the opposite of 30 km East of Delhi is 30 km West of Delhi.

(iii) The opposite of "above sea level" is "below sea level".

Hence, the opposite of 200 m above sea level is 200 m below sea level.

(iv) The opposite of "BC" (Before Christ) is "AD" (Anno Domini).

Hence, the opposite of 325 BC is 325 AD.

(v) The opposite of "Spending" is "Earning".

Hence, the opposite of Spending ₹2700 is Earning ₹2700.

(vi) The opposite of "above freezing point" is "below freezing point".

Hence, the opposite of 25°C above freezing point is 25°C below freezing point.

Gains, earnings, deposits, heights above sea level and altitudes are represented by '+' sign, while losses, expenditures, withdrawals, depths below sea level and depths below the ground are represented by '-' sign.

(i) Gain represents a positive change.

Hence, Gain of 3 kg weight = +3 kg.

(ii) Earning represents money received.

Hence, Earning ₹1340 = +₹1340.

(iii) "Below freezing point" represents a negative temperature.

Hence, 20°C below freezing point = -20°C.

(iv) Loss represents a negative change.

Hence, Loss of ₹470 = -₹470.

(v) Depositing represents adding money to the account.

Hence, Depositing ₹2500 in a bank = +₹2500.

(vi) "Below sea level" represents a depth.

Hence, 240 m below sea level = -240 m.

(vii) "At a height" represents an altitude above the ground.

Hence, A jet plane flying at a height of 9320 m = +9320 m.

(viii) "Down in the basement" represents a depth below the ground.

Hence, 6 m down in the basement of a building = -6 m.

On the number line, the greater number always lies to the right of the smaller number.

(i) Comparing 3 and 5: 5 > 3.

Hence, 5 is to the right of 3 on the number line.

(ii) Comparing 0 and -2: 0 > -2 (since zero is greater than every negative integer).

Hence, 0 is to the right of -2 on the number line.

(iii) Comparing -3 and -5: |-3| = 3 and |-5| = 5. Since 3 < 5, we get -3 > -5.

Hence, -3 is to the right of -5 on the number line.

(iv) Comparing 2 and -7: 2 > -7 (since every positive integer is greater than every negative integer).

Hence, 2 is to the right of -7 on the number line.

On the number line, the smaller number always lies to the left of the greater number.

(i) Comparing -3 and 0: -3 < 0.

Hence, -3 is to the left of 0 on the number line.

(ii) Comparing 2 and -5: -5 < 2.

Hence, -5 is to the left of 2 on the number line.

(iii) Comparing -4 and -7: |-4| = 4 and |-7| = 7. Since 4 < 7, we get -7 < -4.

Hence, -7 is to the left of -4 on the number line.

(iv) Comparing -10 and -16: |-10| = 10 and |-16| = 16. Since 10 < 16, we get -16 < -10.

Hence, -16 is to the left of -10 on the number line.

The number line is shown below:

(i) The integers strictly between -9 and -2 are the integers greater than -9 and less than -2.

Hence, the integers between -9 and -2 are -8, -7, -6, -5, -4 and -3.

(ii) On the number line, the integer farthest to the right among -8, -7, -6, -5, -4 and -3 is -3.

Hence, the largest integer among them is -3.

(iii) On the number line, the integer farthest to the left among -8, -7, -6, -5, -4 and -3 is -8.

Hence, the smallest integer among them is -8.

Four consecutive integers just greater than -9 are obtained by repeatedly adding 1 to -9.

-9 + 1 = -8

-8 + 1 = -7

-7 + 1 = -6

-6 + 1 = -5

Hence, the four consecutive integers just greater than -9 are -8, -7, -6 and -5.

Four consecutive integers just before -2 are obtained by repeatedly subtracting 1 from -2.

-2 - 1 = -3

-3 - 1 = -4

-4 - 1 = -5

-5 - 1 = -6

Listing them in order from smallest to largest, the four consecutive integers just before -2 are -6, -5, -4 and -3.

Hence, the four consecutive integers just before -2 are -6, -5, -4 and -3.

The number line is drawn for reference:

(i) Starting from -1 and moving 6 units to the right:

Hence, after moving 6 units to the right of -1, we reach 5.

(ii) Starting from 2 and moving 7 units to the left:

Hence, after moving 7 units to the left of 2, we reach -5.

(iii) On the number line, 3 lies to the right of -3.

Hence, to reach 3 from -3, we should move in the right direction.

(iv) On the number line, -8 lies to the left of -3.

Hence, to reach -8 from -3, we should move in the left direction.

To find an integer "more than" a given integer, we move to the right on the number line. To find an integer "less than" a given integer, we move to the left on the number line.

(i) Start at -1 and move 5 units to the right:

Hence, the integer which is 5 more than -1 is 4.

(ii) Start at -1 and move 5 units to the left:

Hence, the integer which is 5 less than -1 is -6.

(iii) Start at 2 and move 7 units to the left:

Hence, the integer which is 7 less than 2 is -5.

(iv) Start at -7 and move 3 units to the right:

Hence, the integer which is 3 more than -7 is -4.

By "between" we mean strictly between (the end points are not included).

(i) The integers strictly between -15 and -7 are:

-14, -13, -12, -11, -10, -9, -8.

Counting them gives 7 integers.

Hence, there are 7 integers between -15 and -7.

(ii) The integers strictly between -6 and 3 are:

-5, -4, -3, -2, -1, 0, 1, 2.

Counting them gives 8 integers.

Hence, there are 8 integers between -6 and 3.

(iii) Whole numbers are 0, 1, 2, 3, 4, 5, 6, ... .

The whole numbers strictly between -6 and 6 are:

0, 1, 2, 3, 4, 5.

Counting them gives 6 whole numbers.

Hence, there are 6 whole numbers between -6 and 6.

(iv) Negative integers are -1, -2, -3, ... .

The negative integers strictly between -7 and 4 are:

-6, -5, -4, -3, -2, -1.

Counting them gives 6 negative integers.

Hence, there are 6 negative integers between -7 and 4.

The absolute value of an integer is its numerical value regardless of its sign. We have |a| = a if a ≥ 0 and |a| = -a if a < 0.

(i) |13 - 5| = |8| = 8.

Hence, |13 - 5| = 8.

(ii) |5 - 13| = |-8| = 8.

Hence, |5 - 13| = 8.

(iii) |-11| + |9| = 11 + 9 = 20.

Hence, |-11| + |9| = 20.

(iv) |-8| + |-6| = 8 + 6 = 14.

Hence, |-8| + |-6| = 14.

(v) |7| - |-3| = 7 - 3 = 4.

Hence, |7| - |-3| = 4.

(vi) |-19| - |-13| = 19 - 13 = 6.

Hence, |-19| - |-13| = 6.

We use the rules of comparison of integers: every negative integer is less than every positive integer; zero is greater than every negative integer; for two negative integers, the one with the smaller absolute value is greater.

(i) -3 is negative and 7 is positive. Every negative integer is less than every positive integer.

Hence, -3 < 7.

(ii) Zero is greater than every negative integer.

Hence, 0 > -2.

(iii) Both -10 and -11 are negative. |-10| = 10 and |-11| = 11. Since 10 < 11, the one with smaller absolute value (-10) is greater.

Hence, -10 > -11.

(iv) Both -6 and -2 are negative. |-6| = 6 and |-2| = 2. Since 6 > 2, the one with bigger absolute value (-6) is smaller.

Hence, -6 < -2.

(v) Both -5 and -13 are negative. |-5| = 5 and |-13| = 13. Since 5 < 13, the one with smaller absolute value (-5) is greater.

Hence, -5 > -13.

(vi) Both -30 and -19 are negative. |-30| = 30 and |-19| = 19. Since 30 > 19, the one with bigger absolute value (-30) is smaller.

Hence, -30 < -19.

Ascending order means arranging from the smallest to the greatest.

(i) The given integers are -5, 3, 0, -9, 2.

The negative integers are -5 and -9. |-5| = 5 and |-9| = 9. Since 5 < 9, we have -9 < -5.

The non-negative integers are 0, 3 and 2. In increasing order: 0 < 2 < 3.

Since every negative integer is less than zero (and every positive integer):

-9 < -5 < 0 < 2 < 3.

Hence, the given integers in ascending order are -9, -5, 0, 2, 3.

(ii) The given integers are -28, -33, 9, -4, -31, -2, 35.

The negative integers are -28, -33, -4, -31, -2. Their absolute values are 28, 33, 4, 31, 2.

Arranging the absolute values in increasing order: 2 < 4 < 28 < 31 < 33. So the negatives in increasing order are -33 < -31 < -28 < -4 < -2.

The positive integers are 9 and 35. In increasing order: 9 < 35.

Since every negative integer is less than positive integer :

-33 < -31 < -28 < -4 < -2 < 9 < 35.

Hence, the given integers in ascending order are -33, -31, -28, -4, -2, 9, 35.

Descending order means arranging from the greatest to the smallest.

(i) The given integers are -31, 25, -37, 43, 0, -5.

The positive integers are 25 and 43. In decreasing order: 43 > 25.

Then comes 0, which is greater than every negative integer.

The negative integers are -31, -37 and -5. Their absolute values are 31, 37 and 5.

Arranging the absolute values in decreasing order: 37 > 31 > 5. So the negatives in decreasing order are -5 > -31 > -37.

Combining: 43 > 25 > 0 > -5 > -31 > -37.

Hence, the given integers in descending order are 43, 25, 0, -5, -31, -37.

(ii) The given integers are -101, 95, -3, -8, 36, -7, -84.

The positive integers are 95 and 36. In decreasing order: 95 > 36.

The negative integers are -101, -3, -8, -7 and -84. Their absolute values are 101, 3, 8, 7 and 84.

Arranging the absolute values in decreasing order: 101 > 84 > 8 > 7 > 3. So the negatives in decreasing order are -3 > -7 > -8 > -84 > -101.

Combining: 95 > 36 > -3 > -7 > -8 > -84 > -101.

Hence, the given integers in descending order are 95, 36, -3, -7, -8, -84, -101.

(i) The number 0 is neither positive nor negative. The positive integers are 1, 2, 3, ... and the smallest among them is 1.

Hence, the statement is False.

(ii) Natural numbers are 1, 2, 3, ..., which are all positive integers. Every negative integer is less than every positive integer, hence less than every natural number.

Hence, the statement is True.

(iii) Finding the absolute values,

|-7| = 7 and |-6| = 6. Since 7 > 6, we get -7 < -6, so -7 lies to the left of -6 (not the right).

Hence, the statement is False.

(iv) For a positive integer a, |a| = a (not greater than a). For 0, |0| = 0 (not greater than 0). Only for a negative integer is the absolute value greater than the integer itself.

Hence, the statement is False.

To add a positive integer, we move to the right on the number line. To add a negative integer, we move to the left on the number line.

(i) Start at 4 on the number line and move 5 units to the left:

Hence, 4 + (-5) = -1.

(ii) Start at -4 on the number line and move 5 units to the right:

Hence, (-4) + 5 = 1.

(iii) Start at 7 on the number line and move 3 units to the left:

Hence, 7 + (-3) = 4.

(iv) Start at -6 on the number line and move 2 units to the left:

Hence, -6 + (-2) = -8.

To add two negative integers, we add their absolute values and put the negative sign before the result. To add a positive integer and a negative integer, we subtract the smaller absolute value from the larger absolute value and give the sign of the integer with the larger absolute value.

(i) Both integers are negative. Add their absolute values and put the negative sign:

(-8) + (-14) = -(8 + 14) = -22.

Hence, (-8) + (-14) = -22.

(ii) Both integers are negative. Add their absolute values and put the negative sign:

-35 + (-47) = -(35 + 47) = -82.

Hence, -35 + (-47) = -82.

(iii) Different signs. |91| = 91 and |-48| = 48. Subtract: 91 - 48 = 43. The integer with the larger absolute value (91) is positive, so the answer is positive.

91 + (-48) = +(91 - 48) = 43.

Hence, 91 + (-48) = 43.

(iv) Different signs. |-203| = 203 and |501| = 501. Subtract: 501 - 203 = 298. The integer with the larger absolute value (501) is positive, so the answer is positive.

(-203) + 501 = +(501 - 203) = 298.

Hence, (-203) + 501 = 298.

(v) Different signs. |-36| = 36 and |29| = 29. Subtract: 36 - 29 = 7. The integer with the larger absolute value (-36) is negative, so the answer is negative.

(-36) + 29 = -(36 - 29) = -7.

Hence, (-36) + 29 = -7.

(vi) Different signs. |-131| = 131 and |97| = 97. Subtract: 131 - 97 = 34. The integer with the larger absolute value (-131) is negative, so the answer is negative.

(-131) + 97 = -(131 - 97) = -34.

Hence, (-131) + 97 = -34.

(i) Both integers are negative. Add their absolute values and put the negative sign:

-1083 + (-3974) = -(1083 + 3974) = -5057.

Hence, -1083 + (-3974) = -5057.

(ii) Different signs. |706| = 706 and |-394| = 394. Subtract: 706 - 394 = 312. The integer with the larger absolute value (706) is positive, so the answer is positive.

706 + (-394) = +(706 - 394) = 312.

Hence, 706 + (-394) = 312.

(iii) Different signs. |1309| = 1309 and |-2811| = 2811. Subtract: 2811 - 1309 = 1502. The integer with the larger absolute value (-2811) is negative, so the answer is negative.

1309 + (-2811) = -(2811 - 1309) = -1502.

Hence, 1309 + (-2811) = -1502.

For every integer a, the additive inverse of -a is a, i.e. -(-a) = a. So, the negative of a negative integer is the corresponding positive integer.

(i) -(-5) = 5.

Hence, -(-5) = 5.

(ii) -(-30) = 30.

Hence, -(-30) = 30.

(iii) -(-539) = 539.

Hence, -(-539) = 539.

The additive inverse of an integer a is -a, since a + (-a) = 0.

(i) The additive inverse of 9 is -9.

(Check: 9 + (-9) = 0.)

Hence, the additive inverse of 9 is -9.

(ii) The additive inverse of -11 is -(-11) = 11.

(Check: -11 + 11 = 0.)

Hence, the additive inverse of -11 is 11.

(iii) The additive inverse of -237 is -(-237) = 237.

(Check: -237 + 237 = 0.)

Hence, the additive inverse of -237 is 237.

(iv) The additive inverse of 567 is -567.

(Check: 567 + (-567) = 0.)

Hence, the additive inverse of 567 is -567.

If a is an integer, its additive inverse is -a (and a + (-a) = 0).

(i) Let the required integer be a. Then a is its own additive inverse, so

a = -a

⇒ 2a = 0

⇒ a = 0.

Hence, 0 is the integer which is its own additive inverse.

(ii) Let the required integer be a. Its additive inverse is -a. Then a is 4 more than -a:

a = (-a) + 4

⇒ a + a = 4

⇒ 2a = 4

⇒ a = 2.

(Check: additive inverse of 2 is -2, and 2 = -2 + 4. ✓)

Hence, 2 is the integer which is 4 more than its additive inverse.

(iii) Let the required integer be a. Its additive inverse is -a. Then a is 2 less than -a:

a = (-a) - 2

⇒ a + a = -2

⇒ 2a = -2

⇒ a = -1.

(Check: additive inverse of -1 is 1, and -1 = 1 - 2. ✓)

Hence, -1 is the integer which is 2 less than its additive inverse.

To subtract a positive integer, we move to the left on the number line. To subtract a negative integer, we move to the right on the number line.

(i) To find 4 - (-2), start at 4 and move 2 units to the right:

Hence, 4 - (-2) = 6.

(ii) To find -4 - (-2), start at -4 and move 2 units to the right:

Hence, -4 - (-2) = -2.

(iii) To find 3 - 6, start at 3 and move 6 units to the left:

Hence, 3 - 6 = -3.

(iv) To find -3 - (-5), start at -3 and move 5 units to the right:

Hence, -3 - (-5) = 2.

To subtract one integer from another, we change the sign of the integer to be subtracted and then add. That is, a - b = a + (-b).

(i) Subtract -6 from 9:

9 - (-6) = 9 + 6 = 15.

Hence, the result is 15.

(ii) Subtract 6 from -9:

-9 - 6 = -9 + (-6) = -(9 + 6) = -15.

Hence, the result is -15.

(iii) Subtract -6 from -9:

-9 - (-6) = -9 + 6 = -(9 - 6) = -3.

Hence, the result is -3.

(iv) Subtract -725 from -63:

-63 - (-725) = -63 + 725 = +(725 - 63) = 662.

Hence, the result is 662.

(v) Subtract -376 from 10:

10 - (-376) = 10 + 376 = 386.

Hence, the result is 386.

(vi) Subtract 92 from -620:

-620 - 92 = -620 + (-92) = -(620 + 92) = -712.

Hence, the result is -712.

We use the rule a - b = a + (-b), i.e. change the sign of the integer to be subtracted and add.

(i) -237 - (+1884) = -237 + (-1884)

= -(237 + 1884)

= -2121.

Hence, -237 - (+1884) = -2121.

(ii) -346 - (-1275) = -346 + 1275

= +(1275 - 346)

= 929.

Hence, -346 - (-1275) = 929.

(iii) -190 - (-3512) = -190 + 3512

= +(3512 - 190)

= 3322.

Hence, -190 - (-3512) = 3322.

(iv) -2718 - (+6827) = -2718 + (-6827)

= -(2718 + 6827)

= -9545.

Hence, -2718 - (+6827) = -9545.

If the sum of two integers and one of them are known, the other integer = sum - (the given integer).

(i) Sum = 17 and one integer = -35.

Other integer = 17 - (-35)

= 17 + 35

= 52.

(Check: -35 + 52 = 17. ✓)

Hence, the other integer is 52.

(ii) Sum = -80 and one integer = -90.

Other integer = -80 - (-90)

= -80 + 90

= +(90 - 80)

= 10.

(Check: -90 + 10 = -80. ✓)

Hence, the other integer is 10.

Let x be the integer that must be added to -23 to get -9.

Then, -23 + x = -9.

⇒ x = -9 - (-23)

= -9 + 23

= +(23 - 9)

= 14.

(Check: -23 + 14 = -9. ✓)

Hence, 14 must be added to -23 to get -9.

The predecessor of an integer a is a - 1.

Predecessor of 0 = 0 - 1 = -1.

Hence, the predecessor of 0 is -1.

For an integer a, its successor is a + 1 and its predecessor is a - 1.

(i) For -31:

Successor = -31 + 1 = -30.

Predecessor = -31 - 1 = -32.

Hence, the successor of -31 is -30 and the predecessor of -31 is -32.

(ii) For -735:

Successor = -735 + 1 = -734.

Predecessor = -735 - 1 = -736.

Hence, the successor of -735 is -734 and the predecessor of -735 is -736.

(iii) For -240:

Successor = -240 + 1 = -239.

Predecessor = -240 - 1 = -241.

Hence, the successor of -240 is -239 and the predecessor of -240 is -241.

We group the positive integers and the negative integers separately and then add their results.

(i) 6 - 9 + 4 = (6 + 4) - 9

= 10 - 9

= 1.

Hence, 6 - 9 + 4 = 1.

(ii) -5 - (-3) + 2 = -5 + 3 + 2

= (3 + 2) - 5

= 5 - 5

= 0.

Hence, -5 - (-3) + 2 = 0.

(iii) 7 + (-5) + (-6) = 7 - 5 - 6

= 7 - (5 + 6)

= 7 - 11

= -4.

Hence, 7 + (-5) + (-6) = -4.

(iv) 6 - 3 - (-5) = 6 - 3 + 5

= (6 + 5) - 3

= 11 - 3

= 8.

Hence, 6 - 3 - (-5) = 8.

We group the positive integers and the negative integers separately.

(i) -77 + (-84) + 318 = -77 - 84 + 318

= 318 - (77 + 84)

= 318 - 161

= 157.

Hence, -77 + (-84) + 318 = 157.

(ii) 54 + (-218) - (-76) = 54 - 218 + 76

= (54 + 76) - 218

= 130 - 218

= -(218 - 130)

= -88.

Hence, 54 + (-218) - (-76) = -88.

(iii) -121 - (-78) + (-193) + 576 = -121 + 78 - 193 + 576

= (78 + 576) - (121 + 193)

= 654 - 314

= 340.

Hence, -121 - (-78) + (-193) + 576 = 340.

(iv) -65 + (-76) - (-28) + 32 = -65 - 76 + 28 + 32

= (28 + 32) - (65 + 76)

= 60 - 141

= -(141 - 60)

= -81.

Hence, -65 + (-76) - (-28) + 32 = -81.

We group the positive and the negative integers separately.

(i) 8 - 6 + (-2) - (-3) + 1 = 8 - 6 - 2 + 3 + 1

= (8 + 3 + 1) - (6 + 2)

= 12 - 8

= 4.

Hence, 8 - 6 + (-2) - (-3) + 1 = 4.

(ii) 31 + (-23) - 35 + 18 - 4 - (-3) = 31 - 23 - 35 + 18 - 4 + 3

= (31 + 18 + 3) - (23 + 35 + 4)

= 52 - 62

= -(62 - 52)

= -10.

Hence, 31 + (-23) - 35 + 18 - 4 - (-3) = -10.

A deposit is represented by a positive integer and a withdrawal is represented by a negative integer.

Amount deposited on Monday = +₹4370.

Amount withdrawn on Tuesday = -₹2875.

Amount deposited on Wednesday = +₹1550.

There is no transaction on Thursday, so the balance on Thursday is the same as the balance at the end of Wednesday.

Balance on Thursday = (+4370) + (-2875) + (+1550)

= 4370 + 1550 - 2875

= 5920 - 2875

= ₹3045.

Hence, Rashmi's balance on Thursday was ₹3,045.

(i) The absolute value of an integer is its numerical value regardless of its sign. So |0| = 0.

Hence, the absolute value of 0 is 0.

(ii) When we add two negative integers, we add their absolute values and put a negative sign before the result. So the sum is always negative.

Hence, the sum of two negative integers is always a negative integer.

(iii) The positive integers are 1, 2, 3, 4, ... and the smallest among them is 1.

Hence, the smallest positive integer is 1.

(iv) The negative integers are -1, -2, -3, -4, ... and the largest among them is -1.

Hence, the largest negative integer is -1.

(v) Predecessor of -99 = -99 - 1 = -100.

Hence, the predecessor of -99 is -100.

(i) When a positive integer and a negative integer are added, the sign of the sum is the sign of the integer with the larger absolute value. For example, 8 + (-3) = 5 (positive). So the sum is not always negative.

Hence, the statement is False.

(ii) For every integer a, a + (-a) = 0. So the sum of an integer and its negative (additive inverse) is always zero.

Hence, the statement is True.

(iii) Three integers can sum to zero. For example, 1 + 2 + (-3) = 0.

Hence, the statement is False.

(iv) |-7| = 7 and |-3| = 3. Since 7 > 3, we have |-7| > |-3|.

Hence, the statement is False.

(v) Comparing -20 and -21: |-20| = 20 and |-21| = 21. Since 20 < 21, we have -20 > -21. So -20 lies to the RIGHT of -21, not to the left.

Hence, the statement is False.

(vi) Successor of -29 = -29 + 1 = -28, not -30.

Hence, the statement is False.

(vii) For any two integers a and b, a - b = a + (-b). Since the sum of two integers is always an integer (closure under addition), the difference of two integers is also always an integer.

Hence, the statement is True.

(viii) The additive inverse of a negative integer -a (where a is positive) is -(-a) = a, which is positive.

Hence, the statement is True.

(i) Comparing -8 and -10: |-8| = 8 and |-10| = 10. Since 8 < 10, we have -8 > -10. So -8 lies to the right of -10.

Hence, the statement is True.

(ii) Comparing -100 and -50: |-100| = 100 and |-50| = 50. Since 100 > 50, we have -100 < -50. So -100 lies to the LEFT of -50.

Hence, the statement is False. The correct statement is: "-100 is to the left of -50 on the number line."

(iii) The negative integers are -1, -2, -3, -4, ... and continue indefinitely. So there is no smallest negative integer. (In fact, -1 is the LARGEST negative integer.)

Hence, the statement is False. The correct statement is: "The largest negative integer is -1" (there is no smallest negative integer).

(iv) Comparing -26 and -25: |-26| = 26 and |-25| = 25. Since 26 > 25, we have -26 < -25. So -26 is less than -25.

Hence, the statement is False. The correct statement is: "-26 is less than -25" (or equivalently, -25 is greater than -26).

(v) Predecessor of -188 = -188 - 1 = -189, not -187. (In fact, -187 is the successor of -188.)

Hence, the statement is False. The correct statement is: "-189 is the predecessor of -188" (or equivalently, "-187 is the successor of -188").

Calculating :

-2 + 5 = 3.

Hence, option 3 is the correct option.

The integers strictly between -1 and 1 are the integers greater than -1 and less than 1.

The only such integer is 0.

So, the number of integers between -1 and 1 = 1.

Hence, option 2 is the correct option.

The integers strictly between -3 and 2 are -2, -1, 0 and 1.

Counting them gives 4 integers.

Hence, option 3 is the correct option.

The integers strictly between -10 and -15 are -11, -12, -13 and -14.

For negative numbers, the one with the smaller absolute value (the one closer to zero) is the greater number.

Here, |-11| = 11, is the smallest absolute value. Thus, -11 is the greatest integer.

Hence, option 2 is the correct option.

The integers strictly between -10 and -15 are -11, -12, -13 and -14.

For negative numbers, the one with the smaller absolute value (the one closer to zero) is the greater number.

Among these, |-14| = 14 is the largest absolute value, so -14 is the smallest.

Hence, option 3 is the correct option.

Let additive inverse of -5 be x, then

-5 + x = 0

x = 5.

Only option 2 is true.

Hence, option 2 is the correct option.

Solving,

23 + (-31) = -(31 - 23)

= -8.

Only option 3 is false.

Hence, option 3 is the correct option.

Let us check each option.

1. The sum of two integers is always an integer (closure property of addition). So, this statement (-3) + (-11) is true.

2. Addition is commutative for integers: a + b = b + a. So, this statement(-19) + 13 = 13 + (-19) is true.

3. Zero is the additive identity: a + 0 = a = 0 + a. So, this statement (-15) + 0 = -15 = 0 + (-15) is true.

4. The negative (additive inverse) of -7 is -(-7) = 7, which exists. So the statement "negative of -7 does not exist" is false.

Only option 4 is false.

Hence, option 4 is the correct option.

Other integer = sum - (the given integer)

= -17 - (-9)

= -17 + 9

= -(17 - 9)

= -8.

Hence, option 2 is the correct option.

Subtracting -7 from -4:

-4 - (-7) = -4 + 7

= +(7 - 4)

= 3.

Hence, option 1 is the correct option.

(-12) + 17 - (-10) = -12 + 17 + 10

= (17 + 10) - 12

= 27 - 12

= 15.

Hence, option 3 is the correct option.

Let us evaluate both sides of each option.

1. LHS = -13 and RHS = -8 - (-6) = -8 + 6 = -2. Is -13 > -2? Since |-13| > |-2|, we have -13 < -2. False.

2. LHS = -5 - 4 = -9 and RHS = -12 + 2 = -10. Is -9 > -10? Since |-9| < |-10|, we have -9 > -10. True.

3. LHS = (-8) - 3 = -11 and RHS = (-3) - (-8) = -3 + 8 = 5. Since -11 ≠ 5, False.

4. LHS = (-15) - (-22) = -15 + 22 = 7 and RHS = (-22) - (-15) = -22 + 15 = -7. Is 7 < -7? No, 7 > -7. False.

Only option 2 is true.

Hence, option 2 is the correct option.

Let the integer be a. Then a + a = 2a. We must check when 2a < a.

If a is positive (e.g. a = 5): 2a = 10 and a = 5. Then 2a > a, so the statement is false.

If a = 0: 2a = 0 and a = 0. Then 2a = a, so the statement is false (the sum is not less than the integer).

If a is negative (e.g. a = -5): 2a = -10 and a = -5. Since |-10| > |-5|, we have -10 < -5, i.e. 2a < a, so the statement is true.

So the statement is true only when the integer is negative.

Hence, option 3 is the correct option.

Let us first consider Statement I.

Natural numbers are 1, 2, 3, 4, ... and whole numbers are 0, 1, 2, 3, 4, ... . The sum of two natural numbers is again a natural number, and every natural number is also a whole number.

For example, 3 + 5 = 8, which is both a natural number and a whole number.

∴ Statement I is true.

Now let us consider Statement II.

The closure property of addition for natural numbers states that the sum of two natural numbers is always a natural number. Also, by definition, the set of whole numbers contains all natural numbers (0 being the only whole number that is not a natural number).

∴ Statement II is true.

Statement II correctly explains Statement I, since "sum is a natural number" together with "every natural number is a whole number" gives "sum is a whole number".

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Let us first consider Statement I.

When we add negative integers, we add their absolute values and put a negative sign before the result. So the sum of negative integers is always a negative integer, which is less than zero.

∴ Statement I is true.

Now let us consider Statement II.

Whole numbers are 0, 1, 2, 3, 4, ... . The smallest among them is 0.

∴ Statement II is true.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Let us first consider Statement I.

Distance is a non-negative quantity that measures how much the car has actually moved, regardless of direction. The car travels 500 m and then another 500 m, so the total distance travelled = 500 + 500 = 1000 m, not zero. (Note: the displacement is zero, but the distance is not.)

∴ Statement I is false.

Now let us consider Statement II.

Distances in opposite directions are represented by integers of opposite signs, but distance itself is always non-negative. To find the total distance, we add the absolute values of the individual distances.

∴ Statement II is true.

Thus, Statement I is false but Statement II is true.

Hence, option 2 is the correct option.

Let us first consider Statement I.

For any integer a, |a| = a if a ≥ 0 and |a| = -a if a < 0. In either case |a| ≥ 0, so the absolute value is always non-negative.

∴ Statement I is true.

Now let us consider Statement II.

On the number line, the absolute value of a number represents how far the number is from 0, regardless of direction. Since distance is non-negative, this matches the definition of absolute value.

∴ Statement II is true.

Statement II correctly explains why Statement I is true: distance from 0 is always non-negative, which is exactly why the absolute value is non-negative.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

We first evaluate both sides and then compare them.

(i) LHS = (-3) + (-6) = -(3 + 6) = -9.

RHS = (-3) - (-6) = -3 + 6 = 3.

Since -9 is negative and 3 is positive, -9 < 3.

Hence, (-3) + (-6) < (-3) - (-6).

(ii) LHS = (-21) - (-10) = -21 + 10 = -(21 - 10) = -11.

RHS = (-31) + (-11) = -(31 + 11) = -42.

Comparing -11 and -42: |-11| = 11 and |-42| = 42. Since 11 < 42, we have -11 > -42.

Hence, (-21) - (-10) > (-31) + (-11).

(iii) LHS = 45 - (-11) = 45 + 11 = 56.

RHS = 57 + (-4) = 57 - 4 = 53.

Since 56 > 53,

Hence, 45 - (-11) > 57 + (-4).

(iv) LHS = (-25) - (-42) = -25 + 42 = +(42 - 25) = 17.

RHS = (-42) - (-25) = -42 + 25 = -(42 - 25) = -17.

Since 17 is positive and -17 is negative, 17 > -17.

Hence, (-25) - (-42) > (-42) - (-25).

We group the positive and negative integers separately.

(i) 12 + (-3) + 5 - (-2) = 12 - 3 + 5 + 2

= (12 + 5 + 2) - 3

= 19 - 3

= 16.

Hence, 12 + (-3) + 5 - (-2) = 16.

(ii) 39 - 35 + 7 - (-4) + 21 = 39 - 35 + 7 + 4 + 21

= (39 + 7 + 4 + 21) - 35

= 71 - 35

= 36.

Hence, 39 - 35 + 7 - (-4) + 21 = 36.

(iii) -15 - (-2) - 71 - 8 + 6 = -15 + 2 - 71 - 8 + 6

= (2 + 6) - (15 + 71 + 8)

= 8 - 94

= -(94 - 8)

= -86.

Hence, -15 - (-2) - 71 - 8 + 6 = -86.

(i) |-13| = 13 and |-15| = 15.

|-13| - |-15| = 13 - 15

= -(15 - 13)

= -2.

Hence, |-13| - |-15| = -2.

(ii) 35 - 41 = -(41 - 35) = -6, so |35 - 41| = |-6| = 6.

7 - (-2) = 7 + 2 = 9, so |7 - (-2)| = |9| = 9.

|35 - 41| - |7 - (-2)| = 6 - 9

= -(9 - 6)

= -3.

Hence, |35 - 41| - |7 - (-2)| = -3.

Ascending order means arranging from the smallest to the greatest.

The given integers are -39, 35, -102, 0, -51, -5, -6, 7.

The negative integers are -39, -102, -51, -5 and -6. Their absolute values are 39, 102, 51, 5 and 6.

Arranging the absolute values in increasing order: 5 < 6 < 39 < 51 < 102. So the negatives in increasing order are -102 < -51 < -39 < -6 < -5.

Then comes 0.

The positive integers are 35 and 7. In increasing order: 7 < 35.

Combining: -102 < -51 < -39 < -6 < -5 < 0 < 7 < 35.

Hence, the given integers in ascending order are -102, -51, -39, -6, -5, 0, 7, 35.

For an integer a, its successor is a + 1 and its predecessor is a - 1.

Successor of -199 = -199 + 1 = -198.

Predecessor of -199 = -199 - 1 = -200.

Hence, the successor of -199 is -198 and the predecessor of -199 is -200.

Sum of -235 and 137 = -235 + 137

= -(235 - 137)

= -98.

Now subtract -98 from -152:

-152 - (-98) = -152 + 98

= -(152 - 98)

= -54.

Hence, the required result is -54.

Let x be the integer that must be added to -176 to get -95.

Then, -176 + x = -95.

⇒ x = -95 - (-176)

= -95 + 176

= +(176 - 95)

= 81.

Hence, 81 must be added to -176 to get -95.

Heights above sea level are represented by positive integers and depths below sea level are represented by negative integers.

Height of the first point = +270 m.

Height of the second point = -80 m.

Difference in height = 270 - (-80)

= 270 + 80

= 350 m.

Hence, the difference in height between the two points is 350 m.

Let the integer be a.

Successor of a = a + 1.

Predecessor of a = a - 1.

Sum of successor and predecessor = (a + 1) + (a - 1)

= a + a + 1 - 1

= 2a.

Now 2a is always an even integer (since 2 × any integer is even), regardless of the value of a.

So the sum of the successor and predecessor of an integer is always even, and can never be odd.

Hence, the sum of the successor and predecessor of an integer can never be an odd integer.

On the number line, integers which are 5 units away from -8 lie 5 units to the right of -8 and 5 units to the left of -8.

5 units to the right of -8: -8 + 5 = -3.

5 units to the left of -8: -8 - 5 = -13.

So the integers 5 units away from -8 are -3 and -13.

The additive inverse (opposite) of -3 is -(-3) = 3.

The additive inverse (opposite) of -13 is -(-13) = 13.

Hence, the opposites (additive inverses) of the required integers are 3 and 13.

The integers from -500 to 500 are:

-500, -499, -498, ..., -1, 0, 1, ..., 498, 499, 500.

We can pair each negative integer with its additive inverse on the positive side:

(-500) + 500 = 0,

(-499) + 499 = 0,

(-498) + 498 = 0,

. . .

(-1) + 1 = 0.

The integer 0 is left without a pair, but 0 contributes 0 to the sum.

So the total sum = (sum of all the pairs) + 0 = 0 + 0 + ... + 0 + 0 = 0.

Hence, the sum of all integers from -500 to 500 is 0.

(i) The ground floor (G) is taken as 0. Floors above the ground floor are labelled by positive integers and the parking basements (below the ground floor) are labelled by negative integers, according to how far they are from G.

So the labels are:

P3 → -3, P2 → -2, P1 → -1, G → 0, 1 → +1, 2 → +2, 3 → +3, 4 → +4, 5 → +5, 6 → +6, 7 → +7.

Hence, the floors as integers (from bottom to top) are -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7.

(ii) Movement from P3 to the third floor = (Target floor) - (Starting floor)

= 3 - (-3)

= 3 + 3

= 6.

(The positive sign indicates upward movement.)

Hence, the movement from P3 to the third floor is +6, i.e. 6 floors upward.

(iii) Movement from third floor to P3 = (Target floor) - (Starting floor)

= (-3) - 3

= -6.

(The negative sign indicates downward movement.)

Hence, the movement from the third floor to P3 is -6, i.e. 6 floors downward.

(iv) The bottom-most floor is P3 (= -3) and the top-most floor is the 7th floor (= +7).

The maximum upward movement is from P3 to the 7th floor:

Movement = 7 - (-3) = 7 + 3 = 10.

The maximum downward movement is from the 7th floor to P3:

Movement = (-3) - 7 = -10.

In either case, the magnitude of the movement is 10, which is the maximum possible.

Hence, the maximum movement is 10 floors, between P3 and the 7th floor (i.e. either +10 from P3 to the 7th floor or -10 from the 7th floor to P3).