Negative Numbers & Integers

(i) The negative (opposite) of any integer is obtained by changing its sign. Negative of -20 = -(-20) = 20.

Hence, the negative of -20 is 20.

(ii) Zero is neither positive nor negative, so the negative of 0 = -(0) = 0.

Hence, the negative of 0 is 0.

(iii) Negative of 8 = -(8) = -8.

Hence, the negative of 8 is -8.

(iv) Positive integers represent gain and negative integers represent loss. Since 10 represents a gain of ₹ 10, -10 represents the opposite of a gain, i.e. a loss.

Hence, -10 represents a loss of ₹ 10.

(v) South and North are opposite directions. If going South is negative, then going North (its opposite) is positive.

Hence, going North is positive.

(vi) The greater an integer, the lesser is its negative (opposite). Since 5 < 7, on changing the signs of both sides the inequality reverses, giving -5 > -7.

Hence, -5 > -7.

(vii) On a number line, the greater integer always lies to the right of the smaller integer. Since 3 > -2, integer 3 lies to the right of -2.

Hence, 3 is on the right side of -2.

(viii) On a number line, the smaller integer always lies to the left of the greater integer. Since -8 < -6, integer -8 lies to the left of -6.

Hence, -8 is on the left side of -6.

On a number line, integers increase in value from left to right. So, ascending (increasing) order means reading the marked integers from left to right.

(i) Mark all the given integers on a number line:

Reading the integers from left to right:

Hence, the given integers in ascending order are -14 < -9 < -5 < 0 < 4 < 8 < 12.

(ii) Mark all the given integers on a number line:

Reading the integers from left to right:

Hence, the given integers in ascending order are -9 < -6 < 0 < 5 < 7 < 9.

On a number line, integers decrease in value from right to left. So, descending (decreasing) order means reading the marked integers from right to left.

(i) Mark all the given integers on a number line:

Reading the integers from right to left:

Hence, the given integers in descending order are 12 > 11 > 5 > 3 > 0 > -1 > -4 > -10.

(ii) Mark all the given integers on a number line:

Reading the integers from right to left:

Hence, the given integers in descending order are 6 > 3 > -4 > -7 > -8 > -12.

On a number line, the integer on the right is greater and the integer on the left is smaller.

(i) Every integer that lies to the right is greater than all the numbers on its left.

Hence, an integer, on the given number line, is greater than every number on its left.

(ii) Every integer is greater than the numbers lying to its left.

Hence, an integer on the given number line is greater than every number to its left.

(iii) Since the greater integer lies to the right, 2 > -4 means 2 lies to the right of -4.

Hence, 2 is greater than -4 implies 2 is to the right of -4.

(iv) Comparing -3 and 2: -3 is negative and 2 is positive, so -3 < 2, i.e. -3 is less than 2. Comparing 3 and -2: 3 is positive and -2 is negative, so 3 > -2, i.e. 3 is greater than -2.

Hence, -3 is less than 2 and 3 is greater than -2.

(v) Comparing -4 and -8: |-4| = 4 and |-8| = 8. Since 4 < 8, we get -4 > -8, i.e. -4 is greater than -8. Comparing 4 and 8: 4 < 8, i.e. 4 is less than 8.

Hence, -4 is greater than -8 and 4 is less than 8.

(vi) Comparing 5 and 2: 5 > 2, i.e. 5 is greater than 2. Comparing -5 and -2: |-5| = 5 and |-2| = 2. Since 5 > 2, we get -5 < -2, i.e. -5 is less than -2.

Hence, 5 is greater than 2 and -5 is less than -2.

(vii) Comparing -6 and 3: -6 is negative and 3 is positive, so -6 < 3, i.e. -6 is less than 3. The opposite of -6 is 6 and the opposite of 3 is -3. Comparing 6 and -3: 6 > -3, i.e. 6 is greater than -3.

Hence, -6 is less than 3 and the opposite of -6 is greater than the opposite of 3.

(viii) Comparing 8 and -5: 8 is positive and -5 is negative, so 8 > -5, i.e. 8 is greater than -5. Comparing -8 and 5: -8 is negative and 5 is positive, so -8 < 5, i.e. -8 is less than 5.

Hence, 8 is greater than -5 and -8 is less than 5.

We use the rules of comparison: every positive integer is greater than 0 and greater than every negative integer; zero is greater than every negative integer; and for two negative integers, the one with the smaller absolute value is greater.

(i) Both -15 and -23 are negative. |-15| = 15 and |-23| = 23. Since 15 < 23, the integer with the smaller absolute value is greater.

Hence, -15 is greater.

(ii) -12 is negative and 15 is positive. Every positive integer is greater than every negative integer.

Hence, 15 is greater.

(iii) 8 is a positive integer and every positive integer is greater than 0.

Hence, 8 is greater.

(iv) -3 is a negative integer and zero is greater than every negative integer.

Hence, 0 is greater.

We use the rules of comparison: every negative integer is smaller than 0 and smaller than every positive integer; for two negative integers, the one with the greater absolute value is smaller.

(i) -6 is a negative integer and every negative integer is smaller than 0.

Hence, -6 is smaller.

(ii) -3 is negative and 2 is positive. Every negative integer is smaller than every positive integer.

Hence, -3 is smaller.

(iii) -51 is negative and 15 is positive. Every negative integer is smaller than every positive integer.

Hence, -51 is smaller.

(iv) 0 is smaller than every positive integer, and 13 is positive.

Hence, 0 is smaller.

(i) 3 is a positive integer and every positive integer is greater than 0.

Hence, 3 > 0.

(ii) Zero is greater than every negative integer, and -8 is negative.

Hence, 0 > -8.

(iii) Both -9 and -3 are negative. |-9| = 9 and |-3| = 3. Since 9 > 3, the integer with the greater absolute value (-9) is smaller.

Hence, -9 < -3.

(iv) -3 is negative and 3 is positive. Every negative integer is smaller than every positive integer.

Hence, -3 < 3.

(v) 5 is positive and -1 is negative. Every positive integer is greater than every negative integer.

Hence, 5 > -1.

(vi) -13 is a negative integer and every negative integer is smaller than 0.

Hence, -13 < 0.

(vii) Both -8 and -18 are negative. |-8| = 8 and |-18| = 18. Since 8 < 18, the integer with the smaller absolute value (-8) is greater.

Hence, -8 > -18.

On a number line, integers increase in value from left to right. So, ascending order means reading the marked integers from left to right.

(i) Mark all the given integers on a number line:

Reading the integers from left to right: -8 < -5 < -1 < 0 < 4 < 5.

Hence, the given integers in ascending order are -8, -5, -1, 0, 4, 5.

(ii) Mark all the given integers on a number line:

Reading the integers from left to right: -7 < -6 < -3 < 0 < 2 < 3 < 4.

Hence, the given integers in ascending order are -7, -6, -3, 0, 2, 3, 4.

On a number line, integers decrease in value from right to left. So, descending order means reading the marked integers from right to left.

(i) Mark all the given integers on a number line:

Reading the integers from right to left: 15 > 8 > 0 > -2 > -3 > -5.

Hence, the given integers in descending order are 15, 8, 0, -2, -3, -5.

(ii) Mark all the given integers on a number line:

Reading the integers from right to left: 23 > 12 > 7 > 6 > 0 > -11.

Hence, the given integers in descending order are 23, 12, 7, 6, 0, -11.

(i) The negative integers -1, -2, -3, -4, ... continue indefinitely, so there is no smallest integer. Hence 0 is not the smallest integer.

Hence, the statement is False.

(ii) The opposite of -17 = -(-17) = 17.

Hence, the statement is True.

(iii) The opposite of 0 = -(0) = 0.

Hence, the statement is True.

(iv) Every negative integer lies to the left of 0 on the number line, so it is smaller than 0.

Hence, the statement is True.

(v) Every positive integer is greater than 0, so 0 is smaller (not greater) than every positive integer.

Hence, the statement is False.

(vi) Integers are ..., -3, -2, -1, 0, 1, 2, 3, ... . Zero is included in this collection, so 0 is an integer (a neutral integer).

Hence, the statement is False.

For addition on a number line, the right side of zero is for positive integers and the left side of zero is for negative integers. We first mark the first integer by counting from zero, and then, for the second integer, we move from the position of the first integer.

(i) (+7) + (+4)

For +7, count 7 units to the right of zero. Then, for +4, move 4 units to the right of +7. You reach +11.

Hence, (+7) + (+4) = +11.

(ii) 0 + (+6)

Start at zero. Then, for +6, move 6 units to the right of 0. You reach +6.

Hence, 0 + (+6) = +6.

(iii) (+5) + 0

For +5, count 5 units to the right of zero. Then, adding 0 means there is no further movement, so you stay at +5.

Hence, (+5) + 0 = +5.

For addition on a number line, we move to the right of zero for a positive integer and to the left of zero for a negative integer. We mark the first integer by counting from zero, and then, for the second integer, we move from the position of the first integer.

(i) (-4) + (+5)

For -4, move 4 units to the left of zero. Then, for +5, move 5 units to the right of -4. You reach +1.

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

(ii) 0 + (-2)

Start at zero. Then, for -2, move 2 units to the left of 0. You reach -2.

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

(iii) (-1) + (+4)

For -1, move 1 unit to the left of zero. Then, for +4, move 4 units to the right of -1. You reach +3.

Hence, (-1) + (+4) = +3.

For addition on a number line, we move to the right of zero for a positive integer and to the left of zero for a negative integer. We mark the first integer by counting from zero, and then, for the second integer, we move from the position of the first integer.

(i) (+4) + (-2)

For +4, count 4 units to the right of zero. Then, for -2, move 2 units to the left of +4. You reach +2.

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

(ii) (+3) + (-6)

For +3, count 3 units to the right of zero. Then, for -6, move 6 units to the left of +3. You reach -3.

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

(iii) 3 + (-7)

For +3, count 3 units to the right of zero. Then, for -7, move 7 units to the left of +3. You reach -4.

Hence, 3 + (-7) = -4.

For addition on a number line, we move to the left of zero for a negative integer. When both integers are negative, both moves are to the left. We mark the first integer by counting from zero, and then, for the second integer, we move from the position of the first integer.

(i) (-1) + (-2)

For -1, start from zero and move 1 unit to the left. Then, for -2, move 2 units to the left of -1. You reach -3.

Hence, (-1) + (-2) = -3.

(ii) (-3) + (-4)

For -3, start from zero and move 3 units to the left. Then, for -4, move 4 units to the left of -3. You reach -7.

Hence, (-3) + (-4) = -7.

(iii) (-2) + (-5)

For -2, start from zero and move 2 units to the left. Then, for -5, move 5 units to the left of -2. You reach -7.

Hence, (-2) + (-5) = -7.

To subtract using a number line, mark the positions of both the given numbers on the same number line. Then, starting from the position of the number to be subtracted, count the number of steps and the direction needed to reach the first number. If the steps are towards the right, the answer is positive; if towards the left, the answer is negative.

(i) (+10) - (+2)

Mark the positions of +10 and +2 on the same number line. Starting from +2, count the steps needed to reach +10. We find that 8 steps are needed towards the right.

Hence, (+10) - (+2) = +8.

(ii) (+8) - (-5)

Mark the positions of +8 and -5 on the same number line. Starting from -5, count the steps needed to reach +8. We find that 13 steps are needed towards the right.

Hence, (+8) - (-5) = +13.

(iii) (-6) - (+2)

Mark the positions of -6 and +2 on the same number line. Starting from +2, count the steps needed to reach -6. We find that 8 steps are needed towards the left.

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

(iv) (-7) - (+5)

Mark the positions of -7 and +5 on the same number line. Starting from +5, count the steps needed to reach -7. We find that 12 steps are needed towards the left.

Hence, (-7) - (+5) = -12.

(v) (+4) - (-2)

Mark the positions of +4 and -2 on the same number line. Starting from -2, count the steps needed to reach +4. We find that 6 steps are needed towards the right.

Hence, (+4) - (-2) = +6.

(vi) (-8) - (-4)

Mark the positions of -8 and -4 on the same number line. Starting from -4, count the steps needed to reach -8. We find that 4 steps are needed towards the left.

Hence, (-8) - (-4) = -4.

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 3 units to the right of -1. We reach 2.

Hence, the integer which is 3 more than -1 is 2.

(ii) Start at 2 and move 5 units to the left of 2. We reach -3.

Hence, the integer which is 5 less than 2 is -3.

(iii) Start at -9 and move 5 units to the right of -9. We reach -4.

Hence, the integer which is 5 more than -9 is -4.

(iv) Start at -4 and move 4 units to the left of -4. We reach -8.

Hence, the integer which is 4 less than -4 is -8.

(v) Start at 0 and move 7 units to the right of 0. We reach 7.

Hence, the integer which is 7 more than 0 is 7.

(vi) Start at -8 and move 7 units to the left of -8. We reach -15.

Hence, the integer which is 7 less than -8 is -15.

For addition on a number line, we move to the right of zero for a positive integer and to the left of zero for a negative integer. We mark the first integer by counting from zero, and then, for the second integer, we move from the position of the first integer.

(i) 13 and 15

For +13, count 13 units to the right of zero. Then, for +15, move 15 units to the right of +13. You reach +28.

Hence, 13 + 15 = 28.

(ii) -13 and 15

For -13, move 13 units to the left of zero. Then, for +15, move 15 units to the right of -13. You reach +2.

Hence, -13 + 15 = 2.

(iii) 13 and -15

For +13, count 13 units to the right of zero. Then, for -15, move 15 units to the left of +13. You reach -2.

Hence, 13 + (-15) = -2.

(iv) -13 and -15

For -13, move 13 units to the left of zero. Then, for -15, move 15 units to the left of -13. You reach -28.

Hence, -13 + (-15) = -28.

To subtract using a number line, mark the positions of both the given numbers on the same number line. Then, starting from the position of the number to be subtracted, count the number of steps and the direction needed to reach the first number. If the steps are towards the right, the answer is positive; if towards the left, the answer is negative.

(i) 8 - 5

Mark the positions of 8 and 5 on the same number line. Starting from 5, count the steps needed to reach 8. We find that 3 steps are needed towards the right.

Hence, 5 subtracted from 8 gives 3.

(ii) 8 - (-5)

Mark the positions of 8 and -5 on the same number line. Starting from -5, count the steps needed to reach 8. We find that 13 steps are needed towards the right.

Hence, -5 subtracted from 8 gives 13.

(iii) -7 - 4

Mark the positions of -7 and 4 on the same number line. Starting from 4, count the steps needed to reach -7. We find that 11 steps are needed towards the left.

Hence, 4 subtracted from -7 gives -11.

(iv) -2 - (-8)

Mark the positions of -2 and -8 on the same number line. Starting from -8, count the steps needed to reach -2. We find that 6 steps are needed towards the right.

Hence, -8 subtracted from -2 gives 6.

(v) 12 - (-3)

Mark the positions of 12 and -3 on the same number line. Starting from -3, count the steps needed to reach 12. We find that 15 steps are needed towards the right.

Hence, -3 subtracted from 12 gives 15.

(vi) -3 - (-6)

Mark the positions of -3 and -6 on the same number line. Starting from -6, count the steps needed to reach -3. We find that 3 steps are needed towards the right.

Hence, -6 subtracted from -3 gives 3.

On the given number line, the point A lies 5 units to the right of 0, which fixes the scale of the number line. The point B lies 3 units to the left of 0, so B = -3.

Hence, option 2 is the correct option.

Given, m and n are two negative integers such that m ≥ n.

If m > n, then m - n > 0.

If m = n, then m - n = 0.

Therefore, in both cases, m - n ≥ 0.

So, option 2 is true.

Checking the other options:

m = n need not always be true, because m may be greater than n.

Also, since both m and n are negative integers, m + n is always negative, so m + n ≥ 0 is false.

Hence, option 2 is the correct option.

For every integer P, we have P + (-P) = 0, where -P is the negative (opposite) of P. This holds whether P is positive or negative.

Hence, option 3 is the correct option.

On the given number line, the movement starts at 3 and goes 5 units to the left, ending at -2. This represents 3 - 5.

Hence, option 4 is the correct option.

On the given number line, the movement starts at -1 and goes 3 units to the right, ending at 2. This represents -1 + 3.

Hence, option 4 is the correct option.

On the given number line, the movement starts at 4 and goes 5 units to the left, ending at -1. This represents 4 - 5.

Hence, option 4 is the correct option.

The integers strictly between 5 and 15 are 6, 7, 8, 9, 10, 11, 12, 13 and 14.

Counting them gives 9 integers.

Hence, option 2 is the correct option.

A positive integer represents a gain and a negative integer represents a loss. The integer 0 is neither positive nor negative, so it represents a situation of neither gain nor loss.

Hence, option 2 is the correct option.

For any two integers marked on a number line, the one on the right side of the other is greater and the one on the left side of the other is smaller.

So, if m lies on the left side of n, then m < n. This makes Statement 1 true.

Statement 2 states the same general rule, that the integer on the left of the other is smaller, which is also true.

Thus, both the statements are true.

Hence, option 1 is the correct option.

First, let us check the inequalities given in the Assertion.

-8 < -5 (since |-8| > |-5|), 8 > -5 (positive > negative), -8 < 5 (negative < positive) and 8 > 5. All four are true.

Now changing the signs of both sides of each inequality reverses the sign of inequality:

-8 < -5 becomes 8 > 5; 8 > -5 becomes -8 < 5; -8 < 5 becomes 8 > -5; and 8 > 5 becomes -8 < -5.

These are exactly the four results stated after the ⇒ in the Assertion, so Assertion A is true.

The Reason states that on changing the signs of both sides of an inequation, the sign of inequality reverses. This is a correct property of inequalities (for example, 5 < 7 ⇒ -5 > -7), so Reason R is also true and it correctly explains the Assertion.

Thus, both A and R are true.

Hence, option 3 is the correct option.