Fill in the blanks:
(i) Negative of -20 is ...............
(ii) Negative of 0 is ...............
(iii) Negative of 8 is ...............
(iv) If 10 represents gain of ₹ 10, then -10 represents ...............
(v) If going South is negative, then going North is ...............
(vi) Because 5 < 7, therefore -5 ............... -7.
(vii) If 3 > -2, then 3 is on the ............... of -2.
(viii) If -8 < -6, then -8 is on the ............... of -6.
Answer
(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.
Use a number line to write the following integers in ascending (increasing) order:
(i) -5, 8, 0, -9, 4, -14 and 12
(ii) -6, 7, 0, -9, 5 and 9
Answer
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.
Use a number line to write the following integers in descending (decreasing) order:
(i) -10, 0, 3, -4, 12, 11, -1 and 5
(ii) -4, 3, -8, -12, -7 and 6
Answer
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.
Fill in the blanks, using the following number line:
(i) An integer, on the given number line, is .................... than every number on its left.
(ii) An integer on the given number line is greater than every number to its ...............
(iii) 2 is greater than -4 implies 2 is to the ........................ of -4.
(iv) -3 is .......... than 2 and 3 is ...... than -2.
(v) -4 is .......... than -8 and 4 is ...... than 8.
(vi) 5 is ........... than 2 and -5 is ............ than -2.
(vii) -6 is ................... than 3 and the opposite of -6 is ............ than opposite of 3.
(viii) 8 is ................. than -5 and -8 is ......... than 5.
Answer
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.
In each of the following pairs, state which integer is greater:
(i) -15, -23
(ii) -12, 15
(iii) 0, 8
(iv) 0, -3
Answer
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.
In each of the following pairs, state which integer is smaller:
(i) 0, -6
(ii) 2, -3
(iii) 15, -51
(iv) 13, 0
Answer
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.
In each of the following pairs, replace * with < or > to make the statement true:
(i) 3 * 0
(ii) 0 * -8
(iii) -9 * -3
(iv) -3 * 3
(v) 5 * -1
(vi) -13 * 0
(vii) -8 * -18
Answer
(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.
In each case, arrange the given integers in ascending order, using a number line:
(i) -8, 0, -5, 5, 4, -1
(ii) 3, -3, 4, -7, 0, -6, 2
Answer
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.
In each case, arrange the given integers in descending order, using a number line:
(i) -5, -3, 8, 15, 0, -2
(ii) 12, 23, -11, 0, 7, 6
Answer
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.
For each of the statements given below, state whether it is true or false:
(i) The smallest integer is 0.
(ii) The opposite of -17 is 17.
(iii) The opposite of zero is zero.
(iv) Every negative integer is smaller than 0.
(v) 0 is greater than every positive integer.
(vi) Since zero is neither negative nor positive, it is not an integer.
Answer
(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.