Integers: positive, negative and zero
A tall office tower has six floors above the ground and two parking floors below it. The lift button for the ground floor is marked 0, the upper floors 1, 2, 3 and the parking floors are marked with a minus sign. On a winter morning a thermometer in a hill town also shows a reading below zero. Numbers that go both up and down from zero are everywhere.
The counting numbers, zero, and the opposites of the counting numbers together are called integers: … −3, −2, −1, 0, +1, +2, +3 … Numbers above zero are positive and carry a + sign; numbers below zero are negative and carry a − sign. Zero is neither positive nor negative. Two integers that are the same distance from zero on opposite sides are opposites.
In the figure: Floors above the ground are +, the ground is 0, floors below are −.
Rule 1: Positive, negative and zero — Going up from the starting point is written with +, going down with −. The starting point itself is 0. A lift that rises 4 floors from the ground is at +4; one that sinks 2 floors is at −2.
Integers: positive, negative and zero
In the app you can move this figure with your finger.
Open in the app- Easy: 5 floors above the ground is +5.
- Medium: 3 floors below the ground is −3, and the ground itself is 0.
- Exam-type: A lift at +3 goes down 3 floors and reaches 3 − 3 = 0, the ground floor.
Rule 2: Opposites — The opposite of a positive integer is the negative integer with the same number part, and the other way round. The opposite of 0 is 0 itself.
- Easy: The opposite of +7 is −7.
- Medium: The opposite of −12 is +12.
- Exam-type: A lift at −2 rises 5 floors: it reaches +3, which is 5 steps up from −2.
Zero has no sign. Do not write +0 or −0.
Thinking −9 is bigger than 0 because 9 is a big number; a negative number is always below zero.
Up is +, down is −, the start is 0. Memory clue: “Up plus, down minus, ground zero”.
Spot the difference
| Situation | Integer | Kind |
|---|---|---|
| 4 floors above ground | +4 | positive |
| At the ground floor | 0 | neither |
| 6 floors below ground | −6 | negative |
| Opposite of +8 | −8 | negative |
Practice questions
- 1Write “5 floors below the ground” as an integer.
- 2Is zero positive, negative or neither?
- 3What is the opposite of −9?
- 4A lift is at +2 and goes down 5 floors. Which floor does it reach?
- 5A lift at −4 goes up 7 floors. Which floor does it reach?
- 6At dawn the temperature is 6 degrees below zero. By noon it rises by 9 degrees. What is it at noon?
- 7A lift starts at −3, goes up 8 floors and then down 6 floors. Where does it stop?
- 8How many integers are there from −3 to +3, counting both ends?
Show answers (try them yourself first)
- 11) −5
- 22) Neither
- 33) +9
- 44) −3
- 55) +3
- 66) +3 °C
- 77) −1
- 88) 7
Number line and comparing integers
In a quiz game Meera has +30 points and Arjun has −10 points because he lost some for wrong answers. Everyone can see Meera is ahead. A straight tape with zero in the middle shows the same thing: the person farther to the right is ahead.
A number line has 0 in the middle, positive integers to the right and negative integers to the left, spaced equally. A number on the right is greater than every number on its left. So every positive integer is greater than 0, and 0 is greater than every negative integer.
In the figure: Further right means bigger: −4 < −1 < 0 < +3.
Rule 1: The right-hand number is bigger — To compare two integers, place them on the number line. The one on the right is greater. Between two negative numbers the one nearer to zero is greater, because it lies farther right.
Number line and comparing integers
In the app you can move this figure with your finger.
Open in the app- Easy: +3 > −8, because a positive number is above every negative number.
- Medium: −2 > −7, because −2 is nearer to zero and lies to the right of −7.
- Exam-type: 0 > −1, because zero is to the right of every negative integer.
Rule 2: Ordering and “between” — Ascending order means smallest first, so it runs from left to right on the line. Descending order runs from right to left. Integers between two numbers are the ones that lie strictly between them on the line.
- Easy: Ascending order of −5, +2, −1, 0 is −5, −1, 0, +2.
- Medium: Integers between −3 and +2 are −2, −1, 0, +1: four numbers.
- Exam-type: Of −15, −4 and −9 the greatest is −4 and the smallest is −15.
Both ends are left out when we say “between”: between −3 and +2 does not include −3 or +2.
Saying −9 > −2 because 9 > 2; for negative numbers the bigger number part means the smaller number.
Left is smaller, right is bigger. Memory clue: “Right is bright (bigger)”.
Spot the difference
| Pair | Greater one | Reason |
|---|---|---|
| +4 and −9 | +4 | positive beats negative |
| +6 and +2 | +6 | farther right |
| −3 and −8 | −3 | nearer to zero |
| 0 and −5 | 0 | zero beats negative |
Practice questions
- 1Which is bigger, −6 or +1?
- 2Which is smaller, −8 or −3?
- 3Put the right sign (> or <): −5 ___ −2.
- 4Arrange in ascending order: −7, 3, −1, 0.
- 5How many integers lie between −4 and +3?
- 6Which is the greatest of −12, −2 and −30?
- 7The temperatures of five towns are −4, +6, −11, 0 and −1 degrees. Which is the coldest and which the hottest?
- 8Write the smallest positive integer, the greatest negative integer and zero in ascending order.
Show answers (try them yourself first)
- 11) +1
- 22) −8
- 33) <
- 44) −7, −1, 0, 3
- 55) 6
- 66) −2
- 77) −11 and +6
- 88) −1, 0, +1
Adding integers
A child walks 4 steps forward on a straight path and then 6 steps back. She ends up 2 steps behind where she began. Forward is a positive move and backward is a negative move; adding the two moves tells us where she stands.
To add on the number line, start at the first integer, then move to the right for a positive number and to the left for a negative number. The place where you stop is the sum. A pair of +1 and −1 is a zero pair: together they make 0.
In the figure: (−2) + (+5) = +3 on the line, and (+5) + (−3) = +2 with tokens: three zero pairs cancel.
Rule 1: Same signs: add and keep the sign — When both integers are positive, or both are negative, add the number parts and keep the common sign. Two moves in the same direction add up to a longer move in that direction.
- Easy: (+4) + (+3) = +7.
- Medium: (−5) + (−2) = −7: 5 + 2 = 7, and both are negative.
- Exam-type: (−12) + (−9) = −21: a diver at 12 m below the surface sinks 9 m more.
Rule 2: Different signs: move back or use zero pairs — When the signs differ, the moves pull in opposite directions. Subtract the smaller number part from the bigger and keep the sign of the integer with the bigger number part. Equal opposite moves cancel to 0.
Adding integers
In the app you can move this figure with your finger.
Open in the app- Easy: (+9) + (−4) = +5: 9 − 4 = 5, and the bigger part is positive.
- Medium: (−8) + (+3) = −5: 8 − 3 = 5, and the bigger part is negative.
- Exam-type: (−15) + (+15) = 0, and (−6) + (+10) = +4 because 6 zero pairs cancel and 4 positives remain.
Adding 0 changes nothing, and an integer plus its opposite is always 0.
Writing (−5) + (−3) = −2; two negatives pile up, so the answer is −8.
Same signs add up; different signs fight and the bigger one wins. Memory clue: “Same – add; different – fight”.
Spot the difference
| Case | Example | Answer |
|---|---|---|
| Both positive | (+3) + (+6) | +9 |
| Both negative | (−3) + (−6) | −9 |
| Bigger part positive | (+8) + (−5) | +3 |
| Bigger part negative | (−8) + (+5) | −3 |
Practice questions
- 1Find (+6) + (+7).
- 2Find (−4) + (−9).
- 3Find (+10) + (−3).
- 4Find (−12) + (+5).
- 5Find (−8) + (+8).
- 6A diver is at −14 m and rises 6 m. Where is she now?
- 7Find (−7) + (+12) + (−9).
- 8A lift starts at +2, goes down 6 floors and then up 3 floors. At which floor does it stop?
Show answers (try them yourself first)
- 11) +13
- 22) −13
- 33) +7
- 44) −7
- 55) 0
- 66) −8 m
- 77) −4
- 88) −1
Subtracting integers
A hill-town thermometer reads −2 degrees in the early morning and +5 degrees at noon. How far did it climb? A lift goes from floor −3 to floor +2: how many floors did it travel, and in which direction? Both questions ask for the move that takes us from a starting place to a target.
Subtraction finds the movement needed to go from the starting integer to the target integer: target − start. A very useful fact: subtracting an integer gives the same answer as adding its opposite.
In the figure: From −3 to +2 the move is +5, so (+2) − (−3) = +5, the same as (+2) + (+3).
Rule 1: Subtraction is a movement: target − start — Mark the start and the target on the number line. Count the steps from start to target: to the right is positive, to the left is negative. That count is the answer.
Subtracting integers
In the app you can move this figure with your finger.
Open in the app- Easy: (+7) − (+4) = +3: from 4 to 7 is 3 steps right.
- Medium: (−2) − (+5) = −7: from 5 to −2 is 7 steps left.
- Exam-type: (+2) − (−3) = +5: the lift goes from −3 to +2, 5 floors up.
Rule 2: Subtract = add the opposite — Change the subtraction sign into an addition sign and replace the second integer by its opposite. Then add as usual. Subtracting a negative number is the same as adding the positive number.
- Easy: (+8) − (+3) = (+8) + (−3) = +5.
- Medium: (+5) − (−4) = (+5) + (+4) = +9.
- Exam-type: (−7) − (−10) = (−7) + (+10) = +3.
Subtracting a negative number makes the answer bigger, not smaller.
Writing (+5) − (−4) = +1; the second number must first be turned into its opposite, so the answer is +9.
Flip the second sign, then add. Memory clue: “Subtract: flip and add”.
Spot the difference
| Subtraction | Same as | Answer |
|---|---|---|
| (+6) − (+2) | (+6) + (−2) | +4 |
| (+6) − (−2) | (+6) + (+2) | +8 |
| (−6) − (+2) | (−6) + (−2) | −8 |
| (−6) − (−2) | (−6) + (+2) | −4 |
Practice questions
- 1Find (+9) − (+4).
- 2Find (+3) − (+8).
- 3Find (+5) − (−2).
- 4Find (−6) − (+3).
- 5Find (−4) − (−9).
- 6A lift moves from floor −3 to floor +4. How far and in which direction does it go?
- 7The temperature was +4 degrees at noon and −6 degrees at midnight. What is the change in temperature?
- 8Find (+2) − (−6) − (+10).
Show answers (try them yourself first)
- 11) +5
- 22) −5
- 33) +7
- 44) −9
- 55) +5
- 66) 7 floors up
- 77) −10 (a fall of 10 degrees)
- 88) −2
Integers in real life
A shopkeeper’s passbook shows money put in and money taken out. A mountain camp is high above sea level while a diver’s boat looks down at the sea floor far below it. A weather report gives temperatures above and below the freezing mark. Wherever there are two opposite directions and a natural zero, integers help.
First decide what is zero and which direction is positive. For a bank balance, money in is positive and money owed is negative. For height, above sea level is positive and below is negative. For temperature, above 0 °C is positive and below is negative. Then add the changes to the starting value.
In the figure: Height is measured from sea level and temperature from 0 °C; both go above and below zero.
Rule 1: Choose zero and sign, then add the change — A rise, a deposit or a gain is positive; a fall, a withdrawal or a loss is negative. New value = starting value + change, using the rules for adding integers.
Integers in real life
In the app you can move this figure with your finger.
Open in the app- Easy: Temperature −4 degrees rises by 10: (−4) + (+10) = +6 degrees.
- Medium: Balance +120 and ₹80 spent: (+120) + (−80) = +40, so ₹40 is left.
- Exam-type: Balance +120 and ₹300 spent: (+120) + (−300) = −180, so the account owes ₹180.
Rule 2: Difference between two levels — To find how far apart two readings are, subtract the lower from the higher. Subtracting a negative reading adds its number part, so the gap across zero is the sum of both parts.
- Easy: 40 m and 10 m above sea level differ by 40 − 10 = 30 m.
- Medium: A bird at +20 m and a submarine at −30 m are (+20) − (−30) = 50 m apart.
- Exam-type: A day’s highest temperature is +7 and lowest is −5: the range is (+7) − (−5) = 12 degrees.
Always say what is zero and which way is positive before using a sign.
Reading −50 m as 50 m above sea level; the minus sign means below sea level.
Decide zero and plus direction first, then add. Memory clue: “Zero first, then plus or minus”.
Spot the difference
| Situation | Positive | Negative |
|---|---|---|
| Bank balance | money in | money owed |
| Height | above sea level | below sea level |
| Temperature | above 0 °C | below 0 °C |
| Lift | above ground floor | below ground floor |
Practice questions
- 1A shopkeeper takes ₹75 out of the account. Write the change as an integer.
- 2The temperature is +3 degrees and falls by 8. What is it now?
- 3A submarine is 40 m below sea level. Write its level with a sign.
- 4An account has +500. ₹200 is spent and ₹50 is deposited. What is the balance now?
- 5An account shows −60 (it owes ₹60). ₹100 is deposited. What is the balance?
- 6A mountain camp is at +120 m and a lake is at −15 m. How far apart are they in height?
- 7At night the temperature is −9 degrees and at noon it is +6 degrees. By how many degrees did it rise?
- 8An account starts at +90. ₹140 is spent, ₹70 is deposited and ₹30 is spent. What is the final balance?
Show answers (try them yourself first)
- 11) −75
- 22) −5
- 33) −40 m
- 44) +350
- 55) +40
- 66) 135 m
- 77) 15 degrees
- 88) −10