Integers — Negative Numbers, Number Line & Operations
Have you ever seen the temperature outside drop to -3°C in winter? Or heard someone say “we’re in debt by ₹500”? These situations need numbers smaller than zero — and that’s exactly where integers come in.
We use integers every single day, even without realising it. Let’s understand them properly.
What Are Integers?
We already know the natural numbers: 1, 2, 3, 4, 5 …
We also know whole numbers, which include zero: 0, 1, 2, 3, 4, 5 …
But what about numbers less than zero? Those are negative numbers: -1, -2, -3, -4, -5 …
When we put all of these together — negative numbers, zero, and positive numbers — we get the set of integers.
Integers = …, -4, -3, -2, -1, 0, 1, 2, 3, 4, …
The ”…” means it goes on forever in both directions.
Zero is an integer, but it is neither positive nor negative. It sits right in the middle!
Real Life Situations That Need Integers
Integers aren’t just abstract — they describe the real world.
Temperature: Weather forecasters use -5°C to mean 5 degrees below zero. If Delhi is at 2°C and Shimla is at -3°C, Shimla is colder.
Money (credit and debt): If you have ₹200 in your piggy bank, that’s +200. If you borrowed ₹150 from a friend and haven’t returned it yet, we can say you have -150 (a deficit).
Floors of a building: Many malls have basement levels — B1, B2. If ground floor is 0, then B1 is floor -1, B2 is floor -2.
Sea level: The height of places is measured from sea level. A place 100 m above sea level is +100. A valley 50 m below sea level is -50.
The Number Line
The best way to picture integers is on a number line.
Draw a straight horizontal line. Mark 0 in the centre.
- Numbers to the right of 0 are positive: 1, 2, 3, 4, …
- Numbers to the left of 0 are negative: -1, -2, -3, -4, …
← -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 →A few important facts from the number line:
- The further right a number is, the greater it is. So 3 > 1, and -1 > -4.
- Every negative number is less than every positive number.
- Zero is greater than all negative numbers.
On a number line, moving right means increasing. Moving left means decreasing. Use this to compare integers without confusion.
Adding Integers
Adding integers follows two simple rules depending on the signs.
Rule 1: Same Signs — Add the values, keep the sign
If both integers are positive: just add normally.
(+4) + (+3) = +7
If both integers are negative: add their values, then put a negative sign.
(-4) + (-3) = -(4+3) = -7
Think of it this way: if you’re already ₹4 in debt and you borrow another ₹3, you now owe ₹7. The debt grew.
Rule 2: Different Signs — Subtract the smaller from the larger, keep the sign of the larger
(-7) + (+4) = ?
Ignore the signs for a moment. We have 7 and 4. Subtract: 7 - 4 = 3. Now, which number had the larger value? -7 (the value is 7). So the answer gets a negative sign.
(-7) + (+4) = -3
Another example:
(+9) + (-5) = ? 9 - 5 = 4. Larger value is 9, which is positive. Answer: +4
Number line method: Start at the first number. If adding a positive number, move right. If adding a negative number, move left.
(-3) + (-5): Start at -3, move 5 steps left → land on -8.
- Same signs → Add values, keep the sign
- Different signs → Subtract smaller from larger, take sign of the one with larger value
Subtracting Integers
Here’s the golden rule for subtraction:
Subtracting an integer = Adding its opposite
This means we never actually subtract. We just convert the subtraction into addition.
How to do it: Change the subtraction sign to addition, then change the sign of the number being subtracted.
Examples:
5 - (-3) = 5 + (+3) = 8
(Subtracting -3 is the same as adding +3. When someone removes a debt of ₹3, you’re ₹3 richer!)
(-4) - (+6) = (-4) + (-6) = -10
(-2) - (-5) = (-2) + (+5) = +3
Always rewrite subtraction as addition before solving. It makes life so much easier, and you’ll make far fewer mistakes.
Multiplying Integers
Multiplication has clear sign rules. Let’s build them step by step.
We know: 3 × 4 = 12 (positive × positive = positive)
Now, what is 3 × (-4)?
Think of it as repeated addition:
3 × (-4) = (-4) + (-4) + (-4) = -12
So positive × negative = negative.
By the same logic: negative × positive = negative.
Now, the tricky one: what is (-3) × (-4)?
Let’s look at a pattern:
| Calculation | Result |
|---|---|
| 3 × 4 | 12 |
| 2 × 4 | 8 |
| 1 × 4 | 4 |
| 0 × 4 | 0 |
| (-1) × 4 | -4 |
| (-2) × 4 | -8 |
| (-3) × 4 | -12 |
Each time we go down one row, the result decreases by 4.
Now look at the pattern for multiplying by (-4):
| Calculation | Result |
|---|---|
| 3 × (-4) | -12 |
| 2 × (-4) | -8 |
| 1 × (-4) | -4 |
| 0 × (-4) | 0 |
| (-1) × (-4) | ? |
The results are increasing by 4 each time: -12, -8, -4, 0, … so next must be +4.
That gives us: (-1) × (-4) = +4
Following the same pattern: (-2) × (-4) = +8, and (-3) × (-4) = +12.
Negative × Negative = Positive. This is not arbitrary — it follows logically from the pattern.
- (+) × (+) = (+)
- (+) × (−) = (−)
- (−) × (+) = (−)
- (−) × (−) = (+)
Dividing Integers
Division follows the exact same sign rules as multiplication.
- (+) ÷ (+) = (+)
- (+) ÷ (−) = (−)
- (−) ÷ (+) = (−)
- (−) ÷ (−) = (+)
Examples:
(+12) ÷ (+3) = +4 (+12) ÷ (-3) = -4 (-12) ÷ (+3) = -4 (-12) ÷ (-3) = +4
The same-sign result is always positive. Different-sign result is always negative.
Properties of Integers
Closure Property
When we add or multiply any two integers, the result is always an integer.
- (-3) + 5 = 2 ✓ (integer)
- (-4) × 3 = -12 ✓ (integer)
Commutative Property
Order doesn’t matter for addition and multiplication.
- (-3) + 5 = 5 + (-3) = 2
- (-4) × 3 = 3 × (-4) = -12
Associative Property
Grouping doesn’t matter for addition and multiplication.
- [(-2) + 3] + (-5) = (-2) + [3 + (-5)] = -4
Distributive Property
- a × (b + c) = a×b + a×c
- (-3) × (4 + (-2)) = (-3)×4 + (-3)×(-2) = -12 + 6 = -6
Additive Identity
Adding 0 to any integer gives the same integer.
- (-7) + 0 = -7
Multiplicative Identity
Multiplying any integer by 1 gives the same integer.
- (-7) × 1 = -7
Additive Inverse
Every integer has an opposite. Their sum is zero.
- 5 + (-5) = 0
- (-8) + 8 = 0
The additive inverse of -3 is +3, and the additive inverse of +3 is -3. They cancel each other out completely.
5 Common Mistakes to Avoid
Mistake 1: Forgetting that subtracting a negative = adding a positive
Wrong: 5 - (-3) = 5 - 3 = 2 Right: 5 - (-3) = 5 + 3 = 8
Mistake 2: Wrong sign when adding integers with different signs
Wrong: (-7) + 4 = -11 (accidentally adding instead of subtracting) Right: (-7) + 4 = -3 (subtract 7 - 4 = 3, keep negative sign since |-7| > |4|)
Mistake 3: Thinking negative × negative = negative
Many students write (-4) × (-3) = -12. But the correct answer is +12. Same signs → positive result.
Mistake 4: Confusing -3² and (-3)²
-3² = -(3×3) = -9 (-3)² = (-3) × (-3) = +9
The bracket makes a huge difference!
Mistake 5: Placing integers incorrectly on the number line
Remember: -8 is to the LEFT of -3. So -8 < -3 (not greater!). The more negative, the smaller the number.
Practice Questions
Question 1: Evaluate: (-6) + (-9)
Both signs are negative, so we add the values and keep the negative sign. 6 + 9 = 15 Answer: -15
Question 2: Evaluate: (-12) + 7
Different signs. Subtract: 12 - 7 = 5. Larger value is 12, which is negative. So the answer is negative. Answer: -5
Question 3: Evaluate: 3 - (-8)
Subtracting -8 is the same as adding +8. 3 - (-8) = 3 + 8 = 11
Question 4: Evaluate: (-5) × (-7)
Negative × Negative = Positive. 5 × 7 = 35. Answer: +35
Question 5: Evaluate: (-48) ÷ (+6)
Negative ÷ Positive = Negative. 48 ÷ 6 = 8. Answer: -8
Question 6: The temperature in Shimla on Monday was -2°C. By Wednesday it dropped by 5 more degrees. What was Wednesday’s temperature?
Starting temperature: -2°C Drop of 5 degrees means we subtract 5. (-2) - 5 = (-2) + (-5) = -7°C
Wednesday’s temperature was -7°C.
Question 7: Which is greater: -13 or -9?
On the number line, -9 is to the right of -13. So -9 > -13.
-9 is greater.
Remember: with negative numbers, the one closer to zero is always greater.
Question 8: If a submarine goes 200 m below sea level (represented as -200), and then rises 75 m, at what depth is it now?
Start at -200. Rising 75 m means adding +75. (-200) + 75 = -125.
The submarine is now 125 m below sea level, i.e., at -125 m.
Frequently Asked Questions
Q1: Is zero a positive or negative integer?
Zero is neither positive nor negative. It is the boundary between positive and negative integers. It belongs to the set of integers, but has no sign.
Q2: Why is negative × negative positive? It feels wrong!
It’s natural to feel this way! The best way to think about it: if “negative” means “opposite”, then the opposite of an opposite takes you back to the original. Removing a debt is a gain. The pattern proof we saw in the multiplication section is the clearest mathematical reason.
Q3: What is the additive inverse of 0?
The additive inverse of 0 is 0 itself, because 0 + 0 = 0.
Q4: Can we divide integers and always get an integer?
No! (-7) ÷ 2 = -3.5, which is not an integer. Integers are not closed under division. Division may give fractions or decimals.
Q5: How do we arrange integers in ascending order?
Put them on a number line (or imagine one). Left to right is ascending (smallest to largest). Example: -8, -3, 0, 2, 5 is ascending order.
Q6: What happens when we multiply an integer by 0?
Any integer multiplied by 0 is always 0. So (-500) × 0 = 0. This is the zero product property.
Exam Focus: In CBSE Class 7 exams, the most common integer questions are:
- Addition and subtraction of integers (with word problems like temperature, floors, sea level)
- Multiplication and division sign rules
- Arranging integers on a number line
- Properties of integers (which property is shown by which statement)
Always show your sign rules and steps — don’t just write the final answer!