1. Chapter at a Glance
Until now you have used numbers 0, 1, 2, 3, … But what number tells you that a lift is 2 floors below the ground, that the temperature in Leh is 5 degrees below zero, or that you owe ₹20? We need numbers on the other side of zero — the negative numbers. Together with zero and the positive numbers, they form the integers. In this chapter you will place integers on a number line, compare them and add and subtract them using movements and tokens.
- Negative numbers and integers in real situations.
- The integer number line; comparing and ordering integers.
- Adding and subtracting integers by moving on the number line (the lift model).
- Positive and negative tokens and zero pairs.
- Additive inverses and the rules for adding and subtracting.
- How Indian mathematicians, especially Brahmagupta, first gave rules for negative numbers.
2. Negative Numbers and Integers
Imagine a tall building with floors above the ground and parking floors below it. Ground floor = 0. Floors above: +1, +2, +3, … Floors below: −1, −2, −3, … The minus sign tells us the floor is below the ground.
- Positive integers: 1, 2, 3, … (written with or without +).
- Negative integers: −1, −2, −3, …
- Zero is neither positive nor negative.
- Integers = …, −3, −2, −1, 0, 1, 2, 3, …
| Situation | Positive means | Negative means |
|---|---|---|
| Floors of a building | Above ground | Below ground |
| Temperature | Above 0 °C | Below 0 °C |
| Money | Money you have (credit) | Money you owe (debit, debt) |
| Height | Above sea level | Below sea level |
| Change in score | Gain | Loss |
3. The Integer Number Line
Extend the number line to the left of 0 and mark −1, −2, −3, … at equal gaps. Positive integers lie to the right of zero and negative integers to the left.
Comparing integers
- A number to the right is greater; a number to the left is smaller.
- Every positive integer is greater than every negative integer: 1 > −100.
- Zero is greater than every negative integer and less than every positive integer.
- Among negative integers, the one closer to zero is greater: −2 > −7 (2 floors below is higher than 7 floors below).
Arrange in ascending order: 4, −9, 0, −2, 7, −5.
Solution: From left to right on the number line: −9, −5, −2, 0, 4, 7.
4. Adding Integers — The Lift Model
Think of a lift. Pressing +3 takes it 3 floors up; pressing −3 takes it 3 floors down. Adding an integer means a movement from where you are.
- (+2) + (+3): start at 2, go up 3 → +5.
- (+2) + (−5): start at 2, go down 5 → −3.
- (−4) + (+6): start at −4, go up 6 → +2.
- (−3) + (−2): start at −3, go down 2 → −5.
- Same signs: add the numbers (ignoring signs) and keep the common sign. (−6) + (−7) = −13.
- Different signs: subtract the smaller number from the larger (ignoring signs) and take the sign of the number that is farther from zero. (−9) + 4 = −5; 9 + (−4) = 5.
5. Tokens and Zero Pairs
Another model uses positive tokens (+) and negative tokens (−). One positive token and one negative token together make zero — they form a zero pair and cancel out.
- (+5) + (−3): take 5 positive and 3 negative tokens; remove 3 zero pairs; 2 positive remain → +2.
- (−5) + (+3): 3 zero pairs removed; 2 negative remain → −2.
Additive inverse
A number and its negative add up to zero: 7 + (−7) = 0. So −7 is the additive inverse of 7, and 7 is the additive inverse of −7. The additive inverse of 0 is 0.
6. Subtracting Integers
Subtracting means taking away. With tokens, to take away tokens you don't have, first add zero pairs (which doesn't change the value), then remove.
Example: 3 − (−2). We have 3 positive tokens but no negatives to remove. Add 2 zero pairs (2 positive + 2 negative). Now remove the 2 negatives: 5 positive tokens remain. So 3 − (−2) = 5.
Subtracting an integer is the same as adding its additive inverse:
The temperature in Shimla was 5 °C in the afternoon and −3 °C at night. By how much did it fall?
Solution: Fall = 5 − (−3) = 5 + 3 = 8 °C.
A submarine is at −120 m (below sea level). It rises 45 m and then dives 70 m. Where is it now?
Solution: −120 + 45 − 70 = −75 − 70 = −145 m, i.e. 145 m below sea level.
7. A Pinch of History
Indian mathematicians were among the first in the world to treat zero and negative numbers as numbers that can be added and subtracted. In his book Brāhmasphuṭasiddhānta (628 CE), Brahmagupta described positive numbers as "fortunes" and negative numbers as "debts" and gave rules such as: a debt minus zero is a debt; a fortune minus zero is a fortune; zero minus a debt is a fortune; the sum of a fortune and an equal debt is zero. Similar ideas also appear in old Chinese texts that used red and black rods for positive and negative numbers. European mathematicians accepted negative numbers fully only many centuries later.
8. Properties of Adding Integers
- Closure: the sum of two integers is always an integer.
- Commutative: a + b = b + a. (−4) + 9 = 9 + (−4) = 5.
- Associative: (a + b) + c = a + (b + c). Grouping does not change the sum.
- Additive identity: a + 0 = a. Zero does not change a number.
- Additive inverse: a + (−a) = 0.
- Subtraction is not commutative: 5 − 8 = −3 but 8 − 5 = 3.
Integers in a bank account
A passbook records deposits (money put in) as positive and withdrawals (money taken out) as negative. If a balance is ₹500 and the next entries are −₹750, +₹400 and −₹100, the balance becomes 500 − 750 + 400 − 100 = ₹50. A negative balance would mean the account owes money to the bank.
Use the associative property to find (−37) + 58 + (−63) quickly.
Solution: Group the negatives: (−37) + (−63) = −100. Then −100 + 58 = −42.
Which number is 6 less than −9? Which is 6 more than −9?
Solution: 6 less: −9 − 6 = −15 (move 6 steps left). 6 more: −9 + 6 = −3 (move 6 steps right).
9. Master Formula Table
| Rule | Formula | Example |
|---|---|---|
| Zero pair | a + (−a) = 0 | 8 + (−8) = 0 |
| Same signs | add sizes, keep sign | (−5) + (−6) = −11 |
| Different signs | subtract sizes, sign of larger size | (−12) + 7 = −5 |
| Subtraction | a − b = a + (−b) | 2 − 9 = −7 |
| Subtracting a negative | a − (−b) = a + b | −3 − (−8) = 5 |
| Ordering | right is greater | −1 > −10 |
| Additive identity | a + 0 = a | −6 + 0 = −6 |
| Commutative | a + b = b + a | −3 + 5 = 5 + (−3) |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Integers | …, −2, −1, 0, 1, 2, … | Real-life meaning |
| Number line | Right = greater | Ordering, comparing |
| Addition | Lift movements | Sign rules |
| Tokens | Zero pairs cancel | Explaining results |
| Subtraction | Add the inverse | Temperature, height, money problems |
| History | Brahmagupta's debts and fortunes | Indian contribution |
Practice chart: night temperatures (°C) of four towns
| Town (practice data) | Temperature | Position on number line |
|---|---|---|
| Town A | 4 °C | Right of zero |
| Town B | −2 °C | Just left of zero |
| Town C | −10 °C | Far left |
| Town D | −18 °C | Farthest left (coldest) |
11. Mind Map
- Positive, negative, zero
- Floors, temperature, money
- Right is greater
- Ordering
- Lift model
- Tokens, zero pairs
- a − b = a + (−b)
- Brahmagupta
- Debts and fortunes
12. Tips and Tricks
- Change every subtraction into addition of the opposite before calculating.
- Group all positive numbers and all negative numbers separately, then combine: 7 − 3 − 9 + 4 = (7 + 4) − (3 + 9) = 11 − 12 = −1.
- For "difference" questions (temperature fall, height between two points), subtract the smaller from the larger: higher − lower.
- Draw a quick number line when unsure.
- Thinking −8 > −3 because 8 > 3.
- Writing 4 − (−6) = −2 instead of 10.
- Calling zero a positive number.
- Forgetting the sign of the answer when signs are different.
13. Important One-liners
- Integers include negative numbers, zero and positive numbers.
- Zero is neither positive nor negative.
- On the number line, numbers increase from left to right.
- Every negative integer is less than zero and less than every positive integer.
- Among negative integers, the one closer to zero is greater.
- A positive token and a negative token make a zero pair.
- The additive inverse of a is −a; their sum is 0.
- Subtracting a number is the same as adding its additive inverse.
- a − (−b) = a + b.
- Brahmagupta (628 CE) gave rules for operating with zero and negative numbers.
14. Exam-pattern Questions with Detailed Answers
Which is the greatest: −7, −2, −11, −5?
- −7
- −2
- −11
- −5
Show answer
The negative integer closest to zero is the greatest. −2 is closest.
Answer: (B) −2
Find (−15) + 9.
Show answer
Different signs: 15 − 9 = 6; the larger size (15) is negative, so the answer is −6.
Answer: −6
Write the additive inverse of −23 and of 0.
Show answer
−23 + 23 = 0, so the inverse of −23 is 23. 0 + 0 = 0, so the inverse of 0 is 0.
Answer: 23 and 0
Evaluate: (−8) − (−13).
Show answer
Subtracting −13 is adding 13: −8 + 13 = 5.
Answer: 5
A lift is at floor −2. It goes up 7 floors, then down 9 floors. At which floor does it stop?
- −4
- −2
- 4
- 0
Show answer
−2 + 7 = 5; 5 − 9 = −4.
Answer: (A) Floor −4 (4 floors below ground)
Simplify: 12 − 18 + 5 − 9 + 3.
Show answer
Positives: 12 + 5 + 3 = 20. Negatives: 18 + 9 = 27. Result = 20 − 27 = −7.
Answer: −7
The temperature at noon was 3 °C. By midnight it fell by 11 °C. What was the midnight temperature?
Show answer
3 − 11 = −8 °C.
Answer: −8 °C
Using tokens, explain why (−4) − (−6) = 2.
Show answer
Start with 4 negative tokens. We must remove 6 negatives but have only 4, so add 2 zero pairs (2 positive + 2 negative). Now there are 6 negatives and 2 positives. Remove the 6 negatives: 2 positive tokens remain.
Answer: 2 positive tokens remain, so the result is +2
A diver is at −35 m. A bird is flying 20 m above sea level. What is the vertical distance between them?
Show answer
Distance = higher − lower = 20 − (−35) = 20 + 35 = 55 m.
Answer: 55 m
In a quiz, a correct answer gets +5 and a wrong answer gets −2. Asha answered 12 correctly and 7 wrongly. What is her score?
Show answer
Correct: 12 × 5 = 60. Wrong: 7 × (−2) = −14 (seven losses of 2).
Score = 60 − 14 = 46.
Answer: 46
Find two integers whose sum is −4 and whose difference (larger minus smaller) is 10.
Show answer
Let the integers be a (larger) and b. a + b = −4 and a − b = 10. Adding: 2a = 6, so a = 3. Then b = −4 − 3 = −7.
Check: 3 + (−7) = −4 ✔; 3 − (−7) = 10 ✔.
Answer: 3 and −7
A shopkeeper made a profit of ₹350 on Monday, a loss of ₹520 on Tuesday and a profit of ₹90 on Wednesday. What is the net result for the three days?
Show answer
Write profit as positive and loss as negative: 350 + (−520) + 90 = 440 − 520 = −80.
Answer: A net loss of ₹80
15. Quick Revision Summary
- Integers: negatives, zero, positives; negatives describe below-zero, debts, losses, depths.
- On the number line, right means greater; −1 > −10.
- Adding: same signs → add and keep sign; different signs → subtract and take sign of the larger size.
- Zero pair: a + (−a) = 0; −a is the additive inverse of a.
- Subtracting = adding the inverse: a − b = a + (−b); a − (−b) = a + b.
- Brahmagupta first gave rules for zero and negative numbers (debts and fortunes).