Negative Numbers and Integers
Negative numbers are numbers less than zero. They arise in real life for:
- Temperature below 0°C (e.g., −5°C in Kashmir in winter)
- Depth below sea level (e.g., −200 m)
- Floors below ground in a building (B1, B2 = −1, −2)
- Loss or debt (e.g., a loss of ₹500 = −500)
Integers: The set of all whole numbers and their negatives.
… −4, −3, −2, −1, 0, 1, 2, 3, 4 …
- Positive integers: 1, 2, 3, 4, …
- Negative integers: −1, −2, −3, −4, …
- Zero: neither positive nor negative
Whole numbers are integers, but not all integers are whole numbers (negative integers are not whole numbers).
The Number Line for Integers
←——|——|——|——|——|——|——|——|——→
−4 −3 −2 −1 0 1 2 3 4
- Numbers increase going right.
- Numbers decrease going left.
- Moving right = positive direction.
- Moving left = negative direction.
Ordering Integers
On the number line, a number to the right is always greater.
- −1 > −3 (−1 is to the right of −3)
- −5 < 0 (−5 is to the left of 0)
- −100 < −1 (more negative = smaller)
Examples:
- Arrange in ascending order: −5, 3, −1, 0, −3 → −5, −3, −1, 0, 3
- Arrange in descending order: same → 3, 0, −1, −3, −5
Addition of Integers
On the Number Line
- Add a positive number: move RIGHT
- Add a negative number: move LEFT
Example 1: 3 + 4 = 7 (start at 3, move 4 right → 7)
Example 2: 3 + (−5) = −2 (start at 3, move 5 left → −2)
Example 3: (−2) + (−3) = −5 (start at −2, move 3 left → −5)
Rules for Addition
| Situation | Rule | Example |
|---|---|---|
| Both positive | Add and keep + sign | 5 + 3 = 8 |
| Both negative | Add and keep − sign | (−5) + (−3) = −8 |
| Different signs | Subtract, keep sign of larger | 5 + (−3) = 2; (−5) + 3 = −2 |
Subtraction of Integers
Subtracting an integer = Adding its opposite (additive inverse)
a − b = a + (−b)
Examples:
- 5 − 3 = 5 + (−3) = 2
- 5 − (−3) = 5 + 3 = 8
- (−5) − 3 = (−5) + (−3) = −8
- (−5) − (−3) = (−5) + 3 = −2
Additive Inverse
The additive inverse of a number is the number that, when added, gives 0.
- Additive inverse of 7 = −7 (7 + (−7) = 0)
- Additive inverse of −4 = 4 (−4 + 4 = 0)
- Additive inverse of 0 = 0
Absolute Value (Magnitude)
The absolute value of an integer is its distance from 0 on the number line — always non-negative.
- |5| = 5
- |−5| = 5
- |0| = 0
Key Vocabulary
| Term | Meaning |
|---|---|
| Integer | All positive numbers, negative numbers, and zero |
| Positive integer | Integer greater than 0 |
| Negative integer | Integer less than 0 |
| Additive inverse | The opposite of a number; sum with the number = 0 |
| Absolute value | Distance from 0; always positive |
| Ascending order | Smallest to largest |
| Descending order | Largest to smallest |
Exam Tips
- Zero is neither positive nor negative — common 1-mark question.
- When comparing negative numbers: −1 > −100 (less negative = greater).
- Subtracting a negative number = adding (two negatives make a positive).
- Additive inverse questions: "additive inverse of −7 is 7" — add to get 0.
- On the number line: for adding, move right for (+) and left for (−).
- Integers include fractions? No — −3/2 is not an integer.
Frequently Asked Questions
Integers are the collection of whole numbers and negative numbers: … −3, −2, −1, 0, 1, 2, 3, … Integers include positive numbers, negative numbers, and zero. Fractions and decimals are NOT integers.