Integers on Number Line
Integers are the set of whole numbers and their negatives: ..., −3, −2, −1, 0, 1, 2, 3, ... The best way to understand integers is to see them on a number line. A number line is a straight horizontal line with numbers marked at equal distances.
You have used the number line for whole numbers before. Now, you will extend it to the left side of zero to include negative numbers. This helps you compare integers, find their order, and understand how far apart they are.
Representing integers on a number line is a basic skill that you will use throughout mathematics — in addition, subtraction, inequalities, and even in graphs later on.
What is Integers on Number Line?
Definition: A number line for integers is a horizontal line where:
- Zero (0) is placed at the centre.
- Positive integers (1, 2, 3, ...) are placed to the right of zero at equal gaps.
- Negative integers (−1, −2, −3, ...) are placed to the left of zero at equal gaps.
- The line extends infinitely in both directions (shown by arrows).
Key facts about integers on a number line:
- Every integer has a unique position on the number line.
- Numbers increase as you move right.
- Numbers decrease as you move left.
- Any number to the right is greater than any number to its left.
- The distance between any two consecutive integers is 1 unit.
How to plot an integer on the number line:
- Draw a horizontal line with arrows on both ends.
- Mark 0 at the centre.
- Mark positive integers to the right of 0 at equal gaps.
- Mark negative integers to the left of 0 at equal gaps.
- Put a dot on the number you want to represent.
Types and Properties
1. Positive Integers on the Number Line
Positive integers (1, 2, 3, 4, ...) lie to the right of zero. The farther right you go, the larger the number. For example, 5 is to the right of 3, so 5 > 3.
2. Negative Integers on the Number Line
Negative integers (−1, −2, −3, −4, ...) lie to the left of zero. The farther left you go, the smaller the number. For example, −5 is to the left of −3, so −5 < −3.
- −1 is closer to zero, so it is the largest negative integer closest to zero.
- There is no smallest negative integer — the line goes on forever to the left.
3. Zero on the Number Line
Zero is neither positive nor negative. It sits right in the middle and separates positive integers from negative integers.
4. Comparing Integers Using the Number Line
- The number on the right is always greater.
- The number on the left is always smaller.
- Every positive integer is greater than every negative integer.
- Every positive integer is greater than zero.
- Every negative integer is less than zero.
5. Opposite Integers
Two integers are opposites if they are the same distance from zero but on opposite sides. For example, +4 and −4 are opposites. They are both 4 units away from zero.
Solved Examples
Example 1: Example 1: Plotting positive integers
Problem: Represent 3, 6, and 8 on a number line.
Solution:
- Draw a number line with 0 at the centre.
- Mark points at 3, 6, and 8 to the right of zero.
- All three are positive, so they all lie to the right of zero.
Answer: 3, 6, and 8 are marked to the right of 0, with 3 closest to 0 and 8 farthest.
Example 2: Example 2: Plotting negative integers
Problem: Represent −2, −5, and −7 on a number line.
Solution:
- Draw a number line with 0 at the centre.
- Mark points at −2, −5, and −7 to the left of zero.
- −2 is closest to zero, −7 is farthest.
Answer: −2, −5, and −7 are all to the left of 0. Order from left to right: −7, −5, −2.
Example 3: Example 3: Plotting a mix of integers
Problem: Represent −4, −1, 0, 2, and 5 on a number line.
Solution:
- Draw a number line.
- Mark 0 at the centre.
- −4 and −1 go to the left of 0.
- 2 and 5 go to the right of 0.
- Order from left to right: −4, −1, 0, 2, 5.
Answer: The integers from smallest to largest are: −4, −1, 0, 2, 5.
Example 4: Example 4: Comparing integers using number line
Problem: Which is greater: −3 or −7?
Solution:
- On the number line, −3 is to the right of −7.
- The number on the right is always greater.
Answer: −3 > −7. Even though 7 looks bigger than 3, −7 is farther left (farther from zero on the negative side), making it smaller.
Example 5: Example 5: Arranging integers in ascending order
Problem: Arrange in ascending order: 4, −6, 0, −2, 7, −9.
Solution:
- Plot all on a number line: −9, −6, −2, 0, 4, 7 (left to right).
- Ascending order means smallest to largest (left to right on number line).
Answer: −9, −6, −2, 0, 4, 7
Example 6: Example 6: Finding opposite integers
Problem: Write the opposite of each integer: (a) 5, (b) −8, (c) 0.
Solution:
- (a) Opposite of 5 = −5 (same distance from 0, opposite side).
- (b) Opposite of −8 = +8.
- (c) Opposite of 0 = 0 (zero is its own opposite).
Answer: (a) −5, (b) 8, (c) 0
Example 7: Example 7: Distance between integers on number line
Problem: Find the distance between −3 and 5 on the number line.
Solution:
- Count the steps from −3 to 5: −3, −2, −1, 0, 1, 2, 3, 4, 5.
- That is 8 steps.
- Or calculate: 5 − (−3) = 5 + 3 = 8.
Answer: The distance is 8 units.
Example 8: Example 8: Finding integers between two given integers
Problem: List all integers between −4 and 3.
Solution:
- "Between" means we do not include −4 and 3 themselves.
- Integers between −4 and 3: −3, −2, −1, 0, 1, 2.
Answer: There are 6 integers between −4 and 3: −3, −2, −1, 0, 1, 2.
Example 9: Example 9: Temperature on a number line
Problem: The temperatures of 5 cities are: Delhi 6°C, Shimla −2°C, Srinagar −5°C, Jaipur 10°C, Manali −8°C. Represent on a number line and find the coldest city.
Solution:
- On the number line: −8 (Manali), −5 (Srinagar), −2 (Shimla), 6 (Delhi), 10 (Jaipur).
- The leftmost point is −8 (Manali) — this is the smallest (coldest).
Answer: Manali is the coldest at −8°C.
Example 10: Example 10: Identifying integers from a number line
Problem: Points A, B, C, D are marked on a number line at −6, −2, 1, and 4. Which point is closest to zero? Which is farthest?
Solution:
- Distance of A (−6) from 0 = 6 units.
- Distance of B (−2) from 0 = 2 units.
- Distance of C (1) from 0 = 1 unit.
- Distance of D (4) from 0 = 4 units.
Answer: C (at 1) is closest to zero. A (at −6) is farthest from zero.
Real-World Applications
Thermometers: A thermometer is a vertical number line. Temperatures above zero are positive and temperatures below zero are negative. You read temperatures the same way you read integers on a number line.
Floors in a Building: Basement floors are negative (B1 = −1, B2 = −2) and upper floors are positive. The ground floor is 0. You use the number line idea to figure out how many floors apart two levels are.
Sea Level: Altitudes above sea level are positive (e.g., Mt. Everest at +8,849 m) and depths below sea level are negative (e.g., Dead Sea at −430 m). These are plotted on a vertical number line.
Bank Accounts: A positive balance means you have money. A negative balance (overdraft) means you owe money. The number line helps visualise whether deposits and withdrawals take you above or below zero.
Timeline (History): Years BCE (Before Common Era) can be thought of as negative numbers and years CE as positive numbers. The number line helps understand the gap between historical events.
Key Points to Remember
- A number line for integers has zero at the centre, positive integers to the right, and negative integers to the left.
- Numbers increase as you move right and decrease as you move left.
- Every positive integer is greater than zero and every negative integer.
- Every negative integer is less than zero.
- Among negative integers, the one closer to zero is greater. For example, −2 > −5.
- Zero is neither positive nor negative.
- Opposite integers are the same distance from zero but on opposite sides (e.g., +3 and −3).
- The distance between two integers on the number line = difference of the larger and smaller number.
- There are infinitely many integers — the number line extends forever in both directions.
- The absolute value of an integer is its distance from zero on the number line (always positive).
Practice Problems
- Represent the following on a number line: −5, −3, 0, 2, 6.
- Which is greater: (a) −4 or −9, (b) −3 or 2, (c) 0 or −1?
- Arrange in descending order: 3, −7, 5, −1, 0, −4.
- Write the opposite of: (a) 12, (b) −15, (c) 0, (d) −1.
- How many integers are there between −6 and 6? List them.
- Find the distance between −7 and 3 on the number line.
- The temperatures of 4 cities are: A = −3°C, B = 5°C, C = −8°C, D = 0°C. Which city is the warmest? Which is the coldest?
- True or false: −100 is greater than −1. Explain using the number line.
Frequently Asked Questions
Q1. What are integers?
Integers are all whole numbers and their negatives: ..., −3, −2, −1, 0, 1, 2, 3, ... They do not include fractions or decimals. The set of integers is denoted by Z.
Q2. Is zero a positive or negative integer?
Zero is neither positive nor negative. It is the dividing point between positive and negative integers on the number line.
Q3. How do you compare two negative integers?
The negative integer closer to zero is the greater one. For example, −2 is greater than −5 because −2 is closer to zero (to the right of −5 on the number line). Remember: on the number line, the number on the right is always greater.
Q4. What is the smallest integer?
There is no smallest integer. The number line extends infinitely to the left, so you can always find a smaller integer (−100, −1000, −10000, and so on forever).
Q5. What is the absolute value of an integer?
The absolute value of an integer is its distance from zero on the number line, without considering direction. It is always positive or zero. For example, |−7| = 7 and |7| = 7. Both −7 and 7 are 7 units from zero.
Q6. How many integers are between −3 and 3?
The integers between −3 and 3 (not including −3 and 3) are: −2, −1, 0, 1, 2. That is 5 integers. If you include −3 and 3, there are 7 integers.
Q7. Why is −1 greater than −100?
On the number line, −1 is to the right of −100. The number on the right is always greater. Think of it this way: owing Rs. 1 (−1) is better than owing Rs. 100 (−100). So −1 > −100.
Q8. What are consecutive integers?
Consecutive integers are integers that come one after another without any gap. For example: −2, −1, 0, 1, 2 are five consecutive integers. The difference between any two consecutive integers is always 1.
