Ordering Integers
You already know how to arrange whole numbers in order. You know that 5 is less than 8, and 100 is greater than 50. But what happens when you bring in negative numbers? Is -3 greater or less than -7? Is -5 greater or less than 2?
Integers include positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...), and zero. When you need to put these in order, the rules are a little different from what you are used to with whole numbers.
Think of temperature. On a cold winter day, the temperature might be -5 degrees Celsius. On a colder day, it might be -12 degrees Celsius. Which day is colder? The day with -12 degrees, because -12 is less than -5. Even though 12 is bigger than 5, negative 12 is smaller than negative 5.
In this chapter, you will learn how to compare and order integers using a number line, understand the rules for comparing positive and negative numbers, and arrange integers in ascending and descending order.
What is Ordering Integers?
Definition: Ordering integers means arranging integers from smallest to largest (ascending order) or from largest to smallest (descending order).
What are Integers?
... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...
Rules for Comparing Integers:
- Every positive integer is greater than zero.
- Every negative integer is less than zero.
- Every positive integer is greater than every negative integer.
- Among positive integers, the one with the larger value is greater (same as whole numbers).
- Among negative integers, the one with the smaller absolute value is greater. For example, -3 > -7 because 3 < 7.
- Zero is greater than every negative integer and less than every positive integer.
Key Terms:
| Term | Meaning | Example |
|---|---|---|
| Ascending Order | From smallest to largest | -5, -2, 0, 3, 7 |
| Descending Order | From largest to smallest | 7, 3, 0, -2, -5 |
| Greater than (>) | The first number is bigger | 3 > -1 |
| Less than (<) | The first number is smaller | -4 < 2 |
| Absolute Value | The distance from zero (always positive) | |-7| = 7 |
Ordering Integers Formula
Comparing Integers on a Number Line:
On a number line, the integer to the RIGHT is always GREATER.
The integer to the LEFT is always SMALLER.
Number Line:
... -5 -4 -3 -2 -1 0 1 2 3 4 5 ...
smaller <-------------------------------> greater
Rules Summary:
Positive > 0 > Negative
Comparing Two Negative Integers:
If |a| < |b|, then -a > -b
Where:
- |a| means the absolute value (distance from zero) of a
- Example: |-3| = 3, |-7| = 7. Since 3 < 7, we get -3 > -7.
Types and Properties
There are different types of comparison and ordering problems:
Type 1: Comparing a Positive and a Negative Integer
A positive integer is ALWAYS greater than a negative integer.
- 5 > -8 (even though 8 is bigger than 5, positive beats negative)
- 1 > -100
Type 2: Comparing Two Positive Integers
Same as whole numbers. The bigger number is greater.
- 15 > 9
- 100 > 73
Type 3: Comparing Two Negative Integers
The number closer to zero is greater. The number farther from zero is smaller.
- -2 > -6 (2 is closer to zero than 6)
- -10 > -50 (10 is closer to zero than 50)
- -1 > -1000
Type 4: Comparing with Zero
- Any positive number > 0
- 0 > Any negative number
Type 5: Arranging in Ascending Order
Place integers from smallest (leftmost on number line) to largest (rightmost).
- Given: 3, -1, 5, -4, 0
- Ascending: -4, -1, 0, 3, 5
Type 6: Arranging in Descending Order
Place integers from largest (rightmost on number line) to smallest (leftmost).
- Given: 3, -1, 5, -4, 0
- Descending: 5, 3, 0, -1, -4
Memory Trick for Negative Numbers:
Think of negative numbers as debt. Having a debt of Rs. 7 (-7) is worse than having a debt of Rs. 3 (-3). So -3 is "more" than -7, which means -3 > -7.
Solved Examples
Example 1: Compare a Positive and Negative Integer
Problem: Compare 4 and -9. Which is greater?
Solution:
Rule: Every positive integer is greater than every negative integer.
- 4 is positive, -9 is negative.
- So 4 > -9.
Answer: 4 > -9
Example 2: Compare Two Negative Integers
Problem: Compare -3 and -11. Which is greater?
Solution:
Rule: Among negative integers, the one closer to zero is greater.
- -3 is 3 steps from zero.
- -11 is 11 steps from zero.
- -3 is closer to zero, so -3 > -11.
Answer: -3 > -11
Example 3: Compare with Zero
Problem: Arrange -5, 0, and 3 in ascending order.
Solution:
Rules:
- -5 < 0 (negative is less than zero)
- 0 < 3 (zero is less than positive)
Ascending order (smallest to largest):
- -5, 0, 3
Answer: -5 < 0 < 3
Example 4: Arrange in Ascending Order
Problem: Arrange these integers in ascending order: 7, -2, -8, 3, 0, -5, 1.
Solution:
Step 1: Separate negatives, zero, and positives.
- Negatives: -8, -5, -2
- Zero: 0
- Positives: 1, 3, 7
Step 2: Order the negatives (furthest from zero comes first).
- -8, -5, -2
Step 3: Order the positives (smallest first).
- 1, 3, 7
Step 4: Combine: negatives, then zero, then positives.
- -8, -5, -2, 0, 1, 3, 7
Answer: -8, -5, -2, 0, 1, 3, 7
Example 5: Arrange in Descending Order
Problem: Arrange these integers in descending order: -4, 6, -1, 9, -7, 2.
Solution:
Step 1: Identify the largest to smallest.
- Positives (largest first): 9, 6, 2
- Negatives (closest to zero first): -1, -4, -7
Step 2: Combine in descending order.
- 9, 6, 2, -1, -4, -7
Answer: 9, 6, 2, -1, -4, -7
Example 6: Temperature Comparison
Problem: The temperatures recorded in 5 cities are: Delhi 8 degrees C, Shimla -2 degrees C, Srinagar -6 degrees C, Mumbai 25 degrees C, and Leh -10 degrees C. Arrange the cities from coldest to warmest.
Solution:
Temperatures:
- Delhi: 8
- Shimla: -2
- Srinagar: -6
- Mumbai: 25
- Leh: -10
Ascending order of temperatures:
- -10 < -6 < -2 < 8 < 25
Coldest to warmest:
- Leh (-10), Srinagar (-6), Shimla (-2), Delhi (8), Mumbai (25)
Answer: Leh, Srinagar, Shimla, Delhi, Mumbai (coldest to warmest).
Example 7: Insert Correct Sign
Problem: Insert < or > between each pair: (a) -6 ___ -2, (b) 0 ___ -4, (c) -15 ___ -8, (d) 7 ___ -7.
Solution:
- (a) -6 < -2 (6 is farther from zero, so -6 is smaller)
- (b) 0 > -4 (zero is greater than any negative)
- (c) -15 < -8 (15 is farther from zero, so -15 is smaller)
- (d) 7 > -7 (positive is always greater than negative)
Answer: (a) <, (b) >, (c) <, (d) >
Example 8: Finding Integers Between Two Numbers
Problem: List all integers between -3 and 4.
Solution:
Integers between -3 and 4 (not including -3 and 4):
- -2, -1, 0, 1, 2, 3
Answer: The integers between -3 and 4 are -2, -1, 0, 1, 2, 3 (6 integers).
Example 9: Bank Account Problem
Problem: Ravi's bank balance is Rs. -200 (he owes Rs. 200). Priya's bank balance is Rs. -50. Kiran's balance is Rs. 100. Who has the most money? Who has the least?
Solution:
Balances as integers:
- Ravi: -200
- Priya: -50
- Kiran: 100
Ordering:
- -200 < -50 < 100
Answer: Kiran has the most money (Rs. 100). Ravi has the least (Rs. -200, meaning he owes the most).
Example 10: Find the Smallest and Largest
Problem: From the set {-12, 5, -3, 0, 8, -20, 15}, find the smallest and largest integers.
Solution:
Arrange in ascending order:
- -20, -12, -3, 0, 5, 8, 15
Smallest: -20 (furthest left on number line)
Largest: 15 (furthest right on number line)
Answer: Smallest = -20, Largest = 15.
Real-World Applications
Ordering integers is useful in many real-life situations:
Temperature: Weather reports use negative numbers for temperatures below zero. Comparing -5 degrees C and -15 degrees C tells you which day is colder.
Altitude: Sea level is 0. Mountains have positive altitude (Mount Everest: +8849 m) and ocean trenches have negative altitude (Mariana Trench: -11034 m). Ordering helps compare heights and depths.
Money and Debt: A positive balance means you have money. A negative balance means you owe money. Comparing helps you understand who is better off financially.
Floors in a Building: Ground floor is 0. Floors above are +1, +2, +3. Basement floors are -1, -2, -3. Ordering tells you which floor is higher or lower.
Sports: In golf, scores are measured relative to par. A score of -3 (3 under par) is better than -1 (1 under par), which is better than +2 (2 over par).
History: Years BCE are written as negative numbers. 500 BCE is -500, and 200 BCE is -200. Since -200 > -500, 200 BCE comes after 500 BCE on the timeline.
Key Points to Remember
- Integers include positive numbers, negative numbers, and zero: ..., -3, -2, -1, 0, 1, 2, 3, ...
- On a number line, the number to the right is always greater.
- Every positive integer > 0 > every negative integer.
- Among negative integers, the one closer to zero is greater: -3 > -7.
- Ascending order = smallest to largest: -5, -2, 0, 3, 8.
- Descending order = largest to smallest: 8, 3, 0, -2, -5.
- Think of negative numbers as debt: less debt (-3) is better than more debt (-7), so -3 > -7.
- The absolute value (distance from zero) helps compare negatives: if |a| < |b|, then -a > -b.
- Zero is neither positive nor negative. It is the dividing point.
Practice Problems
- Compare: -9 and -4. Which is greater?
- Compare: -15 and 3. Which is greater?
- Arrange in ascending order: 4, -7, 0, -3, 5, -1.
- Arrange in descending order: -11, 8, -2, 0, 6, -5.
- List all integers between -6 and 3.
- The temperatures in 4 cities are: -8 degrees C, 5 degrees C, -1 degrees C, 12 degrees C. Arrange from coldest to warmest.
- Which is greater: -100 or -1? Explain.
- An elevator goes to floors: -2, -1, 0, 1, 2, 3, 4, 5. List these in descending order.
Frequently Asked Questions
Q1. How do you compare two negative integers?
The negative integer closer to zero is greater. For example, -3 > -7 because -3 is only 3 steps from zero while -7 is 7 steps from zero. Think of it as debt: owing Rs. 3 is better than owing Rs. 7.
Q2. Is zero positive or negative?
Zero is neither positive nor negative. It is the dividing point between positive and negative integers. Zero is greater than all negative integers and less than all positive integers.
Q3. What does ascending order mean?
Ascending order means arranging numbers from the smallest to the largest. For example, -5, -2, 0, 3, 7 is in ascending order.
Q4. What does descending order mean?
Descending order means arranging numbers from the largest to the smallest. For example, 7, 3, 0, -2, -5 is in descending order.
Q5. Is -1 greater than -100?
Yes. -1 is greater than -100 because -1 is closer to zero. On a number line, -1 is to the right of -100. In terms of debt, owing Rs. 1 is much better than owing Rs. 100.
Q6. How does a number line help in ordering integers?
On a number line, numbers increase as you move to the right and decrease as you move to the left. So any number to the right is greater than a number to the left. Place all integers on the line and read from left to right for ascending order.
Q7. Is a positive number always greater than a negative number?
Yes. Every positive number is greater than every negative number. Even 1 is greater than -1000. Positive numbers are to the right of zero, and negative numbers are to the left.
Q8. What are some real-life uses of ordering integers?
Ordering integers is used in comparing temperatures (which day is colder), bank balances (who has more money), altitudes (mountain heights vs ocean depths), building floors (above vs below ground), and dates in history (BCE vs CE).
