Ascending and Descending Order
When we have a group of numbers, we often need to arrange them in a particular order. Arranging numbers from the smallest to the largest is called ascending order. Arranging numbers from the largest to the smallest is called descending order.
Think of climbing stairs. When you go up, each step is higher than the last — that is ascending. When you come down, each step is lower — that is descending. Numbers work the same way.
In Class 6, you will work with large numbers — thousands, lakhs and crores. To arrange them correctly, you need to compare them first using place values. This topic covers how to arrange whole numbers, large numbers and even negative integers in both ascending and descending order.
What is Ascending and Descending Order?
Definition: Ascending order means arranging numbers from the smallest to the largest. We use the symbol < (less than) between them.
Example: 12 < 25 < 37 < 48 < 56
Definition: Descending order means arranging numbers from the largest to the smallest. We use the symbol > (greater than) between them.
Example: 56 > 48 > 37 > 25 > 12
Key symbols:
- < means "less than" (the smaller number comes first)
- > means "greater than" (the larger number comes first)
- = means "equal to"
Types and Properties
Steps to Arrange Numbers in Order:
Step 1: Count the digits
- A number with more digits is always greater than a number with fewer digits.
- Example: 1,234 (4 digits) > 987 (3 digits)
Step 2: If digits are the same, compare from the left
- Start comparing from the leftmost digit (highest place value).
- The number with the larger leftmost digit is greater.
- If the leftmost digits are the same, move to the next digit to the right.
Example: Compare 4,567 and 4,321
- Both have 4 digits. Leftmost digit: both are 4 — same.
- Next digit: 5 vs 3. Since 5 > 3, we get 4,567 > 4,321.
Arranging Large Numbers (Lakhs and Crores):
For very large numbers, follow the same steps:
- Count the number of digits in each number.
- The number with the most digits is the greatest.
- If two numbers have the same number of digits, compare digit by digit from the left.
Example: Arrange in ascending order: 52,31,478; 6,89,345; 52,30,199; 85,12,300
- 6,89,345 has 6 digits — this is the smallest.
- The rest have 7 digits. Compare from the left:
- 52,30,199 and 52,31,478: First four digits 5-2-3-0 vs 5-2-3-1. Since 0 < 1, 52,30,199 < 52,31,478.
- 85,12,300 starts with 8, which is greater than 5. So it is the largest.
Ascending order: 6,89,345 < 52,30,199 < 52,31,478 < 85,12,300
Ascending and Descending Order of Integers:
When negative numbers are included, remember:
- Every positive number is greater than every negative number.
- Zero is greater than every negative number but less than every positive number.
- Among negative numbers, the one with the larger absolute value is smaller.
- Example: −8 < −3 (even though 8 > 3, −8 is further left on the number line).
Solved Examples
Example 1: Example 1: Ascending order of small numbers
Problem: Arrange in ascending order: 45, 12, 78, 33, 56.
Solution:
Compare all numbers. The smallest is 12, then 33, then 45, then 56, then 78.
Answer: 12 < 33 < 45 < 56 < 78
Example 2: Example 2: Descending order of small numbers
Problem: Arrange in descending order: 91, 17, 63, 85, 42.
Solution:
The largest is 91, then 85, then 63, then 42, then 17.
Answer: 91 > 85 > 63 > 42 > 17
Example 3: Example 3: Arranging large numbers in ascending order
Problem: Arrange in ascending order: 7,32,156; 7,31,998; 73,215; 7,32,200.
Solution:
Step 1: Count digits.
- 73,215 has 5 digits — smallest.
- The rest have 6 digits.
Step 2: Among the 6-digit numbers, compare from the left:
- 7,31,998: digits are 7-3-1-9-9-8
- 7,32,156: digits are 7-3-2-1-5-6
- 7,32,200: digits are 7-3-2-2-0-0
First two digits (7-3) are the same for all three. Third digit: 1 vs 2 vs 2. So 7,31,998 is the smallest of these three.
Between 7,32,156 and 7,32,200: fourth digit is 1 vs 2. Since 1 < 2, 7,32,156 < 7,32,200.
Answer: 73,215 < 7,31,998 < 7,32,156 < 7,32,200
Example 4: Example 4: Descending order of large numbers
Problem: Arrange in descending order: 45,00,000; 4,50,000; 45,00,001; 5,00,000.
Solution:
Step 1: Count digits.
- 4,50,000 and 5,00,000 have 6 digits.
- 45,00,000 and 45,00,001 have 7 digits — these are larger.
Step 2: Among 7-digit numbers: 45,00,001 > 45,00,000 (last digit 1 vs 0).
Step 3: Among 6-digit numbers: 5,00,000 > 4,50,000 (first digit 5 vs 4).
Answer: 45,00,001 > 45,00,000 > 5,00,000 > 4,50,000
Example 5: Example 5: Ascending order with integers (negative numbers)
Problem: Arrange in ascending order: −7, 3, −2, 0, 5, −10.
Solution:
On the number line, numbers to the left are smaller.
- −10 is the farthest left — smallest.
- Then −7, then −2, then 0, then 3, then 5.
Answer: −10 < −7 < −2 < 0 < 3 < 5
Example 6: Example 6: Descending order with integers
Problem: Arrange in descending order: 8, −4, 2, −1, 0, −9.
Solution:
Start from the largest: 8, then 2, then 0, then −1, then −4, then −9.
Answer: 8 > 2 > 0 > −1 > −4 > −9
Example 7: Example 7: Finding the greatest and smallest
Problem: From the numbers 3,45,678; 34,56,780; 3,45,670; 34,56,078 — find the greatest and smallest.
Solution:
- 3,45,678 and 3,45,670 have 6 digits.
- 34,56,780 and 34,56,078 have 7 digits.
Greatest must be among the 7-digit numbers: compare 34,56,780 and 34,56,078. Fourth digit: 6 vs 0. Since 6 > 0, greatest = 34,56,780.
Smallest must be among the 6-digit numbers: compare 3,45,678 and 3,45,670. Last digit: 8 vs 0. Since 0 < 8, smallest = 3,45,670.
Example 8: Example 8: Arranging scores in descending order
Problem: In a quiz, five students scored: Anil — 456, Bina — 512, Charu — 498, Devi — 512, Esha — 389. Arrange the scores in descending order.
Solution:
512 appears twice (Bina and Devi). Then 498, then 456, then 389.
Answer: 512 = 512 > 498 > 456 > 389
Bina and Devi are tied at the top.
Example 9: Example 9: Ascending order of populations
Problem: The populations of four towns are: Town A — 1,25,430; Town B — 98,760; Town C — 1,25,340; Town D — 2,00,100. Arrange in ascending order.
Solution:
- 98,760 has 5 digits — smallest.
- 1,25,340 and 1,25,430 have 6 digits. Compare: 1,25,340 < 1,25,430 (fifth digit 4 vs 3... wait, let us check: 1-2-5-3-4-0 vs 1-2-5-4-3-0. Fourth digit: 3 vs 4. Since 3 < 4, 1,25,340 < 1,25,430).
- 2,00,100 has 6 digits, starts with 2 which is > 1. Greatest among 6-digit numbers.
Answer: 98,760 < 1,25,340 < 1,25,430 < 2,00,100
Example 10: Example 10: Temperatures in ascending order
Problem: The temperatures of five cities are: Delhi 38°C, Shimla 12°C, Leh −5°C, Srinagar 2°C, Manali −1°C. Arrange in ascending order.
Solution:
Negative temperatures are colder (smaller). Among negatives: −5 < −1. Then positives: 2 < 12 < 38.
Answer: −5°C < −1°C < 2°C < 12°C < 38°C
(Leh < Manali < Srinagar < Shimla < Delhi)
Real-World Applications
Arranging numbers in order is used everywhere:
- Exam Results: Schools arrange marks in descending order to give ranks. The student with the highest marks gets Rank 1.
- Sports: In a race, finishing times are arranged in ascending order — the smallest time wins. In long jump, distances are arranged in descending order — the longest jump wins.
- Shopping: When comparing prices from different shops, arranging them in ascending order helps find the cheapest option.
- Temperature: Weather reports show cities arranged from coldest to hottest (ascending) or hottest to coldest (descending).
- Data and Computers: Sorting data is one of the most important operations in computers. Search results, contact lists, and file names are all sorted in ascending or descending order.
- Banking: Bank statements show transactions sorted by date — which is a form of ascending order.
Key Points to Remember
- Ascending order: smallest to largest. Use the < symbol.
- Descending order: largest to smallest. Use the > symbol.
- A number with more digits is always greater than a number with fewer digits.
- If two numbers have the same number of digits, compare them digit by digit from the left.
- The first digit where the numbers differ decides which is greater.
- For negative integers: the number farther from zero is smaller. So −10 < −3.
- Every positive number is greater than zero, and every negative number is less than zero.
- When numbers are equal, they can be placed next to each other with the = sign.
- Ascending order is like climbing stairs — numbers go up. Descending order is like coming down.
- Always double-check by reading your answer from left to right to make sure each number is correctly greater or smaller than the next.
Practice Problems
- Arrange in ascending order: 234, 567, 123, 890, 345.
- Arrange in descending order: 6,78,901; 67,890; 6,78,910; 6,79,001.
- Arrange in ascending order: −15, 8, −3, 0, 12, −7.
- Find the greatest and smallest: 4,56,789; 45,67,890; 4,56,798; 45,67,809.
- Five cities have populations: 2,34,500; 23,45,000; 2,34,050; 23,40,500; 3,00,000. Arrange in descending order.
- Arrange these temperatures in ascending order: 5°C, −12°C, 0°C, −3°C, 18°C, −1°C.
- Arrange in ascending order: 99,99,999; 1,00,00,000; 9,99,999; 10,00,000.
- The heights of five students are 152 cm, 148 cm, 155 cm, 148 cm, 160 cm. Arrange in descending order.
Frequently Asked Questions
Q1. What is ascending order?
Ascending order means arranging numbers from the smallest to the largest. For example, 3, 7, 11, 15, 20 is in ascending order. Each number is greater than the one before it.
Q2. What is descending order?
Descending order means arranging numbers from the largest to the smallest. For example, 20, 15, 11, 7, 3 is in descending order. Each number is smaller than the one before it.
Q3. How do you compare two numbers with the same number of digits?
Start comparing from the leftmost digit. If the leftmost digits are the same, move to the next digit. The first position where the digits are different decides which number is greater. The number with the larger digit at that position is the greater number.
Q4. How do you arrange negative numbers in ascending order?
For negative numbers, the one farther from zero is smaller. So −10 is less than −3. In ascending order, −10 comes before −3. Think of the number line: numbers on the left are smaller.
Q5. Is 0 greater than negative numbers?
Yes. Zero is greater than every negative number. On the number line, 0 is to the right of all negative numbers. So −1 < 0, −100 < 0, and so on.
Q6. What symbols are used for ascending and descending order?
For ascending order, use < (less than) between numbers: 2 < 5 < 8. For descending order, use > (greater than): 8 > 5 > 2. The open side of the symbol always faces the larger number.
Q7. Can two equal numbers appear in ascending or descending order?
Yes. If two numbers are equal, place them next to each other. Use = between them. For example: 3 < 5 = 5 < 9 is valid in ascending order.
Q8. How do you arrange numbers in crores and lakhs?
First count the digits. A number in crores (8+ digits) is always greater than a number in lakhs (6-7 digits). If two numbers have the same number of digits, compare them digit by digit from the left.
