Comparing Large Numbers
Imagine you and your friend are arguing about which city has more people. Mumbai has a population of 1,24,12,478 and Delhi has a population of 1,11,07,835. Which city has more people? To answer this, you need to know how to compare large numbers. Comparing numbers is something you do every day without even thinking about it. When you check cricket scores, you compare runs. When you look at prices in a shop, you compare costs. But what happens when the numbers get really big, like lakhs and crores? That is where you need a proper method. In this chapter, we will learn simple rules to compare any two large numbers, no matter how many digits they have. We will also learn how to arrange numbers in ascending and descending order. This is an important part of the Knowing Our Numbers chapter in Grade 6 mathematics, and once you understand it, you will feel confident working with big numbers in everyday life.
What is Comparing Large Numbers - Grade 6 Maths (Knowing Our Numbers)?
When we compare two numbers, we find out which one is greater (bigger) and which one is smaller. We use two special symbols for this:
The symbol > means greater than. For example, 58 > 43 means 58 is greater than 43.
The symbol < means less than. For example, 27 < 61 means 27 is less than 61.
The symbol = means equal to. For example, 50 = 50 means both numbers are the same.
Here is a fun trick to remember: Think of the symbol as the mouth of a hungry crocodile. The crocodile always opens its mouth towards the bigger number because it wants to eat the bigger meal! So in 58 > 43, the crocodile's mouth opens towards 58 because 58 is bigger.
When numbers are small, like 7 and 12, comparing is easy. You just know that 12 is bigger. But when numbers have 6, 7, or 8 digits, you need a step-by-step method. That is what the rules of comparison are for.
Comparing numbers is the foundation of many things in maths. It helps you arrange numbers in order, estimate results, and understand the size of quantities in real life, like populations, distances, and amounts of money.
Comparing Large Numbers Formula
There is no single formula for comparing numbers, but there are clear rules that work every time:
Rule 1: Count the digits. A number with more digits is always greater. For example, a 5-digit number is always greater than a 4-digit number. So 10,000 > 9,999.
Rule 2: If both numbers have the same number of digits, compare digit by digit from the left. Start from the leftmost digit (the highest place value). The number with the greater digit in the leftmost place is the greater number.
Rule 3: If the leftmost digits are the same, move to the next digit to the right and compare again. Keep doing this until you find a digit that is different. The number with the greater digit at that position is the greater number.
Rule 4: If all digits are the same, the numbers are equal.
These rules work for numbers of any size, whether they have 2 digits or 10 digits. The key idea is that the leftmost digit has the highest value, so it matters the most when comparing.
Derivation and Proof
Let us understand why these rules work by thinking about place value.
Consider two numbers: 4,52,187 and 3,98,654. Both are 6-digit numbers. The first number has 4 in the lakhs place, and the second has 3 in the lakhs place. Since 4 lakhs is more than 3 lakhs, the first number is greater, no matter what the other digits are. Even if the second number had 9 in every other position, 4,00,000 is still bigger than 3,99,999.
This is because of how our number system works. Each position has a value that is 10 times the position to its right. The lakhs place is worth 1,00,000, while the ten-thousands place is worth only 10,000. So one extra in the lakhs place (1,00,000) is bigger than the maximum possible in all the places to its right combined (99,999).
That is why we always compare from left to right. The leftmost digit controls the biggest chunk of the number's value. If it is bigger, the whole number is bigger, regardless of what comes after.
When the leftmost digits are equal, those big chunks cancel out, and we move to the next position to find the difference. This process continues until we find a mismatch.
Let us see this with a table for 4,52,187 vs 3,98,654:
| Position | 4,52,187 | 3,98,654 | Result |
| Lakhs | 4 | 3 | 4 > 3 |
Since 4 > 3 in the lakhs place, we stop here. 4,52,187 > 3,98,654.
Now compare 7,23,456 and 7,45,123:
| Position | 7,23,456 | 7,45,123 | Result |
| Lakhs | 7 | 7 | Equal, move right |
| Ten-thousands | 2 | 4 | 2 < 4 |
Since 2 < 4 in the ten-thousands place, 7,23,456 < 7,45,123.
Types and Properties
Comparing large numbers can appear in different forms in your textbook and exams. Here are the main types:
Type 1: Direct Comparison Using Symbols - You are given two numbers and asked to put >, < or = between them. You use the rules of comparison: first check the number of digits, then compare digit by digit from the left.
Type 2: Ascending Order (Smallest to Largest) - You are given a set of numbers and asked to arrange them from the smallest to the largest. This is called ascending order. Think of climbing stairs, going up from the smallest step. First compare all the numbers using the rules, then write them with the smallest first.
Type 3: Descending Order (Largest to Smallest) - You are given a set of numbers and asked to arrange them from the largest to the smallest. This is called descending order. Think of going down a slide, starting from the top. Write the biggest number first and the smallest last.
Type 4: Finding the Greatest and Smallest Number - From a given set of numbers, find which one is the greatest and which one is the smallest. Use the comparison rules to identify them.
Type 5: Forming the Greatest and Smallest Number - You are given a set of digits and asked to form the greatest or smallest possible number using all of them. To make the greatest number, arrange the digits in descending order. To make the smallest, arrange them in ascending order. If 0 is one of the digits, remember it cannot be the first digit of a number.
Type 6: Comparison Word Problems - Real-life problems where you compare populations, distances, amounts of money, or other large quantities and determine which is greater or arrange them in order.
Solved Examples
Example 1: Example 1: Comparing Numbers with Different Number of Digits
Problem: Compare 98,765 and 1,23,456.
Solution:
Step 1: Count the digits.
98,765 has 5 digits.
1,23,456 has 6 digits.
Step 2: A number with more digits is always greater.
Since 6 digits > 5 digits:
98,765 < 1,23,456
No matter what the digits are, a 6-digit number is always bigger than a 5-digit number.
Example 2: Example 2: Comparing Numbers with Same Number of Digits
Problem: Compare 5,43,217 and 5,67,892.
Solution:
Step 1: Both numbers have 6 digits, so we compare digit by digit from the left.
| Position | 5,43,217 | 5,67,892 |
| Lakhs | 5 | 5 | Same, move right |
| Ten-thousands | 4 | 6 | 4 < 6 |
Step 2: Since 4 < 6 in the ten-thousands place:
5,43,217 < 5,67,892
Example 3: Example 3: Comparing When Several Digits Match
Problem: Compare 8,34,521 and 8,34,578.
Solution:
Both have 6 digits. Let us compare position by position:
| Position | 8,34,521 | 8,34,578 |
| Lakhs | 8 | 8 | Same |
| Ten-thousands | 3 | 3 | Same |
| Thousands | 4 | 4 | Same |
| Hundreds | 5 | 5 | Same |
| Tens | 2 | 7 | 2 < 7 |
Since 2 < 7 in the tens place:
8,34,521 < 8,34,578
Example 4: Example 4: Arranging in Ascending Order
Problem: Arrange in ascending order: 45,321; 4,53,210; 4,532; 45,231.
Solution:
Step 1: Count digits of each number:
4,532 has 4 digits, 45,321 has 5 digits, 45,231 has 5 digits, 4,53,210 has 6 digits.
Step 2: The 4-digit number is the smallest, and the 6-digit number is the largest.
Step 3: Compare the two 5-digit numbers: 45,321 and 45,231.
First three digits (4, 5, 2 and 4, 5, 3) - wait, let us write them properly: 45,321 and 45,231.
| Position | 45,321 | 45,231 |
| Ten-thousands | 4 | 4 | Same |
| Thousands | 5 | 5 | Same |
| Hundreds | 3 | 2 | 3 > 2 |
So 45,231 < 45,321.
Ascending order: 4,532 < 45,231 < 45,321 < 4,53,210
Example 5: Example 5: Arranging in Descending Order
Problem: Arrange in descending order: 7,89,432; 78,943; 7,89,423; 7,90,000.
Solution:
Step 1: Count digits: 78,943 has 5 digits. The other three have 6 digits each.
So 78,943 is the smallest.
Step 2: Compare the three 6-digit numbers:
7,89,432 vs 7,89,423 vs 7,90,000
Compare 7,90,000 with the others: Lakhs digit is 7 for all. Ten-thousands digit: 9, 8, 8. Since 9 > 8, 7,90,000 is the greatest.
Now compare 7,89,432 and 7,89,423:
| Position | 7,89,432 | 7,89,423 |
| Lakhs | 7 | 7 | Same |
| Ten-thousands | 8 | 8 | Same |
| Thousands | 9 | 9 | Same |
| Hundreds | 4 | 4 | Same |
| Tens | 3 | 2 | 3 > 2 |
So 7,89,432 > 7,89,423.
Descending order: 7,90,000 > 7,89,432 > 7,89,423 > 78,943
Example 6: Example 6: Finding Greatest and Smallest from a Set
Problem: Find the greatest and smallest number from: 3,45,678; 34,56,780; 34,567; 3,45,687.
Solution:
Step 1: Count digits:
34,567 = 5 digits, 3,45,678 = 6 digits, 3,45,687 = 6 digits, 34,56,780 = 7 digits.
Step 2: The number with the most digits is the greatest: 34,56,780 (7 digits).
The number with the fewest digits is the smallest: 34,567 (5 digits).
Greatest number: 34,56,780
Smallest number: 34,567
Example 7: Example 7: Forming Greatest and Smallest Numbers from Digits
Problem: Using the digits 7, 2, 9, 5, form the greatest and smallest 4-digit numbers. (Each digit used only once.)
Solution:
To form the greatest number, arrange digits in descending order (biggest first): 9, 7, 5, 2
Greatest number = 9,752
To form the smallest number, arrange digits in ascending order (smallest first): 2, 5, 7, 9
Smallest number = 2,579
Example 8: Example 8: Forming Smallest Number When 0 Is a Digit
Problem: Using the digits 3, 0, 8, 1, form the smallest 4-digit number.
Solution:
Ascending order of digits: 0, 1, 3, 8. But wait! If we write 0138, that is not a 4-digit number. A number cannot start with 0.
So we put the smallest non-zero digit first, then arrange the rest in ascending order.
Smallest non-zero digit = 1. Remaining digits in ascending order: 0, 3, 8.
Smallest number = 1,038
Example 9: Example 9: Comparing Populations (Word Problem)
Problem: The population of City A is 23,45,678 and the population of City B is 23,54,678. Which city has a larger population?
Solution:
Both numbers have 7 digits. Compare from the left:
| Position | 23,45,678 | 23,54,678 |
| Ten-lakhs | 2 | 2 | Same |
| Lakhs | 3 | 3 | Same |
| Ten-thousands | 4 | 5 | 4 < 5 |
Since 4 < 5 in the ten-thousands place:
23,45,678 < 23,54,678
City B has a larger population.
Example 10: Example 10: Ordering Distances (Word Problem)
Problem: The distances of four planets from the Sun are: Mercury = 5,79,09,175 km, Venus = 10,82,08,930 km, Earth = 14,95,97,870 km, Mars = 22,79,39,100 km. Arrange these planets in ascending order of their distance from the Sun.
Solution:
Step 1: Count digits:
Mercury: 5,79,09,175 = 8 digits. Venus: 10,82,08,930 = 9 digits. Earth: 14,95,97,870 = 9 digits. Mars: 22,79,39,100 = 9 digits.
Step 2: Mercury has only 8 digits, so it is the smallest. The other three have 9 digits each.
Step 3: Compare the 9-digit numbers by leftmost digit: Venus starts with 1, Earth starts with 1, Mars starts with 2. So Mars is the largest.
Step 4: Compare Venus (10,82,08,930) and Earth (14,95,97,870): Second digit is 0 vs 4. Since 0 < 4, Venus < Earth.
Ascending order: Mercury < Venus < Earth < Mars
Real-World Applications
Comparing large numbers is useful in many real-life situations. When you read about the populations of different countries, you compare numbers to find which country has more people. India has a population of about 1,40,00,00,000 (140 crore) and the population of Australia is about 2,60,00,000 (2.6 crore). By comparing, you can immediately see that India has a much larger population.
In sports, comparing numbers helps you decide who scored more runs, who won by more goals, or which team has a better record. If Virat Kohli has scored 12,344 runs and Sachin Tendulkar scored 18,426 runs in ODIs, comparing these numbers tells you who scored more.
When your family goes shopping, comparing prices helps you get the best deal. If one TV costs Rs. 32,999 and another costs Rs. 34,500, comparing helps you see which one is cheaper.
Banks and businesses compare amounts of money to make decisions. The government compares budgets of different states, revenues of different years, and expenditure across departments. Even in geography, we compare the areas of countries, heights of mountains, and lengths of rivers using large numbers.
In science, comparing large numbers helps us understand distances in space, sizes of cells, and quantities in chemistry. All of these require the same skill of comparing numbers that you are learning now.
Key Points to Remember
- 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 starting from the leftmost position (highest place value).
- The number with the greater digit in the leftmost differing position is the greater number.
- Use the symbol > for greater than, < for less than, and = for equal to.
- Ascending order means arranging numbers from smallest to largest.
- Descending order means arranging numbers from largest to smallest.
- To form the greatest number from given digits, arrange the digits in descending order.
- To form the smallest number from given digits, arrange the digits in ascending order. If 0 is one of the digits, the smallest non-zero digit comes first.
- The leftmost digit is the most important because it represents the highest place value.
- Comparing numbers is essential in daily life for making decisions about prices, scores, distances, and populations.
Practice Problems
- Compare and put the correct symbol (>, < or =): 6,78,345 and 6,87,345.
- Compare: 99,999 and 1,00,001.
- Arrange in ascending order: 4,56,789; 45,678; 4,56,798; 4,567.
- Arrange in descending order: 12,34,567; 12,43,567; 1,23,456; 12,34,576.
- Using the digits 6, 3, 0, 8, 1, form the greatest and smallest 5-digit numbers.
- The areas of three states are: Rajasthan = 3,42,239 sq km, Maharashtra = 3,07,713 sq km, Madhya Pradesh = 3,08,252 sq km. Arrange them in descending order of area.
- Find the greatest and smallest numbers from: 5,00,000; 4,99,999; 5,00,001; 50,000.
- Which is greater: the number formed by digits 4, 7, 2, 9 in descending order, or the number formed by digits 5, 3, 8, 1 in descending order?
Frequently Asked Questions
Q1. Why do we compare digits from left to right?
We compare from left to right because the leftmost digit has the highest place value. In a 6-digit number, the leftmost digit is in the lakhs place, which is worth 1,00,000. Even if all the other digits are smaller, one extra in the lakhs place makes the number bigger than all the rest combined. So the leftmost digit matters the most.
Q2. Is a 5-digit number always greater than a 4-digit number?
Yes, always. The smallest 5-digit number is 10,000 and the largest 4-digit number is 9,999. Since 10,000 is greater than 9,999, every 5-digit number is greater than every 4-digit number. This rule applies to any comparison between numbers with different numbers of digits.
Q3. What does ascending order mean?
Ascending order means arranging numbers from the smallest to the largest. Think of it as climbing stairs, where each step goes higher. For example, 12, 25, 38, 47 is in ascending order. The word ascending comes from ascend, which means to go up.
Q4. What does descending order mean?
Descending order means arranging numbers from the largest to the smallest. Think of it as coming down a slide. For example, 89, 67, 45, 23 is in descending order. The word descending comes from descend, which means to come down.
Q5. How do I remember the greater than and less than symbols?
Think of the symbol as a hungry crocodile's mouth. The crocodile always opens its mouth towards the bigger number because it wants the bigger meal. In 15 > 8, the mouth opens towards 15. In 3 < 10, the mouth opens towards 10. Another trick: the pointed (smaller) end points to the smaller number.
Q6. Can two different-looking numbers be equal?
In terms of value, numbers can look different but be equal. For example, 007 and 7 are the same number. Leading zeros do not change a number's value. But if both numbers have the same digits in the same positions, they are equal. For example, 5,43,210 = 5,43,210.
Q7. Why can't a number start with zero?
By convention, we do not write leading zeros in a number. If we wrote 0523, it would really just be 523, a 3-digit number, not a 4-digit number. So when forming the smallest number from given digits that include 0, we put the smallest non-zero digit first, then put 0 and the remaining digits after it.
Q8. How is comparing numbers useful in daily life?
Comparing numbers helps you make decisions every day. You compare prices when shopping, scores when watching cricket, marks in exams, and distances when travelling. Even when choosing a mobile phone plan, you compare data limits and costs. It is one of the most basic and useful maths skills.
Q9. What if I need to compare more than two numbers?
When comparing more than two numbers, first sort them by the number of digits. Numbers with fewer digits are smaller. Then among numbers with the same number of digits, compare them pair by pair using the digit-by-digit method. This way you can arrange any number of values in ascending or descending order.
Q10. Is 1,00,000 greater than 99,999?
Yes. 1,00,000 is a 6-digit number and 99,999 is a 5-digit number. Since a 6-digit number is always greater than a 5-digit number, 1,00,000 > 99,999. In fact, 1,00,000 is exactly 1 more than 99,999.
