What are Roman numerals?

A system of notation using letters: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

How is the Indian number system different from the International system?

Indian system groups digits as lakhs and crores; International system uses millions and billions.

Finding an approximate value close to the actual value when exact calculation is not needed.

NCERT Solutions for Class 6 Maths Chapter 1 — Knowing Our Numbers

Q: How do you compare large numbers?

Compare the number of digits first. If equal, compare digit by digit from left to right.

Understanding numbers is the foundation of mathematics. In this chapter, you will learn to work with large numbers, compare them, arrange them in order, estimate values, and read Roman numerals. Every solution below follows a step-by-step approach so you can understand the reasoning, not just the answer.

Key Concepts

Concept	What It Means	Example
Large Numbers	Numbers with many digits — thousands, lakhs, crores, millions	5,67,89,432 (Indian), 56,789,432 (International)
Indian Number System	Groups digits as Ones, Tens, Hundreds, Thousands, Lakhs, Crores	23,45,678 reads as twenty-three lakh forty-five thousand six hundred seventy-eight
International Number System	Groups digits as Ones, Thousands, Millions, Billions	2,345,678 reads as two million three hundred forty-five thousand six hundred seventy-eight
Comparison of Numbers	Decide which number is bigger by checking digit count or leftmost digits	9,87,654 > 8,76,543 because 9 > 8 at the leftmost place
Ascending / Descending Order	Arranging numbers from smallest to largest (ascending) or largest to smallest (descending)	Ascending: 123, 345, 567; Descending: 567, 345, 123
Estimation	Approximating a value by rounding off to the nearest ten, hundred, thousand, etc.	4,867 rounded to nearest hundred = 4,900
Roman Numerals	An ancient system using letters: I, V, X, L, C, D, M	XIV = 14, XL = 40, XC = 90

Exercise 1.1 — Comparing Numbers and Forming Numbers

Question 1: Fill in the blanks

Answer

(a) 1 lakh = _______ ten thousand.

Solution

Step 1: 1 lakh = 1,00,000
Step 2: 1 ten thousand = 10,000
Step 3: Divide 1,00,000 by 10,000 = 10

Answer1 lakh = 10 ten thousand

(b) 1 million = _______ hundred thousand.

Solution

Step 1: 1 million = 1,000,000
Step 2: 1 hundred thousand = 100,000
Step 3: Divide 1,000,000 by 100,000 = 10

Answer1 million = 10 hundred thousand

(c) 1 crore = _______ ten lakh.

Solution

Step 1: 1 crore = 1,00,00,000
Step 2: 1 ten lakh = 10,00,000
Step 3: Divide 1,00,00,000 by 10,00,000 = 10

Answer1 crore = 10 ten lakh

(d) 1 crore = _______ million.

Solution

Step 1: 1 crore = 1,00,00,000
Step 2: 1 million = 10,00,000
Step 3: Divide 1,00,00,000 by 10,00,000 = 10

Answer1 crore = 10 million

Question 2: Place commas correctly and write the number

Answer

Write each of the following in the Indian and International system:

(a) Seventeen lakh fifty-seven thousand nine hundred twenty-three

Solution

The number is: 17,57,923
In the Indian system: 17,57,923 — Seventeen lakh fifty-seven thousand nine hundred twenty-three
In the International system: 1,757,923 — One million seven hundred fifty-seven thousand nine hundred twenty-three

(b) Ninety-nine lakh ninety-nine thousand nine hundred ninety-nine

Solution

The number is: 99,99,999
In the Indian system: 99,99,999 — Ninety-nine lakh ninety-nine thousand nine hundred ninety-nine
In the International system: 9,999,999 — Nine million nine hundred ninety-nine thousand nine hundred ninety-nine

Question 3: Insert commas suitably and write the number names

Answer

(a) 87595762 (Indian system)

Solution

Step 1: Start from the right and group the digits: 8,75,95,762
Step 2: Read the number: Eight crore seventy-five lakh ninety-five thousand seven hundred sixty-two

Answer8,75,95,762 — Eight crore seventy-five lakh ninety-five thousand seven hundred sixty-two

(b) 8546283 (Indian system)

Solution

Step 1: Group from the right: 85,46,283
Step 2: Read: Eighty-five lakh forty-six thousand two hundred eighty-three

Answer85,46,283 — Eighty-five lakh forty-six thousand two hundred eighty-three

Question 4: Which is greater?

Answer

(a) 93,05,002 or 79,21,345

Solution

Step 1: Both numbers have 7 digits.
Step 2: Compare the leftmost digit: 9 (in 93,05,002) and 7 (in 79,21,345).
Step 3: Since 9 > 7, the first number is greater.

Answer93,05,002 is greater.

(b) 5,67,89,432 or 5,67,89,340

Solution

Step 1: Both numbers have 9 digits and the first 7 digits are the same (5,67,89,4).
Step 2: Compare the next digit: 3 (in both).
Step 3: Compare the last two digits: 32 vs 40. Since 32 < 40... Wait, let us recheck.
Step 4: Actually, the digits after 5,67,89,4 are: the first number has "32" and the second has "40". But let us compare digit by digit properly.
5,67,89,432 vs 5,67,89,440 — Wait, the second number is 5,67,89,340 which has digits ...9,3,4,0.
Actually let us write them aligned: 5,67,89,432 and 5,67,89,340.
All digits match until the tens place: 3 vs 4. Since 4 > 3, the second number is larger.

Answer5,67,89,340 is greater. (Corrected: actually 5,67,89,432 > 5,67,89,340 because 432 > 340.)

Let me redo: 5,67,89,432 — the hundreds digit after 89 is 4. In 5,67,89,340, after 89 we have 340. So we compare 432 vs 340. Since 4 > 3 in the hundreds place, 5,67,89,432 is greater.

Question 5: Arrange the following in ascending order

Answer

8,47,351 ; 9,47,351 ; 5,47,351 ; 1,47,351 ; 4,47,351

Solution

Step 1: All numbers have 6 digits.
Step 2: Compare the leftmost digit: 8, 9, 5, 1, 4.
Step 3: Arrange in increasing order of the leftmost digit: 1, 4, 5, 8, 9.
Answer (ascending): 1,47,351 ; 4,47,351 ; 5,47,351 ; 8,47,351 ; 9,47,351

Question 6: Arrange the following in descending order

Answer

3,45,678 ; 6,78,901 ; 2,34,567 ; 9,01,234 ; 5,67,890

Solution

Step 1: All numbers have 6 digits.
Step 2: Compare the leftmost digit: 3, 6, 2, 9, 5.
Step 3: Arrange in decreasing order: 9, 6, 5, 3, 2.
Answer (descending): 9,01,234 ; 6,78,901 ; 5,67,890 ; 3,45,678 ; 2,34,567

Question 7: Make the greatest and smallest 4-digit numbers using the digits 5, 3, 7, 1 (without repeating)

Solution

Greatest number: Arrange digits in descending order: 7, 5, 3, 1 → 7531
Smallest number: Arrange digits in ascending order: 1, 3, 5, 7 → 1357

Question 8: Make the greatest and smallest 4-digit numbers using the digits 6, 2, 9, 0 (without repeating)

Solution

Greatest number: Arrange in descending order: 9, 6, 2, 0 → 9620
Smallest number: Arrange in ascending order, but 0 cannot be at the leftmost place. So swap 0 with the next smallest digit: 2, 0, 6, 9 → 2069

Exercise 1.2 — Place Value, Expanded Form, and Number Systems

Question 1: Write the following numbers in the expanded form

Answer

(a) 4,57,923

Solution

Step 1: Identify each digit's place value:
4 → 4,00,000 (four lakh)
5 → 50,000 (fifty thousand)
7 → 7,000 (seven thousand)
9 → 900 (nine hundred)
2 → 20 (twenty)
3 → 3 (three)
Expanded form: 4,00,000 + 50,000 + 7,000 + 900 + 20 + 3 = 4,57,923

(b) 23,07,659

Solution

Place values:
2 → 20,00,000 (twenty lakh)
3 → 3,00,000 (three lakh)
0 → 0 (ten thousand place is zero)
7 → 7,000 (seven thousand)
6 → 600 (six hundred)
5 → 50 (fifty)
9 → 9 (nine)
Expanded form: 20,00,000 + 3,00,000 + 0 + 7,000 + 600 + 50 + 9 = 23,07,659

Question 2: Write the place value and face value of 5 in 5,43,210

Solution

Step 1: The digit 5 is in the lakhs place.
Step 2: Place value of 5 = 5 × 1,00,000 = 5,00,000
Step 3: Face value of 5 = 5 (the digit itself)

AnswerPlace value = 5,00,000; Face value = 5

Question 3: Write the number name of 7,84,52,103 in the Indian system

Solution

Step 1: Place commas in the Indian system: 7,84,52,103
Step 2: Read from left: 7 crore, 84 lakh, 52 thousand, 1 hundred three

AnswerSeven crore eighty-four lakh fifty-two thousand one hundred three

Question 4: Write the number name of 8,345,672 in the International system

Solution

Step 1: Place commas in the International system: 8,345,672
Step 2: Read from left: 8 million, 345 thousand, 6 hundred seventy-two

AnswerEight million three hundred forty-five thousand six hundred seventy-two

Question 5: Population of a city was 2,35,471 in the year 2001. In 2011, it was found to be 2,87,654. What was the increase in population?

Solution

Step 1: Population in 2011 = 2,87,654
Step 2: Population in 2001 = 2,35,471
Step 3: Increase = 2,87,654 − 2,35,471
Step 4: Subtract:
2,87,654
2,35,471
--------
52,183

AnswerThe population increased by 52,183.

Exercise 1.3 — Estimation and Rounding Off

Question 1: Round off the following numbers to the nearest ten

Answer

(a) 4,867

Solution

Step 1: Look at the ones digit: 7.
Step 2: Since 7 ≥ 5, round up. Add 1 to the tens digit: 6 → 7.
Step 3: Replace ones digit with 0.

Answer4,867 rounded to the nearest ten = 4,870

(b) 3,542

Solution

Step 1: Look at the ones digit: 2.
Step 2: Since 2 < 5, round down. Keep the tens digit the same.
Step 3: Replace ones digit with 0.

Answer3,542 rounded to the nearest ten = 3,540

Question 2: Round off the following to the nearest hundred

Answer

(a) 7,845

Solution

Step 1: Look at the tens digit: 4.
Step 2: Since 4 < 5, round down. Keep the hundreds digit the same.
Step 3: Replace tens and ones with 00.

Answer7,845 rounded to the nearest hundred = 7,800

(b) 9,563

Solution

Step 1: Look at the tens digit: 6.
Step 2: Since 6 ≥ 5, round up. Add 1 to the hundreds digit: 5 → 6.
Step 3: Replace tens and ones with 00.

Answer9,563 rounded to the nearest hundred = 9,600

Question 3: Round off the following to the nearest thousand

Answer

(a) 2,56,700

Solution

Step 1: Look at the hundreds digit: 7.
Step 2: Since 7 ≥ 5, round up. Add 1 to the thousands digit: 6 → 7.
Step 3: Replace hundreds, tens, and ones with 000.

Answer2,56,700 rounded to the nearest thousand = 2,57,000

(b) 4,23,100

Solution

Step 1: Look at the hundreds digit: 1.
Step 2: Since 1 < 5, round down. Keep the thousands digit the same.
Step 3: Replace hundreds, tens, and ones with 000.

Answer4,23,100 rounded to the nearest thousand = 4,23,000

Question 4: Estimate: 5,290 + 17,986

Solution (rounding to nearest thousand)

Step 1: Round 5,290 to nearest thousand → 5,000
Step 2: Round 17,986 to nearest thousand → 18,000
Step 3: Estimated sum = 5,000 + 18,000 = 23,000
Verification: Actual sum = 5,290 + 17,986 = 23,276. The estimate 23,000 is reasonably close.

Question 5: Estimate: 8,785 − 4,321

Solution (rounding to nearest thousand)

Step 1: Round 8,785 to nearest thousand → 9,000
Step 2: Round 4,321 to nearest thousand → 4,000
Step 3: Estimated difference = 9,000 − 4,000 = 5,000
Verification: Actual difference = 8,785 − 4,321 = 4,464. The estimate of 5,000 is a reasonable approximation.

Question 6: Estimate the product: 389 × 52

Solution (rounding to nearest hundred and ten)

Step 1: Round 389 to nearest hundred → 400
Step 2: Round 52 to nearest ten → 50
Step 3: Estimated product = 400 × 50 = 20,000
Verification: Actual product = 389 × 52 = 20,228. The estimate is close.

Exercise 1.4 — Roman Numerals

Key Rules for Roman Numerals

Roman	Value
I	1
V	5
X	10
L	50
C	100
D	500
M	1000

Rules: 1. A symbol can be repeated up to 3 times (III = 3, XXX = 30, CCC = 300). 2. V, L, D are never repeated. 3. A smaller symbol before a larger one means subtraction: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900. 4. A smaller symbol after a larger one means addition: VI = 6, XI = 11, LX = 60.

Question 1: Write the Roman numerals for

Answer

(a) 34

Solution

Step 1: Break 34 into tens and ones: 30 + 4
Step 2: 30 = XXX, 4 = IV

Answer34 = XXXIV

(b) 76

Solution

Step 1: Break 76: 70 + 6
Step 2: 70 = LXX, 6 = VI

Answer76 = LXXVI

(c) 99

Solution

Step 1: Break 99: 90 + 9
Step 2: 90 = XC, 9 = IX

Answer99 = XCIX

(d) 148

Solution

Step 1: Break 148: 100 + 40 + 8
Step 2: 100 = C, 40 = XL, 8 = VIII

Answer148 = CXLVIII

Question 2: Write the Hindu-Arabic numerals for

Answer

(a) XLII

Solution

Step 1: XL = 40, II = 2
Step 2: 40 + 2 = 42

AnswerXLII = 42

(b) LXXXV

Solution

Step 1: L = 50, XXX = 30, V = 5
Step 2: 50 + 30 + 5 = 85

AnswerLXXXV = 85

(c) CXLIV

Solution

Step 1: C = 100, XL = 40, IV = 4
Step 2: 100 + 40 + 4 = 144

AnswerCXLIV = 144

(d) CCIX

Solution

Step 1: CC = 200, IX = 9
Step 2: 200 + 9 = 209

AnswerCCIX = 209

Question 3: Which of the following are meaningless Roman numerals?

Answer

(a) VV

AnswerMeaningless. The symbol V (5) cannot be repeated.

(b) IC

AnswerMeaningless. I can only be placed before V and X for subtraction, not before C.

(c) XVV

AnswerMeaningless. V cannot be repeated.

(d) XCIX

AnswerValid. XC = 90, IX = 9 → 99. This is a correct Roman numeral.

Common Mistakes

1. Confusing place value and face value: The face value is the digit itself (e.g., 5 in 5,43,210 has face value 5), but the place value depends on position (5,00,000). Always identify the place first.

2. Incorrect comma placement: In the Indian system, commas go after every two digits from the right (except the first three digits). For example, write 1,23,456 (Indian) not 123,456 (International format in Indian context).

3. Rounding in the wrong direction: When the digit is exactly 5, always round up. For example, 4,850 rounded to the nearest hundred is 4,900, not 4,800.

4. Misreading Roman numerals: Remember that a smaller symbol before a larger one means subtraction, not addition. IV = 4 (not 6), and XC = 90 (not 110).

5. Forgetting that zero cannot start a number: When forming the smallest number, do not place 0 in the leftmost position. For digits 3, 0, 7, 1, the smallest number is 1037, not 0317.

6. Estimation without rounding first: Always round off both numbers before estimating. Do not calculate the exact answer and then round — that defeats the purpose of estimation.

Exam Checklist

[ ] I can read and write large numbers in both the Indian and International number systems with correct comma placement.
[ ] I can compare two large numbers by checking digit count first, then comparing digit by digit from the left.
[ ] I can arrange numbers in ascending and descending order.
[ ] I can form the greatest and smallest numbers from given digits, remembering that zero cannot be the leftmost digit.
[ ] I can write numbers in expanded form showing each digit's place value.
[ ] I can state the place value and face value of any digit in a number.
[ ] I can round off numbers to the nearest ten, hundred, thousand, and lakh.
[ ] I can estimate sums, differences, and products by rounding first, then computing.
[ ] I can convert between Roman numerals and Hindu-Arabic numerals.
[ ] I can identify invalid Roman numerals based on the repetition and subtraction rules.

Frequently Asked Questions

Compare the number of digits first. If equal, compare digit by digit from left to right.

NCERT Solutions for Class 6 Maths Chapter 1 Knowing Our Numbers | KalviGavani

NCERT Solutions for Class 6 Maths Chapter 1 — Knowing Our Numbers

Key Concepts

Exercise 1.1 — Comparing Numbers and Forming Numbers

Exercise 1.2 — Place Value, Expanded Form, and Number Systems

Exercise 1.3 — Estimation and Rounding Off

Exercise 1.4 — Roman Numerals

Key Rules for Roman Numerals

Common Mistakes

Exam Checklist

Frequently Asked Questions

Other question types — same chapter

CBSE Class 6 Maths Chapter 1 Notes — Knowing Our Numbers

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NCERT Solutions for Class 6 Maths Chapter 4 Basic Geometrical Ideas | KalviGavani