Class 6 — Mathematics (NCERT)
Knowing Our Numbers
Class 6
- ✓By the end of this lesson students will be able to understand and use the concept of place value for large numbers.
- ✓By the end of this lesson students will be able to read and write large numbers using both the Indian and International systems of numeration.
- ✓By the end of this lesson students will be able to perform estimation by rounding off numbers to the nearest tens, hundreds, and thousands.
- ✓By the end of this lesson students will be able to understand and use Roman numerals up to 1000.
Key concepts
In the Indian system, we group digits into periods: ones, thousands, lakhs, and crores. The places are Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, Crores, Ten Crores, and so on. Commas are used to separate these periods. The first comma comes after three digits from the right (hundreds place), and then after every two digits. For example, 5,67,89,123 is read as Five crore sixty-seven lakh eighty-nine thousand one hundred twenty-three.
In the International system, we group digits into periods: ones, thousands, and millions. The places are Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions, and so on. Commas are used after every three digits from the right. For example, 56,789,123 is read as Fifty-six million seven hundred eighty-nine thousand one hundred twenty-three.
To compare two numbers, first count the number of digits. The number with more digits is greater. If the number of digits is the same, compare the digits from the leftmost place value. The number with the greater digit at the first differing place is the greater number.
Estimation means finding a value that is close enough to the exact answer. Rounding off helps in estimation.\n* Rounding to the nearest tens: Look at the digit in the ones place. If it is 0, 1, 2, 3, or 4, replace the ones digit with 0 and keep the tens digit as it is. If it is 5, 6, 7, 8, or 9, replace the ones digit with 0 and increase the tens digit by 1.\n* Rounding to the nearest hundreds: Look at the digit in the tens place. If it is 0, 1, 2, 3, or 4, replace the tens and ones digits with 0s and keep the hundreds digit as it is. If it is 5, 6, 7, 8, or 9, replace the tens and ones digits with 0s and increase the hundreds digit by 1.\n* Rounding to the nearest thousands: Look at the digit in the hundreds place. If it is 0, 1, 2, 3, or 4, replace the hundreds, tens, and ones digits with 0s and keep the thousands digit as it is. If it is 5, 6, 7, 8, or 9, replace the hundreds, tens, and ones digits with 0s and increase the thousands digit by 1.
Roman numerals are a system of numerical notation used by the Romans. There are seven basic symbols:\n* I = 1\n* V = 5\n* X = 10\n* L = 50\n* C = 100\n* D = 500\n* M = 1000\nRules for forming Roman Numerals:\n1. Repetition: A symbol can be repeated up to three times to add its value (e.g., II = 2, XXX = 30). V, L, and D are never repeated.\n2. Addition: If a symbol of smaller value is written to the right of a symbol of greater value, their values are added (e.g., VI = 5 + 1 = 6, LX = 50 + 10 = 60).\n3. Subtraction: If a symbol of smaller value is written to the left of a symbol of greater value, the smaller value is subtracted from the greater value (e.g., IV = 5 - 1 = 4, XL = 50 - 10 = 40).\n * V, L, D are never subtracted.\n * I can be subtracted only from V and X.\n * X can be subtracted only from L and C.\n * C can be subtracted only from D and and M.
Key facts to remember
- 1In the Indian system, commas are placed after 3 digits from the right, then after every 2 digits (e.g., 1,23,45,678).
- 2In the International system, commas are placed after every 3 digits from the right (e.g., 123,456,789).
- 3The seven basic Roman numeral symbols are I (1), V (5), X (10), L (50), C (100), D (500), M (1000).
- 4When rounding, if the digit to the right of the rounding place is 5 or greater, round up; otherwise, round down.
- 5The smallest 4-digit number is 1000. The largest 4-digit number is 9999.
Worked examples
Example 1
Write the number 78921092 in words in both the Indian and International systems of numeration.
Answer
Indian System: Seven crore eighty-nine lakh twenty-one thousand ninety-two.\nInternational System: Seventy-eight million nine hundred twenty-one thousand ninety-two.
Pay close attention to the placement of commas in each system.
Example 2
Estimate the sum 5290 + 17986 by rounding off to the nearest thousands.
Answer
The estimated sum is 23000.
Always check the digit to the right of the place you are rounding to.
Example 3
Convert 94 to Roman numeral and XLIX to Hindu-Arabic numeral.
Answer
94 in Roman numeral is XCIV.\nXLIX in Hindu-Arabic numeral is 49.
Remember the subtraction rule: a smaller value symbol placed before a larger value symbol means subtraction.
Common mistakes
- ✗Confusing the placement of commas between the Indian and International systems of numeration.
- ✗Incorrectly applying rounding rules, especially when the digit to be rounded is 5.
- ✗Violating Roman numeral rules, such as repeating V, L, or D, or subtracting I from L or C.
- ✗Writing numbers like 99 as IC instead of XCIX.
- ✗Not identifying the correct place value when asked to round off to a specific place.
Exam tips
- ★Always read the question carefully to determine whether the Indian or International system of numeration is required.
- ★Practice writing numbers in expanded form to strengthen your understanding of place value.
- ★Memorise the basic Roman numeral symbols and their rules thoroughly.
- ★When estimating, clearly state the rounded numbers before performing the operation.
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