1. The Indian System of Numeration
This system is primarily used in India. It groups digits into periods: Units, Thousands, Lakhs, and Crores.
- Place Value Chart: Units, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, Crores, Ten Crores.
- Commas: The first comma is placed after the hundreds place (three digits from the right). Subsequent commas are placed after every two digits.
- Example: 12,34,56,789 (Twelve crore, thirty-four lakh, fifty-six thousand, seven hundred eighty-nine).
2. The International System of Numeration
This system is used globally. It groups digits into periods of threes: Units, Thousands, and Millions.
- Place Value Chart: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions.
- Commas: Commas are placed after every three digits from the right.
- Example: 123,456,789 (One hundred twenty-three million, four hundred fifty-six thousand, seven hundred eighty-nine).
3. Comparison & Conversion Table
The most important part of this chapter is knowing where the two systems meet and where they differ.
| Indian System | International System | Power of 10 |
| 1 Lakh | 100 Thousand | 10^5 |
| 10 Lakh | 1 Million | 10^6 |
| 1 Crore | 10 Million | 10^7 |
| 10 Crore | 100 Million | 10^8 |
4. Face Value vs. Place Value
- Face Value: The actual value of the digit itself (The face value of 5 in 5,000 is just 5).
- Place Value: The value represented by a digit in a number on the basis of its position.
- Formula: Place Value = Face Value $\times$ Value of the Place
5. Expanded Form
Writing a number as the sum of the place values of its digits.
- Example (Indian): 5,43,210 = $(5 \times 1,00,000) + (4 \times 10,000) + (3 \times 1,000) + (2 \times 100) + (1 \times 10) + 0$
EXERCISE 3
Question 1.
Write the following numerals using Indian System words or International system (as
required in words) :
(i) 4, 35, 342 = …………..
(ii) 36, 71, 430 = …………..
(iii) 4, 28, 30, 004 = …………..
(iv) 75, 132, 684 = ……………
(v) 815, 906 = …………….
(vi) 5, 420, 700 = ……………..
Solution :
(i) 4, 35, 342
- System: Indian (Comma after 3 digits, then 2).
- In Words: Four lakh, thirty-five thousand, three hundred forty-two.
(ii) 36, 71, 430
- System: Indian (Comma after 3 digits, then 2).
- In Words: Thirty-six lakh, seventy-one thousand, four hundred thirty.
(iii) 4, 28, 30, 004
- System: Indian (Grouped in lakhs and crores).
- In Words: Four crore, twenty-eight lakh, thirty thousand, four.
(iv) 75, 132, 684
- System: International (Groups of 3).
- In Words: Seventy-five million, one hundred thirty-two thousand, six hundred eighty-four.
(v) 815, 906
- System: International (Group of 3).
- In Words: Eight hundred fifteen thousand, nine hundred six.
(vi) 5, 420, 700
- System: International (Groups of 3).
- In Words: Five million, four hundred twenty thousand, seven hundred.
Question 2.
Write the following numbers, placing the commas, according to Indian system :
(i) 835629 = …………
(ii) 35640254 = …………
(iii) 2826040 = ………
Solution :
(i) 835629
- Count 3 digits from the right: 629
- Count 2 digits from there: 35
- Result: 8,35,629
- (In words: Eight lakh, thirty-five thousand, six hundred twenty-nine)
(ii) 35640254
- Count 3 digits from the right: 254
- Count 2 digits: 40
- Count 2 digits: 56
- Result: 3,56,40,254
- (In words: Three crore, fifty-six lakh, forty thousand, two hundred fifty-four)
(iii) 2826040
- Count 3 digits from the right: 040
- Count 2 digits: 26
- Result: 28,26,040
- (In words: Twenty-eight lakh, twenty-six thousand, forty)
Question 3.
Write the following numbers, by placing the commas, according to International system :
(i) 6509820 = ………..
(ii) 428140584 = …………
(iii) 63560981 = ……….
Solution :
(i) 6509820
- Count 3 digits from the right: 820
- Count the next 3 digits: 509
- Result: 6,509,820
- (In words: Six million, five hundred nine thousand, eight hundred twenty)
(ii) 428140584
- Count 3 digits from the right: 584
- Count the next 3 digits: 140
- Result: 428,140,584
- (In words: Four hundred twenty-eight million, one hundred forty thousand, five hundred eighty-four)
(iii) 63560981
- Count 3 digits from the right: 981
- Count the next 3 digits: 560
- Result: 63,560,981
- (In words: Sixty-three million, five hundred sixty thousand, nine hundred eighty-one)
Question 4.
Fill in the blanks :
(i) Four lakh sixty seven thousand three hundred six.
= ……………… (In numeral form)
= …………… (In International System)
= ……………. (In International numeration)
(ii) Thirteen lakh forty five.
= ……………. (In numeral form)
= ……………. (In International System)
= ……………. (In International numerals)
Solution :
(i) Four lakh sixty-seven thousand three hundred six.
- In Numeral Form (Indian): First, write the digits: 4 (Lakhs), 67 (Thousands), 306 (Units). = 4,67,306
- In International System (Commas): Take the digits 467306 and place commas every 3 digits from the right. = 467,306
- In International Numeration (Words): Read the International grouping (467 thousand + 306). = Four hundred sixty-seven thousand three hundred six.
(ii) Thirteen lakh forty-five.
- In Numeral Form (Indian): Be careful with the placeholders! We have 13 (Lakhs), 00 (Thousands), and 045 (Units). = 13,00,045
- In International System (Commas): Take the digits 1300045 and place commas every 3 digits from the right. = 1,300,045
- In International Numerals (Words): Read the International grouping (1 million + 300 thousand + 45). = One million three hundred thousand forty-five.
Question 5.
Fill in the blanks :
(i) Six hundred four thousand eight hundred forty seven.
= ……………. (In numeral form)
= ……………. (In Indian System)
= ……………… (Number name in Indian System)
(ii) Two million three hundred ten thousand one hundred four
= …………….. (In numeral form)
= …………… (In Indian System)
= ……………. (In Indian numeration)
= …………….. (In international numerals)
Solution :
(i) Six hundred four thousand eight hundred forty-seven.
- In Numeral Form (International): Combine 604 (Thousands) and 847 (Units). = 604,847
- In Indian System (Commas): Take the digits 604847 and place commas after 3 digits, then 2 digits. = 6,04,847
- Number Name in Indian System: Read the Indian grouping (6 Lakh + 04 Thousand + 847). = Six lakh four thousand eight hundred forty-seven.
(ii) Two million three hundred ten thousand one hundred four.
- In Numeral Form (International): Combine 2 (Millions), 310 (Thousands), and 104 (Units). = 2,310,104
- In Indian System (Commas): Take the digits 2310104 and place commas after 3 digits, then 2, then 2. = 23,10,104
- In Indian Numeration (Words): Read the Indian grouping (23 Lakh + 10 Thousand + 104). = Twenty-three lakh ten thousand one hundred four.
- In International Numerals: This is the figure with International commas. = 2,310,104
