Number System
NUMERATION SYSTEM
Indian System of Numeration
In Indian number system, ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used to write a number.
Each of these ten symbols is called a digit. Out of these digits:
i) 0, 2, 4, 6 and 8 are even numbers
ii) 1, 3, 5, 7 and 9 are odd numbers
According to Indian Numeration System, first comma comes after three digits from the right and the next comma comes after every two digit.
For a given number, we start from the extreme right as Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, etc.
In Indian system to read and write a large number with ease, number is split up into groups or periods.
PERIOD | Place Value | Number | Zeroes |
---|---|---|---|
Ones | Ones | 1 | 0 |
Tens | Tens | 10 | 1 |
Hundreds | Hundreds | 100 | 2 |
Thousands | Thousands | 1,000 | 3 |
Ten- Thousands | 10,000 | 4 | |
Lakhs | Lakhs | 1,00,000 | 5 |
Ten-Lakhs | 10,00,000 | 6 | |
Crores | Crores | 1,00,00,000 | 7 |
Ten-Crores | 10,00,00,000 | 8 |
1. Periods are not written in plural
2. The word ‘and’ is not used before tens and ones
3. Do not put commas while writing the numeral.
4. While reading a number, all digits in the same period are read
together and the name of the period (except the ones) is read
along with it.
Example: Write the following in words according to Indian
Numeration system
i) 3673458 ii) 945329
PERIOD | Lakhs | Thousands | Ones | ||||
---|---|---|---|---|---|---|---|
Ten Lakhs |
Lakhs | Ten Thousands |
Thousands | Hundreds | Tens | Ones | |
i) | 3 | 6 | 7 | 3 | 4 | 5 | 8 |
ii) | 9 | 4 | 5 | 3 | 2 | 9 |
i) Thirty six lakh seventy three thousand four hundred fifty eight
ii) Nine lakh forty five thousand three hundred twenty nine
Example: Write the numerals and place the commas.
i) Six crores two lakh twenty one thousand two hundred sixty
three.
ii) Seven lakh nine thousand nine hundred nineteen
PERIOD | Crores | Lakhs | Thousands | Ones | ||||||
Ten Crores |
Crores | Ten Lakhs |
Lakhs | Ten Thousands |
Thousands | Hundreds | Tens | Ones | ||
i) | 6 | 0 | 2 | 2 | 1 | 2 | 6 | 3 | ||
ii) | 7 | 0 | 9 | 9 | 1 | 9 |
i) 6,02,21,263 ii) 7,09,919
International System of Numeration
According to International System of Numeration, the first comma is placed after the hundreds place and the next commas are placed after every three digits and this grouping of number is called period.
In International System of Numeration we use ones, tens, hundreds, thousands, millions and billions.
PERIOD | Place Value | Number | Zeroes |
---|---|---|---|
Ones | Ones | 1 | 0 |
Tens | Tens | 10 | 1 |
Hundreds | Hundreds | 100 | 2 |
Thousands | Thousands | 1,000 | 3 |
Ten- Thousands | 10,000 | 4 | |
Hundred-Thousands | 100,000 | 5 | |
Millions | Million | 1,000,000 | 6 |
Ten Million | 10,000,000 | 7 | |
Hundred Million | 100,000,000 | 8 | |
Billions | Billion | 1,000,000,000 | 9 |
Ten Billion | 10,000,000,000 | 10 | |
Hundred Billion | 100,000,000,000 | 11 |
While reading a number in this system, all digits in the same period
are read together and the name of the period (except the ones) is
read along with it.
Example: Write the following in words according to International
Numeration system
i) 45145879632 ii) 61346987
PERIOD | Billions | Millions | Thousands | Ones | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Ten Billions |
Billions | Hundred Millions |
Ten Millions |
Millions | Hundred Thousands |
Ten Thousands |
Thousands | Hundreds | Tens | Ones | |
i) | 4 | 5 | 1 | 4 | 5 | 8 | 7 | 9 | 6 | 3 | 2 |
ii) | 6 | 1 | 3 | 4 | 6 | 9 | 8 | 7 |
i) Forty five billion one hundred forty five million eight hundred seventy nine thousand six hundred thirty two
ii) Sixty one million three hundred forty six thousand nine hundred eighty seven.
Example: Write the numerals and place the commas.
i) Forty two million two hundred eighty seven thousand four hundred fifty seven
ii) Five million thirty six thousand nine
PERIOD | Millions | Thousands | Ones | ||||||
---|---|---|---|---|---|---|---|---|---|
Hundred Millions |
Ten Millions |
Millions | Hundred Thousands |
Ten Thousands |
Thousands | Hundreds | Tens | Ones | |
i) | 4 | 2 | 2 | 8 | 7 | 4 | 5 | 7 | |
ii) | 5 | 0 | 3 | 6 | 0 | 0 | 9 |
i) 42,287,457 ii) 5,036,009
Important Question
Example: Use the given digits without repetition and make the largest and the smallest number
i) 5, 4, 3, 7
- To get the largest number we will arrange the digits in descending order. i.e. 7, 5, 4, 3. Therefore, the largest number is 7543.
- Now, to get the smallest number, we will arrange the given digits in ascending order .i.e. 3, 4, 5, 7. Therefore, the smallest number is 3457.
ii) 1, 4, 9, 2
- To get the largest number we will arrange the digits in descending order. i.e. 9, 4, 2, 1. Therefore, the largest number is 9421.
- Now, to get the smallest number, we will arrange the given digits in ascending order .i.e. 1, 2, 4, 9. Therefore, the smallest number is 1249
Roman Numerals
System of Roman Numerals is one of the earliest systems of writing numerals.
There are seven basic symbols to write any numeral.
Rules for Roman Numerals
Rule 1: If a symbol is repeated, its value is added as many times it occurs.
a) A symbol cannot be repeated more than three times.
b) Only I, X, C, and M can be repeated.
c) V, L and D are never repeated.
Example: Write the following in Hindu Arabic numeral
i) II = 1 + 1 = 2
ii) XXX = 10 + 10 + 10 = 30
iii) CC = 100 + 100 = 200
iv) MM = 1000 + 1000 = 2000
Rule 2:
If a symbol of smaller value is written to the right of the symbol of greater value, its value gets added to the value of the greater symbol.
Example: Write the following in Hindu Arabic numeral
i) VI = 5 + 1= 6 i
ii) LXII = 50 + 10 + 1 + 1 = 62
iii) MCV = 1000 + 100 + 5 = 1105
iv) XXXV = 10 + 10 + 10 + 5 = 35
v) LXXI = 50 + 10 + 10 + 1 = 71
Rule 3:
If a symbol of smaller value is written to the left of the symbol of greater value, its value gets subtracted from the value of the greater symbol.
a) V, L and D are never subtracted.
b) I can be subtracted from V and X only.
c) X can be subtracted from L and C only.
d) C can be subtracted from D and M only.
Example: Write the following in Hindu Arabic numeral
i) IV = 5 – 1 = 4
ii) XL = 50 – 10 = 40
iii) CD = 500 – 100 = 400
Rule 4:
If a symbol of smaller value is placed between two symbols of larger value, then its value is always subtracted from the value of the symbol immediately following it.
Example: Write the following in Hindu Arabic numeral
i) XIX = 10 + (10 – 1) = 10 + 9 = 19
ii) CXL = 100 + (50 – 10) = 100 + 40 = 140
iii) MCD = 1000 + (500 – 100) = 1000 + 400 = 1400
iv) CCCXIV = 100 + 100 + 100 + 10 + 4 = 314
v) CDXV = 500 – 100 + 10 + 5 = 400 + 10 + 5 = 415
Example: Write the following in Roman numeral
i) 411 = 400 + 10 + 1 = (500-100) + 10 + 1 = CDXI
ii) 265 = 200 + 60 + 5 = (100 + 100) + (50+10) + 5 = CCLXV
iii) 501 = 500 + 1 = DI
iv) 421 = 400 + 20 + 1 = (500 – 100) + (10 + 10) + 1 = CDXXI
v) 753 = 700 + 50 + 3 = (500 + 100 + 100) + 50 + 3 = DCCLIII
1) If 30% of a certain number is 12.6, what is the number?
- 24
- 42
- 23
- 32
2) One-fourth of one-third of two-fifth of a number is 15. What will be 40% of that number?
- 120
- 180
- 270
- 350
3) A number exceeds 20% of itself by 40. What is the number?
- 50
- 60
- 80
- 320
4) The difference between two numbers is 14, and their sum is 40. Find out the product of these two numbers?
- 960
- 280
- 351
- Data inadequate.
5) A number, when divided by 4, is reduced by 21. What is the number?
- 18
- 20
- 28
- 38
6) Two different natural numbers are such that their product is less than their sum. One of the numbers must be
- 1
- 2
- 3
- None of these.
7) Two-third of one-fifth of one-fourth of a number is 10. What is 30% of that number?
- 60
- 100
- 270
- 90
8) The sum of two consecutive odd numbers in a set of three consecutive odd numbers is 5 more than the third number. What is the second of these numbers?
- 5
- 7
- 9
- 11
9) The ratio between a two-digit number and the sum of the digits of that number is 4:1. If the digit in the unit place is 3 more than the digit in the ten’s place, what is that number?
- 24
- 63
- 36
- None of these.
10) The difference between a two-digit number and the number obtained by interchanging the digits is 27. What is the difference between the digits of the number?
- 3
- 6
- 9
- 5
11) The difference between the digits of a two-digit number is 4. What is the digit in the unit’s place? To find out the answer, which of the information given in the statements P and Q is/are sufficient?
P: The difference between the number and the number obtained by interchanging the positions of the digits is 36.
Q: The sum of the digits of that number is 12.
- Only P is sufficient
- Only Q is sufficient
- Both P and Q are needed
- Either P or Q is sufficient.
12) The sum of the digits of a two-digit number is 14, and the difference between the digits is 2. Which of the following can be that number?
- 95
- 68
- 86
- 68 or 86
13) If one-seventh of a number exceeds its eleventh part by 100, what is the number?
- 770
- 1100
- 1825
- 1925
14) The difference between two numbers is 1550. If the larger number is divided by the smaller one, the quotient is 6, and the remainder is 20. What is the smaller number?
- 306
- 308
- 310
- 312
15) The sum and difference of two numbers are 25 and 10 respectively. Find the difference of their squares.
- 190
- 200
- 210
- 220
16) The product of two numbers is 300, and the sum of their squares is 625. What are the numbers?
- 1025
- 1125
- 1225
- 1325
17) The difference between a two-digit number and the number obtained by interchanging the position of the digits is 45. What is the difference between digits of that number?
- 6
- 5
- 7
- Data inadequate.
18) When the numerator of a certain fraction is increased by 1 and the denominator by 2, it changes to ; but when the numerator is increased by 5 and the denominator by 1, the value of the fraction changes to . What is the original fraction?
19) The difference between a two-digit number and the number obtained by changing the positions of its digits is 36. What is the difference between the two digits of that number?
- 3
- 4
- 9
- Cannot be determined.
- None of these.
20) A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. Find the number.
- 18
- 24
- 42
- 81