ARITHMETIC
NUMBER SYSTEM
1. Introduction
The Number System is one of the most important topics in Arithmetic. It deals with different types of numbers, their properties, relationships, and methods of performing calculations. A strong understanding of the Number System is essential for solving problems in competitive examinations, school mathematics, and everyday calculations.
Numbers are symbols used for counting, measuring, comparing, and performing mathematical operations.
These numbers can be classified into different groups according to their properties.
2. Types of Numbers
2.1 Natural Numbers
The numbers used for counting are called Natural Numbers.
The smallest natural number is 1. Natural numbers do not include zero.
2.2 Whole Numbers
Natural numbers together with zero are called Whole Numbers.
The smallest whole number is 0.
2.3 Integers
Integers consist of positive numbers, negative numbers, and zero.
Positive integers: 1, 2, 3, 4, …
Negative integers: −1, −2, −3, −4, …
Zero: 0
2.4 Rational Numbers
A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is called a Rational Number.
Every integer is also a rational number because:
2.5 Irrational Numbers
Numbers that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0, are called Irrational Numbers.
Their decimal expansions are non-terminating and non-repeating.
2.6 Real Numbers
The combination of rational numbers and irrational numbers is called the set of Real Numbers.
3. Even and Odd Numbers
Even Numbers
Numbers divisible by 2 are called even numbers.
An even number can be written as 2n, where n is an integer.
Odd Numbers
Numbers that are not divisible by 2 are called odd numbers.
An odd number can be written as 2n + 1.
4. Prime and Composite Numbers
Prime Number
A number greater than 1 having exactly two factors, 1 and itself, is called a Prime Number.
2 is the only even prime number.
Composite Number
A number greater than 1 having more than two factors is called a Composite Number.
1 is neither prime nor composite.
5. Factors and Multiples
Factors
A number that divides another number exactly is called a factor.
1, 2, 3, 4, 6, 12
Multiples
The numbers obtained by multiplying a given number by 1, 2, 3, 4, … are called its multiples.
ROMAN NUMERALS
ROMAN NUMBER SYSTEM
6. Introduction to Roman Numerals
Roman Numerals are a number system that originated in ancient Rome. Instead of using the digits 0–9, Roman Numerals use specific letters of the English alphabet to represent numbers.
These are the seven basic Roman numeral symbols.
7. Basic Roman Numeral Values
| Roman Symbol | Value | Meaning / Use |
|---|---|---|
| I | 1 | One |
| V | 5 | Five |
| X | 10 | Ten |
| L | 50 | Fifty |
| C | 100 | One Hundred |
| D | 500 | Five Hundred |
| M | 1000 | One Thousand |
8. Rules of Roman Numerals
Rule 1 — Addition
When a smaller or equal value comes after a larger value, the values are generally added.
XI = 10 + 1 = 11
XV = 10 + 5 = 15
Rule 2 — Subtraction
When a smaller value comes before a larger value in an allowed combination, the smaller value is subtracted from the larger value.
IX = 10 − 1 = 9
XL = 50 − 10 = 40
XC = 100 − 10 = 90
Rule 3 — Repetition
The symbols I, X, C and M can generally be repeated up to three times consecutively.
The symbols V, L and D are not normally repeated.
9. Important Roman Numerals
| Number | Roman | Number | Roman |
|---|---|---|---|
| 1 | I | 10 | X |
| 2 | II | 20 | XX |
| 3 | III | 30 | XXX |
| 4 | IV | 40 | XL |
| 5 | V | 50 | L |
| 6 | VI | 90 | XC |
| 7 | VII | 100 | C |
| 8 | VIII | 400 | CD |
| 9 | IX | 500 | D |
| 10 | X | 900 | CM |
| 11 | XI | 1000 | M |
ROMAN NUMERAL CONVERSION
NATURAL ↔ ROMAN
10. Natural Number → Roman Numeral
To convert a natural number into a Roman numeral, break the number into thousands, hundreds, tens and units. Then replace each part with its corresponding Roman symbol.
5 = V and 3 = III
Therefore, 8 = VIII
10 = X and 4 = IV
Therefore, 14 = XIV
40 = XL and 9 = IX
Therefore, 49 = XLIX
2000 = MM, 20 = XX and 4 = IV
Therefore, 2024 = MMXXIV
11. Roman Numeral → Natural Number
To convert a Roman numeral into a natural number, read the symbols from left to right. Add values when a symbol is followed by a symbol of equal or smaller value. Subtract when a smaller value comes before a larger value in a valid subtractive pair.
Therefore, XVI = 16
Therefore, XIX = 19
Therefore, XLV = 45
Therefore, CMXLIV = 944
12. Important Conversion Table
| Natural | Roman | Natural | Roman |
|---|---|---|---|
| 4 | IV | 40 | XL |
| 9 | IX | 50 | L |
| 14 | XIV | 90 | XC |
| 19 | XIX | 100 | C |
| 24 | XXIV | 400 | CD |
| 29 | XXIX | 500 | D |
| 49 | XLIX | 900 | CM |
| 99 | XCIX | 1000 | M |
13. Twenty Practice Questions
Convert the following numbers and Roman numerals as directed. Try to solve them yourself before checking the answers.
| No. | Question | Type |
|---|---|---|
| 1 | Convert 7 into Roman Numeral. | Natural → Roman |
| 2 | Convert 12 into Roman Numeral. | Natural → Roman |
| 3 | Convert 18 into Roman Numeral. | Natural → Roman |
| 4 | Convert 25 into Roman Numeral. | Natural → Roman |
| 5 | Convert 39 into Roman Numeral. | Natural → Roman |
| 6 | Convert 44 into Roman Numeral. | Natural → Roman |
| 7 | Convert 58 into Roman Numeral. | Natural → Roman |
| 8 | Convert 75 into Roman Numeral. | Natural → Roman |
| 9 | Convert 99 into Roman Numeral. | Natural → Roman |
| 10 | Convert 145 into Roman Numeral. | Natural → Roman |
| 11 | Convert VII into Natural Number. | Roman → Natural |
| 12 | Convert XIII into Natural Number. | Roman → Natural |
| 13 | Convert XXIV into Natural Number. | Roman → Natural |
| 14 | Convert XXXIX into Natural Number. | Roman → Natural |
| 15 | Convert XLVI into Natural Number. | Roman → Natural |
| 16 | Convert LXIX into Natural Number. | Roman → Natural |
| 17 | Convert XCIV into Natural Number. | Roman → Natural |
| 18 | Convert CXXV into Natural Number. | Roman → Natural |
| 19 | Convert CDXL into Natural Number. | Roman → Natural |
| 20 | Convert CMXCIX into Natural Number. | Roman → Natural |
