Number System : 1a
NUMBER SYSTEM
Knowing about Numbers & Four Fundamental Operations
| Operation | Symbol | Meaning | Example |
|---|---|---|---|
| Addition | + | Combining quantities | 35 + 15 = 50 |
| Subtraction | − | Finding difference | 50 − 18 = 32 |
| Multiplication | × | Repeated addition | 6 × 7 = 42 |
| Division | ÷ | Equal sharing | 48 ÷ 6 = 8 |
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | Yes | Yes |
| Associative | Yes | Yes |
| Distributive | No | Yes (× over +, −) |
BODMAS rule helps in correct order of operations in mixed calculations.
Always solve brackets first before applying multiplication or division.
Hindu-Arabic & International System of Numeration
| Indian System | Period | International System | Period |
|---|---|---|---|
| Ones | 1 | Ones | 1 |
| Thousands | 1,000 | Thousands | 1,000 |
| Lakhs | 1,00,000 | Millions | 1,000,000 |
| Crores | 1,00,00,000 | Billions | 1,000,000,000 |
| Number | Indian System | International System |
|---|---|---|
| 12345678 | 1,23,45,678 | 12,345,678 |
Indian system uses Lakhs and Crores, while International system uses Millions and Billions.
Always place commas correctly before reading large numbers in exams.
Place Value and Face Value
| Digit | Place | Face Value | Place Value |
|---|---|---|---|
| 7 (in 7,253) | Thousands | 7 | 7000 |
| 5 (in 7,253) | Tens | 5 | 50 |
| Term | Meaning |
|---|---|
| Face Value | Digit itself |
| Place Value | Face value × position value |
Face value never changes, but place value depends on position.
In comparison problems, start checking from leftmost digit.
Whole Numbers, Even & Odd Numbers
| Type | Definition | Examples |
|---|---|---|
| Whole Numbers | Natural numbers + 0 | 0, 1, 2, 3… |
| Even Numbers | Divisible by 2 | 4, 6, 10 |
| Odd Numbers | Not divisible by 2 | 3, 5, 9 |
| Property | Even + Even | Even + Odd | Odd + Odd |
|---|---|---|---|
| Result | Even | Odd | Even |
Zero is the smallest whole number and neither positive nor negative.
Check last digit to quickly identify even or odd numbers.
Factors, Multiples & Divisibility Tests
| Term | Meaning | Example |
|---|---|---|
| Factor | Divides a number exactly | 4 is factor of 20 |
| Multiple | Product of number and integer | 20 is multiple of 4 |
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit even | 246 ✅ |
| 3 | Sum of digits divisible by 3 | 183 ✅ |
| 5 | Ends in 0 or 5 | 205 ✅ |
| 9 | Sum of digits divisible by 9 | 729 ✅ |
Every factor of a number is less than or equal to the number.
Use divisibility tests to simplify LCM and HCF quickly.
Prime & Composite Numbers, Prime Factorization
| Type | Definition | Examples |
|---|---|---|
| Prime | Exactly 2 factors | 2, 3, 5, 7 |
| Composite | More than 2 factors | 4, 6, 8, 9 |
| Number | Prime Factorization |
|---|---|
| 60 | 2² × 3 × 5 |
| 84 | 2² × 3 × 7 |
2 is the only even prime number.
Prime factorization helps find both HCF and LCM faster.
HCF (GCD) and LCM
| Method | HCF | LCM |
|---|---|---|
| Factor Method | Common lowest powers | Highest powers |
| Division Method | Repeated division | Product ÷ HCF |
| Relation | Formula |
|---|---|
| LCM × HCF | Product of two numbers |
LCM of two numbers ≥ larger number, HCF ≤ smaller number.
Use division method for large numbers in competitive exams.
Fractions and Decimals
| Type | Definition | Example |
|---|---|---|
| Proper Fraction | Numerator < Denominator | 3/5 |
| Improper Fraction | Numerator ≥ Denominator | 7/4 |
| Decimal | Fraction with denominator powers of 10 | 0.75 |
| Operation | Rule |
|---|---|
| Add/Subtract | Same denominator |
| Multiply | Numerators × Denominators |
| Divide | Multiply by reciprocal |
0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4
Convert decimals to fractions during tricky calculations.
Rational Numbers
| Term | Meaning | Example |
|---|---|---|
| Rational Number | p/q where q ≠ 0 | 5/7, −3/2 |
| Recurring Decimal | Repeating digits | 0.333… |
| Property | Result |
|---|---|
| Closure | Closed for +, −, × |
| Multiplicative Inverse | Reciprocal exists (except 0) |
All terminating and recurring decimals are rational numbers.
Always check denominator is not zero in rational forms.
Irrational Numbers & Real Numbers
| Type | Definition | Example |
|---|---|---|
| Irrational Numbers | Non-terminating, non-repeating | √2, π |
| Real Numbers | Rational + Irrational | All points on number line |
| Exponent Rule | Law |
|---|---|
| am × an | am+n |
| am ÷ an | am−n |
Irrational numbers cannot be written in p/q form.
Memorize exponent laws for faster simplification.
Squares, Square Roots, Cubes and Cube Roots
| Term | Meaning | Example |
|---|---|---|
| Square | n × n | 7² = 49 |
| Cube | n × n × n | 3³ = 27 |
| Number | Square Root | Cube Root |
|---|---|---|
| 64 | 8 | 4 |
Square of even number is even, square of odd is odd.
Remember squares and cubes up to 25 for competitive exams.
Sets and Venn Diagrams
| Type | Definition | Example |
|---|---|---|
| Empty Set | No elements | {} |
| Universal Set | Contains all elements | U |
| Subset | Contained within another set | A ⊆ B |
| Operation | Symbol | Meaning |
|---|---|---|
| Union | ∪ | All elements |
| Intersection | ∩ | Common elements |
Venn diagrams visually show relationships between sets.
Always draw diagrams to solve set-based word problems.