Number System
Master important formulas and concepts with our comprehensive guide
Important Formulas & Concepts
Study MaterialNumber System
The Number System is one of the most important topics in Quantitative Aptitude. It forms the foundation for arithmetic, algebra, simplification, divisibility, HCF & LCM, remainder theorem, decimal fractions, and advanced aptitude problems asked in SSC, Banking, Railway, CAT, UPSC, and campus placement exams.
A number system is a mathematical system used to represent numbers in a consistent and logical manner. It includes different categories of numbers along with their properties, operations, and relationships.
Why Number System is Important?
- Frequently asked in SSC, Railway, Banking, CAT, CDS, and State exams.
- Improves calculation speed and logical thinking.
- Forms the base of arithmetic and algebra.
- Useful in simplification, divisibility, remainder, and approximation questions.
- Important for quantitative aptitude and reasoning sections.
Digits and Numerals
In the Hindu-Arabic Number System, we use the symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
These symbols are called digits.
✔ 0 is called an insignificant digit.
✔ Digits from 1 to 9 are called significant digits.
A mathematical representation formed using digits is called a numeral.
Example: 4589, 92314, 78654
Face Value and Place Value
Face Value
The face value of a digit is the value of the digit itself, irrespective of its position in the number.
Example: In the number 486729, the face value of 8 is 8 and the face value of 7 is 7.
Place Value
The place value of a digit depends on its position in the numeral.
Example: In the number 72843016:
| Digit | Place | Place Value |
|---|---|---|
| 7 | Ten Lakhs | 70,00,000 |
| 2 | Lakhs | 2,00,000 |
| 8 | Ten Thousands | 80,000 |
| 4 | Thousands | 4,000 |
| 3 | Hundreds | 300 |
| 1 | Tens | 10 |
| 6 | Ones | 6 |
✔ Place Value = Digit × Value of Position
Types of Numbers
1. Natural Numbers
Natural numbers are counting numbers and are denoted by N.
Example: N = {1, 2, 3, 4, 5, ...}
- All natural numbers are positive.
- 1 is the smallest natural number.
- 0 is not a natural number.
2. Whole Numbers
Whole numbers include all natural numbers together with zero.
Example: W = {0, 1, 2, 3, 4, ...}
- 0 is the smallest whole number.
- Whole numbers are non-negative integers.
3. Integers
Integers include positive numbers, negative numbers, and zero.
Example: I = {... -3, -2, -1, 0, 1, 2, 3 ...}
Types of Integers
- Positive Integers: 1, 2, 3, 4...
- Negative Integers: -1, -2, -3, -4...
✔ 0 is neither positive nor negative.
4. Even Numbers
A number divisible by 2 is called an even number.
Example: 2, 4, 6, 8, 10...
- The unit digit of every even number is 0, 2, 4, 6, or 8.
5. Odd Numbers
A number that is not divisible by 2 is called an odd number.
Example: 1, 3, 5, 7, 9...
- The unit digit of every odd number is 1, 3, 5, 7, or 9.
6. Prime Numbers
A prime number has exactly two factors: 1 and the number itself.
Example: 2, 3, 5, 7, 11...
- 2 is the only even prime number.
- Every prime number is greater than 1.
- 1 is not a prime number.
- The smallest odd prime number is 3.
Important Property:
Every prime number greater than 3 can be represented in the form:
6n ± 1
7. Composite Numbers
Composite numbers are natural numbers having more than two factors.
Example: 4, 6, 8, 9, 12...
- Composite numbers can be odd or even.
- 1 is neither prime nor composite.
8. Co-prime Numbers
Two numbers are called co-primes if their HCF is 1.
Example: (7, 9), (15, 16)
- Co-prime numbers may or may not be prime.
9. Rational Numbers
A number that can be written in the form p/q where q ≠ 0 is called a rational number.
Example: 3/5, 7/9, 13/15
10. Irrational Numbers
Numbers that cannot be represented in the form p/q are called irrational numbers.
Example: √2, √3, √5, π
- Non-terminating and non-repeating decimals are irrational numbers.
- π is an irrational number.
11. Real Numbers
The combination of rational and irrational numbers is called real numbers.
Example: 5/9, √2, π
- Real numbers are represented by R.
Important Formulas of Number System
Formulas of Number Series
- 1 + 2 + 3 + 4 + ... + n = n(n + 1) / 2
- 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1) / 6
- 1³ + 2³ + 3³ + ... + n³ = [n(n + 1) / 2]²
- Sum of first n odd numbers = n²
- Sum of first n even numbers = n(n + 1)
Important Algebraic Identities
| Identity | Formula |
|---|---|
| Difference of Squares | (a + b)(a − b) = a² − b² |
| Square of Sum | (a + b)² = a² + b² + 2ab |
| Square of Difference | (a − b)² = a² + b² − 2ab |
| Cube Sum Identity | a³ + b³ = (a + b)(a² − ab + b²) |
| Cube Difference Identity | a³ − b³ = (a − b)(a² + ab + b²) |
Divisibility Rules
Divisibility rules help in solving aptitude questions quickly without performing long division.
| Number | Divisibility Rule |
|---|---|
| 2 | Last digit must be 0, 2, 4, 6, or 8 |
| 3 | Sum of digits must be divisible by 3 |
| 4 | Last two digits must be divisible by 4 |
| 5 | Last digit must be 0 or 5 |
| 6 | Number must be divisible by both 2 and 3 |
| 8 | Last three digits must be divisible by 8 |
| 9 | Sum of digits must be divisible by 9 |
| 10 | Last digit must be 0 |
| 11 | Difference of alternate digit sums must be divisible by 11 |
Recurring Decimal to Fraction Conversion
Recurring decimals can easily be converted into fractions using standard formulas.
- 0.3333... = 3/9 = 1/3
- 0.232323... = 23/99
- 0.125125125... = 125/999
Important Exam Tips
- Learn divisibility rules thoroughly.
- Memorize prime numbers up to 100.
- Practice squares and cubes regularly.
- Understand algebraic identities properly.
- Focus on remainder and unit digit concepts.
- Revise number series formulas frequently.
The Number System chapter is the backbone of Quantitative Aptitude. Strong command over concepts, formulas, divisibility rules, and number properties helps in solving aptitude questions faster and accurately in competitive examinations.