📚 Bihar Police Sub‑Inspector (SI) – Quantitative Aptitude
Topic: Number System
Objective – Understand the different kinds of numbers, their properties, and how to work with them in everyday quantitative problems.
1. Simple Explanation
The number system is the set of all numbers that we can use in mathematics. It is organised into several layers, each adding more types of numbers:
| Layer | What it contains | Key Examples |
|---|---|---|
| Natural Numbers | Counting numbers, no zero | 1, 2, 3, … |
| Whole Numbers | Natural numbers + 0 | 0, 1, 2, … |
| Integers | Whole numbers + negative counterparts | … –3, –2, –1, 0, 1, 2, 3 … |
| Rational Numbers | Numbers that can be written as a fraction of two integers (denominator ≠ 0) | ½, 4, –3/7 |
| Irrational Numbers | Cannot be expressed as a fraction; infinite non‑repeating decimal | √2, π, e |
| Real Numbers | All rationals + irrationals | All numbers that can be represented on the number line |
| Complex Numbers | Numbers of the form a + bi (i² = –1) | 3 + 4i, –2i |
Why it matters – SI exams ask you to identify the type of a number, test divisibility, work with fractions, and solve problems that involve converting between different forms (e.g., fractional to decimal or decimal to fraction).
2. Key Points
| Concept | What to Remember | Quick Tip |
|---|---|---|
| Place Value | Every digit’s value depends on its position (units, tens, hundreds, etc.) | 4 385 = 4 × 10³ + 3 × 10² + 8 × 10¹ + 5 × 10⁰ |
| Divisibility Rules | Quick ways to check if a number is divisible by another (useful for prime checks, factorisation) | • 2 – last digit even • 3 – sum of digits divisible by 3 • 5 – last digit 0 or 5 • 9 – sum of digits divisible by 9 |
| Prime vs. Composite | Prime: only divisible by 1 and itself. Composite: has other divisors. | Use divisibility rules or trial division up to √n |
| Factor & Multiple | Factor: number that divides another without remainder. Multiple: number that is divisible by the given number. | If a is a factor of b, then b = a × k |
| Greatest Common Divisor (GCD) | Largest number that divides two numbers. | Use prime factorisation or Euclidean algorithm |
| Least Common Multiple (LCM) | Smallest number that is a multiple of two numbers. | LCM = ( |
| Decimal Expansion | Terminating (ends) vs. repeating (cycle). | Fraction with denominator having only 2 & 5 → terminating |
| Fraction to Decimal | Divide numerator by denominator. | Keep track of repeating pattern |
| Decimal to Fraction | Multiply by power of 10 to clear decimals, then reduce. | 0.75 = 75/100 = 3/4 |
| Number Bases | Binary (base‑2), Octal (base‑8), Decimal (base‑10), Hex (base‑16). | Useful for computer‑related questions |
3. Worked Example
Problem
Find the sum of all positive integers less than 30 that are divisible by 4 or 6.
Step‑by‑Step Solution
-
Identify multiples of 4 under 30
4, 8, 12, 16, 20, 24, 28
Sum₄ = 4 + 8 + 12 + 16 + 20 + 24 + 28 = 112 -
Identify multiples of 6 under 30
6, 12, 18, 24
Sum₆ = 6 + 12 + 18 + 24 = 60 -
Avoid double‑counting – numbers divisible by both 4 and 6 are divisible by LCM(4,6)=12.
Multiples of 12 under 30: 12, 24
Sum₁₂ = 12 + 24 = 36 -
Apply Inclusion‑Exclusion
Total = Sum₄ + Sum₆ – Sum₁₂
Total = 112 + 60 – 36 = 136
Answer: 136
Quick Check
- 112 (from 4‑multiples)
- 60 (from 6‑multiples)
- Subtract 36 (shared) → 136 ✔️
4. Quick Revision Summary
| Topic | What to Review | Sample Question |
|---|---|---|
| Divisibility Rules | 2,3,4,5,6,9,10 | Is 1,368 divisible by 6? |
| Prime vs. Composite | Identify primes up to 100 | Is 91 prime? |
| GCD & LCM | Euclidean algorithm, LCM formula | GCD(84, 90) = ? |
| Decimal ↔ Fraction | Conversion methods | Convert 0. |