Odisha Forest Guard – Quantitative Aptitude
Topic: Number System
Aim:
Understand the basics of different number systems (decimal, binary, octal, hexadecimal, etc.), how to convert between them, and key properties that are frequently tested in the exam.
📌 1. Simple Explanation
| Concept | Definition |
|---|---|
| Number System | A way of representing numbers using a set of symbols (digits) and a base (radix). |
| Base (radix) | The number of unique digits, including zero, that a number system uses. E.g., decimal (base‑10) uses digits 0–9. |
| Places / Positions | Each digit in a number has a place value equal to the base raised to an exponent. |
| Conversion | Translating a number from one base to another using repeated division (to base‑10) or repeated multiplication (from base‑10). |
Why it matters for the Forest Guard exam
- Many questions involve converting numbers, checking divisibility, or interpreting codes written in different bases (binary, octal, hex).
- Knowing quick tricks saves time on the test.
📚 2. Key Points
| # | Topic | Important Notes |
|---|---|---|
| 1 | Decimal (Base‑10) | Most common; digits 0–9. |
| 2 | Binary (Base‑2) | Digits 0,1. Place values: 1, 2, 4, 8, 16… |
| 3 | Octal (Base‑8) | Digits 0–7. Place values: 1, 8, 64, 512… |
| 4 | Hexadecimal (Base‑16) | Digits 0–9, A–F (A=10, B=11,…F=15). Place values: 1, 16, 256, 4096… |
| 5 | Conversion to Decimal | Multiply each digit by its base‑power and sum. |
| 6 | Conversion from Decimal | Repeated division by the target base; remainders form the digits (right‑to‑left). |
| 7 | Divisibility Rules |
- Binary: Even = last bit 0, odd = last bit 1.
- Octal: Divisible by 8 if last three digits form a number divisible by 8.
- Hex: Divisible by 16 if last two hex digits are 00. | | 8 | Common Tricks |
- Binary ↔ Decimal: 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16.
- Octal ↔ Decimal: 8⁰=1, 8¹=8, 8²=64.
- Hex ↔ Decimal: 16⁰=1, 16¹=16, 16²=256. | | 9 | Digital Representation | In computing, binary is used for memory addresses and file permissions; hexadecimal is popular for color codes and memory dumps. | |10 | Common Exam Patterns |
- Convert a given number from one base to another.
- Find the decimal equivalent of a binary/oct/hex number.
- Identify the base of a given number using given clues. |
🧮 3. Worked Example
Problem (Typical Forest Guard Question)
Convert the binary number 110101₍₂₎ to its decimal and hexadecimal equivalents.
Step‑by‑Step Solution
| Digit | Position (from right, starting at 0) | Value (Binary × 2^pos) |
|---|---|---|
| 1 | 0 | 1 × 2⁰ = 1 |
| 0 | 1 | 0 × 2¹ = 0 |
| 1 | 2 | 1 × 2² = 4 |
| 0 | 3 | 0 × 2³ = 0 |
| 1 | 4 | 1 × 2⁴ = 16 |
| 1 | 5 | 1 × 2⁵ = 32 |
Sum: 32 + 16 + 4 + 1 = 53₍₁₀₎
Decimal → Hexadecimal
53 ÷ 16 = 3 remainder 5
So, 53₍₁₀₎ = 35₍₁₆₎
Answer:
- Decimal: 53₍₁₀₎
- Hexadecimal: 35₍₁₆₎
🔄 4. Quick Revision Summary
| Concept | Quick Check |
|---|---|
| Decimal | Digits 0‑9, base 10. |
| Binary | Only 0 and 1; even ⇔ last bit 0, odd ⇔ last bit 1. |
| Octal | Digits 0‑7; divisible by 8 ⇔ last 3 octal digits divisible by 8. |
| Hexadecimal | Digits 0‑9, A‑F; 10=A, …, 15=F. |
| To Decimal | Σ (digit × base^position). |
| From Decimal | Repeated division by target base; keep remainders. |
| Common Base Powers | 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16; 8⁰=1, 8¹=8, 8²=64; 16⁰=1, 16¹=16, 16²=256. |
| Trick for Hex ↔ Binary | 1 hex digit ↔ 4 binary digits. |
Tip: Memorize the first few powers of 2, 8, and 16. They are the backbone of quick conversions.
📘 5. Practice Problem (Try it Yourself)
Convert 237₍₈₎ to decimal and then to hexadecimal.
Solution Outline
- Decimal: 2×64 + 3×8 + 7×1 = 128