📚 Number System – Study Notes (Himachal Pradesh PSC – Quantitative Aptitude)
Why this topic matters?
The number system forms the backbone of all quantitative problems. Knowing how to convert between bases, apply divisibility rules, and quickly spot patterns saves time and boosts accuracy in the exam.
1️⃣ Simple Explanation
| Term | Meaning | Example |
|---|---|---|
| Number System | A way of representing numbers using a set of symbols (digits) and a base. | The decimal system uses digits 0‑9 and base 10. |
| Base (radix) | The number of unique digits, including zero, used to represent numbers. | Binary → base 2; Octal → base 8; Hexadecimal → base 16. |
| Place Value | Value contributed by a digit depending on its position. | In 4 587 (decimal) the digit 8 is in the “tens” place → 8 × 10 = 80. |
| Positional Notation | A number system where each digit’s value depends on its position. | All common numeric systems (decimal, binary, octal, hexadecimal) are positional. |
Key idea: Converting between systems is just a matter of repeatedly dividing (or multiplying) by the target base and recording remainders (or quotients).
2️⃣ Key Points
| Topic | What to Remember |
|---|---|
| Decimal System (Base 10) | 10 unique digits (0–9). Most everyday arithmetic is done here. |
| Binary (Base 2) | Digits: 0, 1. Used in computers. |
| Octal (Base 8) | Digits: 0–7. Useful in early computing. |
| Hexadecimal (Base 16) | Digits: 0–9, A–F (A=10, B=11,…, F=15). Compact representation of binary. |
| Converting Decimal → Other Bases | Repeated division by the base + record remainders (bottom‑up). |
| Converting Other Bases → Decimal | Sum of digit × (base^position). |
| Divisibility Rules | Quick tests to decide if a number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 20. |
| Prime Numbers | Numbers > 1 with only two divisors (1 and itself). Useful for factorisation. |
| Special Numbers | Perfect squares, cubes, triangular, factorial, powers of 2, powers of 10. |
| Modular Arithmetic | Useful for remainder‑based problems (e.g., “What is the last digit of 7^100?”). |
| Number Patterns | Recognise arithmetic/geometric progressions, digit patterns, palindromes, etc. |
3️⃣ Worked Example
Problem
Convert the decimal number 156 to binary, octal, and hexadecimal.
Then convert the binary number 110101 back to decimal.
Finally, check if 156 is divisible by 3, 4, 7, and 13.
Step 1: Decimal → Binary
| Division | Quotient | Remainder |
|---|---|---|
| 156 ÷ 2 | 78 | 0 |
| 78 ÷ 2 | 39 | 0 |
| 39 ÷ 2 | 19 | 1 |
| 19 ÷ 2 | 9 | 1 |
| 9 ÷ 2 | 4 | 1 |
| 4 ÷ 2 | 2 | 0 |
| 2 ÷ 2 | 1 | 0 |
| 1 ÷ 2 | 0 | 1 |
Read remainders upwards → 10011100.
156 (decimal) = 10011100₂
Step 2: Decimal → Octal
| Division | Quotient | Remainder |
|---|---|---|
| 156 ÷ 8 | 19 | 4 |
| 19 ÷ 8 | 2 | 3 |
| 2 ÷ 8 | 0 | 2 |
Read remainders upward → 234₈.
156 (decimal) = 234₈
Step 3: Decimal → Hexadecimal
| Division | Quotient | Remainder |
|---|---|---|
| 156 ÷ 16 | 9 | 12 (C) |
| 9 ÷ 16 | 0 | 9 |
Read upward → 9C₁₆.
156 (decimal) = 9C₁₆