📚 Number System – Study Notes for Bihar Police SI (Quantitative Aptitude)
Goal – Understand how numbers are written and changed in different “systems” (bases) so you can solve quick mental math questions in the exam.
1️⃣ What is a Number System?
A number system is just a way to write numbers using a set of symbols (digits) and a rule for where each symbol belongs.
Everyday analogy
| Situation | Symbol set | How it works |
|---|---|---|
| Counting apples on a table | 0‑9 | You write one apple as 1, ten apples as 10. |
| Using a digital clock | 0‑9 (hours) | 12:45 means 12 hours and 45 minutes. |
In math we mostly use:
| Base | Digits used | Example (decimal) |
|---|---|---|
| 10 (decimal) | 0‑9 | 345 |
| 2 (binary) | 0,1 | 1011 |
| 8 (octal) | 0‑7 | 345 |
| 16 (hexadecimal) | 0‑9, A‑F | 1A3 |
Tip: The “base” tells you how many distinct digits you need.
2️⃣ Place Value – The Backbone of All Systems
-
Hundreds, Tens, Units
- In 345
- 3 = hundreds (3 × 100)
- 4 = tens (4 × 10)
- 5 = units (5 × 1)
- In 345
-
In other bases the “place” values change:
- Binary: 1 0 1 1 = 1×8 + 0×4 + 1×2 + 1×1
- Octal: 3 4 5 = 3×64 + 4×8 + 5×1
3️⃣ Common Number Systems & Quick Conversions
3.1 Decimal ➜ Binary
- Divide the number by 2.
- Write down the remainder (0 or 1).
- Repeat with the quotient until it becomes 0.
- The binary number is the remainders read bottom‑to‑top.
| Decimal | Binary |
|---|---|
| 13 | 1101 |
| 7 | 111 |
Example
13 ÷ 2 = 6 R 1
6 ÷ 2 = 3 R 0
3 ÷ 2 = 1 R 1
1 ÷ 2 = 0 R 1
→ Binary: 1101
3.2 Binary ➜ Decimal
- List the binary digits from right to left.
- Multiply each digit by 2ⁿ (n = position starting at 0).
- Add all the products.
| Binary | Decimal |
|---|---|
| 1101 | 13 |
| 111 | 7 |
Example
1×8 + 1×4 + 0×2 + 1×1 = 13
3.3 Decimal ➜ Octal
- Divide by 8, keep remainders.
- Read remainders from bottom to top.
| Decimal | Octal |
|---|---|
| 70 | 106 |
| 15 | 17 |
Example
70 ÷ 8 = 8 R 6
8 ÷ 8 = 1 R 0
1 ÷ 8 = 0 R 1
→ Octal: 106
3.4 Decimal ➜ Hexadecimal
- Divide by 16, keep remainders.
- Remainders 10–15 are written as A–F.
- Read remainders bottom‑to‑top.
| Decimal | Hex |
|---|---|
| 254 | FE |
| 31 | 1F |
Example
254 ÷ 16 = 15 R 14 (E)
15 ÷ 16 = 0 R 15 (F)
→ Hex: FE
4️⃣ Quick Divisibility Rules (Decimal)
| Divisible by | Check |
|---|---|
| 2 | Last digit even (0,2,4,6,8) |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit 0 or 5 |
| 6 | Divisible by 2 and 3 |
| 9 | Sum of digits divisible by 9 |
| 10 | Last digit 0 |
Tiny Example
Check 1234 for divisibility by 4: last two digits 34 → 34 ÷ 4 = 8.5 → Not divisible.
5️⃣ Prime Numbers & Factors
-
Prime number – only divisible by 1 and itself.
Example: 2, 3, 5, 7, 11, 13… -
Factors – numbers that divide evenly into a given number.
Example: Factors of 12 → 1, 2, 3, 4, 6, 12.
Quick test for primes (up to 100)
- If the number is even → not prime (except 2).
- If it ends in 5 → not prime