📚 Number System – UPSC EPFO Study Notes
(Quantitative Aptitude – Beginner Friendly)
1️⃣ What is a Number System?
- A way to write numbers.
- Think of it as a language for numbers.
- Every system has a base – the number of different digits it uses.
| System | Base | Digits used | Everyday Analogy |
|---|---|---|---|
| Decimal (our everyday) | 10 | 0‑9 | 10 fingers on both hands |
| Binary | 2 | 0,1 | On‑off switches |
| Octal | 8 | 0‑7 | 8‑color crayons |
| Hexadecimal | 16 | 0‑9, A‑F | 16‑flavor ice‑cream |
2️⃣ Decimal System (Base‑10)
- The system you use every day.
- Each place is 10 times the one to its right.
Example
2 345
- 2 thousands (2 × 1000)
- 3 hundreds (3 × 100)
- 4 tens (4 × 10)
- 5 ones (5 × 1)
So 2 345 = 2×1000 + 3×100 + 4×10 + 5×1.
3️⃣ Binary System (Base‑2)
- Only digits 0 and 1.
- Each place is 2 times the one to its right.
Analogy
Like a light switch: off = 0, on = 1.
Example
1101₂
- 1×2³ = 8
- 1×2² = 4
- 0×2¹ = 0
- 1×2⁰ = 1
Sum = 8 + 4 + 0 + 1 = 13₁₀.
4️⃣ Octal System (Base‑8)
- Digits 0 to 7.
- Each place is 8 times the one to its right.
Analogy
Think of 8‑color crayons: you have 8 distinct colors.
Example
257₈
- 2×8² = 128
- 5×8¹ = 40
- 7×8⁰ = 7
Total = 128 + 40 + 7 = 175₁₀.
5️⃣ Hexadecimal System (Base‑16)
- Digits 0‑9 and letters A‑F (10‑15).
- Each place is 16 times the one to its right.
Analogy
16‑flavor ice‑cream sundae: flavors A–F stand for 10–15.
Example
1A3₁₆
- 1×16² = 256
- A (10)×16¹ = 160
- 3×16⁰ = 3
Sum = 256 + 160 + 3 = 419₁₀.
6️⃣ Converting Between Systems
6.1 Decimal ➜ Other Bases
Step 1 – Divide the number by the new base.
Step 2 – Record the remainder (digit).
Step 3 – Repeat with the quotient until it becomes 0.
Result – Remainders read bottom‑to‑top.
Tiny Example – Convert 13₁₀ to binary.
| Division | Quotient | Remainder |
|---|---|---|
| 13 ÷ 2 | 6 | 1 |
| 6 ÷ 2 | 3 | 0 |
| 3 ÷ 2 | 1 | 1 |
| 1 ÷ 2 | 0 | 1 |
Read remainders upside‑down → 1101₂.
6.2 Other Bases ➜ Decimal
Step 1 – Write the number with place values.
Step 2 – Multiply each digit by (base)ⁿ where n is the position from right (starting at 0).
Step 3 – Add all results.
Tiny Example – Convert 101₈ to decimal.
| Digit | Position (from right) | Base | Value |
|---|---|---|---|
| 1 | 0 | 8 | 1×8⁰ = 1 |
| 0 | 1 | 8 | 0×8¹ = 0 |
| 1 | 2 | 8 | 1×8² = 64 |
Sum = 64 + 0 + 1 = 65₁₀.
6.3 Direct Binary ↔ Hexadecimal
- Group binary digits into sets of 4 (from right).
- Convert each 4‑bit group to one hex digit.
Example – 11001101₂
- Group →
1100 1101 1100₂ = C₁₆,1101₂ = D₁₆- Result → CD₁₆.