📚 Number System – Quick Study Notes
(For Tripura Patwari / Lekhpal – Quantitative Aptitude)
1️⃣ Simple Explanation
A number system is a way of representing numbers using a set of symbols (digits) and a base (radix).
| Base | Digits used | Common name | Example |
|---|---|---|---|
| 2 | 0,1 | Binary | 1011 |
| 8 | 0‑7 | Octal | 175 |
| 10 | 0‑9 | Decimal (our everyday system) | 423 |
| 16 | 0‑9, A‑F | Hexadecimal | 1A3F |
- Base‑N system uses N different digits.
- Each digit’s value depends on its place (powers of the base).
- Conversions let us move numbers between systems—essential for quick mental math and digital electronics.
2️⃣ Key Points
| Concept | What to Know | Quick Tip |
|---|---|---|
| Place Value | In base b, the digit at position k (from right, starting at 0) represents digit × b^k. |
“Right‑most = 1, next = base, next = base², …” |
| Decimal (Base‑10) | Digits 0–9; powers of 10. | Most common; used in exams. |
| Binary (Base‑2) | Digits 0,1; powers of 2. | Useful for mental “powers of 2” checks. |
| Octal (Base‑8) | Digits 0–7; powers of 8. | Handy for grouping binary digits (3 bits). |
| Hexadecimal (Base‑16) | Digits 0–9, A–F (10–15); powers of 16. | Widely used in computing; each hex digit = 4 binary bits. |
| Conversion Method | - Decimal → Other: divide by base, note remainders. - Other → Decimal: multiply digits by powers of the base and sum. |
Remember “Remainder first → last digit” for Decimal → Binary. |
| Remainder Sequence | When converting, the first remainder (from the first division) is the least significant digit. | Works for any base. |
| Common Powers | 2⁰=1, 2¹=2, …, 10⁰=1, 10¹=10, 16⁰=1, 16¹=16. | Memorise a few powers for quick checks. |
| Rational vs. Irrational | Rational numbers can be expressed as finite or repeating decimals. Irrational numbers (π, √2) cannot. | Not usually tested in basic aptitude but good to know. |
3️⃣ Worked Example
Convert the decimal number 154 to binary, octal, and hexadecimal.
Step‑by‑Step
| Conversion | Process | Result |
|---|---|---|
| Binary (Base‑2) | Divide 154 by 2, record remainders. | |
| 154 ÷ 2 = 77 remainder 0 | ||
| 77 ÷ 2 = 38 remainder 1 | ||
| 38 ÷ 2 = 19 remainder 0 | ||
| 19 ÷ 2 = 9 remainder 1 | ||
| 9 ÷ 2 = 4 remainder 1 | ||
| 4 ÷ 2 = 2 remainder 0 | ||
| 2 ÷ 2 = 1 remainder 0 | ||
| 1 ÷ 2 = 0 remainder 1 | ||
| Read remainders from bottom to top | 10011010₂ |
|
| Octal (Base‑8) | Divide 154 by 8. | |
| 154 ÷ 8 = 19 remainder 2 | ||
| 19 ÷ 8 = 2 remainder 3 | ||
| 2 ÷ 8 = 0 remainder 2 | ||
| Read remainders | 232₈ |
|
| Hexadecimal (Base‑16) | Divide 154 by 16. | |
| 154 ÷ 16 = 9 remainder 10 (A) | ||
| 9 ÷ 16 = 0 remainder 9 | ||
| Read remainders | 9A₁₆ |
Check (Optional):
10011010₂→
1·2⁷ + 0·2⁶ + 0·2⁵ + 1·2⁴ + 1·2³ + 0·2² + 1·2¹ + 0·2⁰
=128 + 16 + 8 + 2=154. ✓
4️⃣ Quick Revision Summary
- Number System = Base + Digits
- Place value rule: digit × base^position.