📖 Notes
1. Number System
- A way to represent numbers in computers.
- Two types:
- Non‑positional: Symbols have fixed values regardless of position (e.g., tally marks, Roman numerals).
- Positional: Value depends on position (decimal, binary, octal, hexadecimal).
2. Decimal Number System (Base 10)
- Uses digits 0–9.
- Place values: powers of 10 (units, tens, hundreds…).
- Example: (123 = 1 x 100 + 2 x 10 + 3 x 1).
3. Binary Number System (Base 2)
- Uses digits 0 and 1.
- Place values: powers of 2 (1, 2, 4, 8…).
- Example: (1001_2 = 8 + 1 = 9_{10}).
- Bit: Smallest unit (0 or 1).
- Byte: 8 bits grouped together.
- Nibble: 4 bits (half a byte).
4. Octal Number System (Base 8)
- Digits: 0–7.
- Place values: powers of 8.
- Each octal digit = 3 binary digits.
- Example: (112_8 = 74_{10}).
5. Hexadecimal Number System (Base 16)
- Digits: 0–9 and A–F (A=10, B=11, C=12, D=13, E=14, F=15).
- Each hex digit = 4 binary digits.
- Compact way to represent binary.
- Example: (4F_{16} = 79_{10}).
6. Conversions
- Decimal → Binary: Divide by 2, record remainders.
- Binary → Decimal: Multiply digits by powers of 2, add results.
- Decimal → Octal: Divide by 8, record remainders.
- Decimal → Hexadecimal: Divide by 16, record remainders.
- Binary ↔ Octal/Hex: Group bits (3 for octal, 4 for hex).
7. Important Rules
- Binary numbers: only 0 or 1.
- Octal numbers: digits 0–7.
- Hexadecimal numbers: digits 0–9, A–F.
- Last binary digit = parity check (0 = even, 1 = odd).
- Zero power rule: any number raised to power 0 = 1.
