Binary, Number Systems & Digital Codes
Complete, exam-ready learning module explaining why binary is used, how other number systems relate to it, and how conversions work internally.
1. Why Computers Use Binary
Digital computers are built using transistors that have two stable states. Binary maps naturally to these states.
- 0 → OFF (Low voltage)
- 1 → ON (High voltage)
- Noise resistant
- Hardware efficient
2. Number Systems in Digital Computers
Decimal
Base-10, human friendly
Binary
Base-2, machine language
Octal
Base-8, 3-bit grouping
Hexadecimal
Base-16, 4-bit grouping
3. Relationship with Binary
| Binary Bits | Equivalent |
|---|---|
| 3 | 1 Octal digit |
| 4 | 1 Hex digit |
| 8 | 1 Byte |
4. Positional Weight & Fractional Binary
Each position has a weight based on the base. Binary also supports fractions.
Example: (101.101)2
= 1×2² + 0×2¹ + 1×2⁰ + 1×2⁻¹ + 0×2⁻² + 1×2⁻³ = 4 + 1 + 0.5 + 0.125 = 5.625
5. Decimal → Binary (Step-by-Step)
6. Binary → Decimal (Weight Method)
7. Binary → Octal → Hex (Grouping)
8. Digital Codes
BCD
Decimal digits separately encoded
Gray Code
Single-bit transition, error reduction
ASCII
Text representation standard
9. Signed vs Unsigned Numbers
Binary numbers can be interpreted in two ways:
- Unsigned: All bits represent magnitude
- Signed: MSB represents sign (2’s complement)
| Type | Range (8-bit) |
|---|---|
| Unsigned | 0 to 255 |
| Signed | -128 to +127 |
Exam Tip: MSB = 1 → negative number (signed)
Signed / Unsigned Visualizer (8-bit)
10. Binary Addition (With Carry Animation)
11. Signed Overflow Detection
Overflow occurs when:
- Two positive numbers give negative result
- Two negative numbers give positive result
Overflow is ignored in unsigned arithmetic.
12. Binary Subtraction (2’s Complement)
13. Gray ↔ Binary Conversion
14. Binary Multiplication (Shift & Add)
Binary multiplication follows the shift-and-add principle. Each bit of the multiplier decides whether a shifted multiplicand is added.
- Left shift by 1 → multiply by 2
- Bit = 1 → add shifted value
- Bit = 0 → skip
Hardware Insight: Implemented using shift registers and adders in the ALU.
15. Binary Division (Long Division)
Binary division is similar to decimal long division. It uses repeated comparison, subtraction, and shifting.
- Compare shifted divisor
- Subtract when possible
- Shift right step-by-step
Hardware Insight: Uses restoring or non-restoring division algorithms.
16. Binary Arithmetic – Exam Summary
- Binary addition uses carry propagation
- Subtraction uses 2’s complement
- Multiplication = shift + add
- Division = subtract + shift
- Overflow applies only to signed numbers