Unit III — Binary Number System
The binary number system (Base-2) uses only two digits: 0 and 1. Every piece of data inside a computer — instructions, numbers, text, images — is ultimately stored and processed as binary values. The digits 0 and 1 map directly to off (low voltage) and on (high voltage) in digital circuits.
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Binary Terminology
| Term | Meaning |
|---|---|
| Bit | A single binary digit (0 or 1) — smallest unit of data |
| Nibble | 4 bits (e.g., 1010) |
| Byte | 8 bits (e.g., 10110011) |
| Word | 16, 32, or 64 bits depending on architecture |
| Kilobyte (KB) | 1024 bytes = 2¹⁰ bytes |
| Megabyte (MB) | 1024 KB = 2²⁰ bytes |
| Gigabyte (GB) | 1024 MB = 2³⁰ bytes |
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Binary to Decimal Conversion
Method: Multiply each bit by its place value (power of 2) and sum.
Place values (right to left): 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128 …
Example: Convert (10110101)₂ to decimal
| Bit | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
|---|---|---|---|---|---|---|---|---|
| Position | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| Place Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| Value | 128 | 0 | 32 | 16 | 0 | 4 | 0 | 1 |
Sum = 128 + 32 + 16 + 4 + 1 = 181 So (10110101)₂ = (181)₁₀
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Decimal to Binary Conversion
Method: Repeated Division by 2 — divide by 2 repeatedly and note remainders.
Example: Convert (45)₁₀ to binary
| Division | Quotient | Remainder |
|---|---|---|
| 45 ÷ 2 | 22 | 1 ← LSB |
| 22 ÷ 2 | 11 | 0 |
| 11 ÷ 2 | 5 | 1 |
| 5 ÷ 2 | 2 | 1 |
| 2 ÷ 2 | 1 | 0 |
| 1 ÷ 2 | 0 | 1 ← MSB |
Read remainders bottom to top: (45)₁₀ = (101101)₂
Verification: 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 = 32+8+4+1 = 45 ✓
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Binary Fractions
For fractional binary numbers, positions to the right of the binary point use negative powers of 2:
- 2⁻¹ = 0.5, 2⁻² = 0.25, 2⁻³ = 0.125
Example: (0.101)₂ = 1×0.5 + 0×0.25 + 1×0.125 = 0.625
Decimal fraction to binary: Multiply repeatedly by 2, record integer part.
Example: Convert (0.625)₁₀ to binary:
- 0.625 × 2 = 1.25 → 1
- 0.25 × 2 = 0.5 → 0
- 0.5 × 2 = 1.0 → 1
Read top to bottom: (0.625)₁₀ = (0.101)₂
Key Takeaway: Binary is the language of computers. The key conversions are binary↔decimal using place value (B→D) and repeated division/multiplication by 2 (D→B). Memorise powers of 2 up to 2¹⁰ = 1024 for quick exam work.