Unit III — Number System Inter-Conversions
Inter-conversion between binary, octal, decimal, and hexadecimal is a core exam skill. This lesson provides a structured reference for all 12 possible conversion directions.
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Conversion Reference Map
Binary (Base-2)
↗ ↖
Octal Hexadecimal
(Base-8) (Base-16)
↘ ↙
Decimal (Base-10)---
All 12 Conversions — Quick Methods
| From | To | Method |
|---|---|---|
| Binary | Decimal | Multiply each bit by 2^position, sum all |
| Decimal | Binary | Divide repeatedly by 2, read remainders upward |
| Binary | Octal | Group 3 bits from right; convert each group |
| Octal | Binary | Replace each digit with 3-bit binary equivalent |
| Binary | Hex | Group 4 bits from right; convert each group |
| Hex | Binary | Replace each digit with 4-bit binary equivalent |
| Decimal | Octal | Divide repeatedly by 8 |
| Octal | Decimal | Multiply each digit by 8^position, sum |
| Decimal | Hex | Divide repeatedly by 16 |
| Hex | Decimal | Multiply each digit by 16^position, sum |
| Octal | Hex | Octal → Binary → Hex |
| Hex | Octal | Hex → Binary → Octal |
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Worked Example 1: Decimal to All Others
Convert (175)₁₀
→ Binary: 175÷2=87R1, 87÷2=43R1, 43÷2=21R1, 21÷2=10R1, 10÷2=5R0, 5÷2=2R1, 2÷2=1R0, 1÷2=0R1 = (10101111)₂
→ Octal: From binary: 010|101|111 → 2|5|7 = (257)₈ Or: 175÷8=21R7, 21÷8=2R5, 2÷8=0R2 → (257)₈
→ Hex: From binary: 1010|1111 → A|F = (AF)₁₆ Or: 175÷16=10R15(F), 10÷16=0R10(A) → (AF)₁₆
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Worked Example 2: Hex to All Others
Convert (1B3)₁₆
→ Binary: 1=0001, B=1011, 3=0011 → (000110110011)₂ = (110110011)₂
→ Decimal: 1×256 + 11×16 + 3 = 256 + 176 + 3 = (435)₁₀
→ Octal: Binary grouped as 3: 000|110|110|011 → 0|6|6|3 = (0663)₈ = (663)₈
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Worked Example 3: Octal to All Others
Convert (374)₈
→ Binary: 3=011, 7=111, 4=100 → (011111100)₂ = (11111100)₂
→ Decimal: 3×64 + 7×8 + 4 = 192 + 56 + 4 = (252)₁₀
→ Hex: Binary grouped as 4: 1111|1100 → F|C = (FC)₁₆
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Summary Powers Table (Memorise These)
| Power | Value |
|---|---|
| 2⁰ | 1 |
| 2¹ | 2 |
| 2² | 4 |
| 2³ | 8 |
| 2⁴ | 16 |
| 2⁵ | 32 |
| 2⁶ | 64 |
| 2⁷ | 128 |
| 2⁸ | 256 |
| 8⁰ | 1 |
| 8¹ | 8 |
| 8² | 64 |
| 8³ | 512 |
| 16⁰ | 1 |
| 16¹ | 16 |
| 16² | 256 |
Key Takeaway: The fastest conversions use binary as a bridge — Octal↔Binary (3-bit groups) and Hex↔Binary (4-bit groups). For decimal, use repeated division (going into a new base) or multiply by place values (coming from a new base). Practice at least two worked examples from each conversion type before the exam.