Unit III — Octal & Hexadecimal Number Systems

Octal (Base-8) and Hexadecimal (Base-16) are compact representations of binary numbers. They are used extensively in computing because they map cleanly to groups of binary digits.

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Octal Number System (Base-8)

• Uses digits: 0, 1, 2, 3, 4, 5, 6, 7
• Each octal digit represents exactly 3 binary bits.
• Used in Unix/Linux file permissions (e.g., chmod 755).

Octal ↔ Binary mapping:

Octal to Decimal: (347)₈ = 3×8² + 4×8¹ + 7×8⁰ = 192 + 32 + 7 = (231)₁₀

Decimal to Octal (repeated division by 8): Convert (231)₁₀:

• 231 ÷ 8 = 28 R 7
• 28 ÷ 8 = 3 R 4
• 3 ÷ 8 = 0 R 3

Read bottom to top: (231)₁₀ = (347)₈ ✓

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Hexadecimal Number System (Base-16)

• Uses digits: 0–9 and A–F (where A=10, B=11, C=12, D=13, E=14, F=15).
• Each hex digit represents exactly 4 binary bits (one nibble).
• Used extensively for memory addresses, colour codes (#FF5733), and byte values.

Hex ↔ Decimal mapping:

Hexadecimal to Decimal: (2AF)₁₆ = 2×16² + 10×16¹ + 15×16⁰ = 512 + 160 + 15 = (687)₁₀

Decimal to Hexadecimal (repeated division by 16): Convert (687)₁₀:

• 687 ÷ 16 = 42 R 15 (F)
• 42 ÷ 16 = 2 R 10 (A)
• 2 ÷ 16 = 0 R 2

Read bottom to top: (687)₁₀ = (2AF)₁₆ ✓

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Binary ↔ Octal (Group 3 bits)

Binary to Octal: Group bits into sets of 3 from right; convert each group. (110101011)₂ → 110 | 101 | 011 → 6 | 5 | 3 = (653)₈

Octal to Binary: Replace each octal digit with its 3-bit equivalent. (725)₈ → 7=111 | 2=010 | 5=101 = (111010101)₂

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Binary ↔ Hexadecimal (Group 4 bits)

Binary to Hex: Group bits into sets of 4 from right; convert each group. (1011 1101)₂ → 1011=B | 1101=D = (BD)₁₆

Hex to Binary: Replace each hex digit with its 4-bit equivalent. (3F)₁₆ → 3=0011 | F=1111 = (00111111)₂

Key Takeaway: Octal and hexadecimal are shorthand for binary — each octal digit = 3 bits, each hex digit = 4 bits. Hexadecimal is ubiquitous in computing (memory addresses, colour codes, debugging). Master the grouping technique for instant binary↔octal and binary↔hex conversions.