Octal Number System — Digital Electronics

The octal (base-8) number system uses eight digits and provides a compact way to represent binary numbers, with each octal digit corresponding to exactly three binary bits.

Fundamentals

Octal Counting

Positional Value

Octal to Decimal Conversion

Example: Convert 745₈ to decimal

Example: Convert 36.5₈ to decimal

Decimal to Octal Conversion

Example: Convert 259₁₀ to octal

Verify: 4×64 + 0×8 + 3×1 = 256 + 0 + 3 = 259 ✓

Fractional part: Convert 0.6875₁₀ to octal

Octal to Binary Conversion (Direct Method)

Each octal digit maps to exactly 3 binary bits:

Example: Convert 572₈ to binary

Example: Convert 64.31₈ to binary

Binary to Octal Conversion

Group binary digits in sets of 3 (from the decimal point outward):

Example: Convert 10110101₂ to octal

Example: Convert 1101.10112 to octal

Octal Arithmetic

Octal Addition

Octal Subtraction

Applications of Octal System

1. Unix/Linux File Permissions:

1. Older Computer Systems: PDP-8 (12-bit words displayed as 4 octal digits)
2. Aviation Transponder Codes: Squawk codes are 4-digit octal (0000-7777)
3. Compact Binary Notation: Shorter than binary for documentation

Why Octal is Less Common Today

With 8-bit bytes becoming standard, hexadecimal (base-16) became more practical since each hex digit represents exactly 4 bits (one nibble), and two hex digits represent one byte perfectly. Octal's 3-bit grouping doesn't align cleanly with 8-bit bytes.

Interview Questions

Q1: Why does each octal digit correspond to exactly 3 binary bits? Because 8 = 2³. The octal system has 8 symbols (0-7), and 3 binary bits can represent exactly 8 values (000 to 111). This mathematical relationship makes conversion direct and trivial — no arithmetic needed, just group and substitute.

Q2: Convert 255₁₀ to octal without converting to binary first. 255 ÷ 8 = 31 remainder 7; 31 ÷ 8 = 3 remainder 7; 3 ÷ 8 = 0 remainder 3. Reading remainders upward: 255₁₀ = 377₈. (Note: 3×64 + 7×8 + 7 = 192 + 56 + 7 = 255 ✓)

Q3: Why is hexadecimal preferred over octal in modern computing? Modern computers use 8-bit bytes. Hexadecimal (base-16, 4 bits per digit) divides evenly into bytes (2 hex digits = 1 byte), while octal (3 bits per digit) does not cleanly represent bytes. A 16-bit address is 4 hex digits but requires 5+ octal digits with awkward boundaries.

Q4: In Unix, what does chmod 644 mean in binary? 6 = 110 (rw-), 4 = 100 (r--), 4 = 100 (r--). Full binary: 110 100 100. Owner has read+write, group has read-only, others have read-only.

Q5: Can you have a fractional octal number that doesn't terminate when converted to decimal? No. Every finite octal fraction converts to a terminating decimal fraction because 8 = 2³ and the only prime factor of 8 is 2, which is also a factor of 10. However, some terminating decimal fractions (like 0.1₁₀) do NOT terminate in octal.