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Octal (Base 8)
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Sources & Methodology
Octal (base 8) is defined in POSIX standards for Unix file permissions and in foundational computer science mathematics references.
POSIX Standard — sys/stat.h File Permission Bits
Official POSIX definition of octal permission bits used in Unix/Linux chmod commands
Khan Academy — Number Systems in Computing
Educational reference for base conversion including decimal to octal using repeated division by 8
Method: Repeated division by 8. Remainders are always 0–7 (octal digits only). Read remainders bottom to top.
Permission display: When result is 3 digits (0–777), each octal digit maps to 3 binary bits (rwx) for Unix permission visualization.
JavaScript: n.toString(8) — exact for all safe integers.
Permission display: When result is 3 digits (0–777), each octal digit maps to 3 binary bits (rwx) for Unix permission visualization.
JavaScript: n.toString(8) — exact for all safe integers.
⏱ Last reviewed: April 2026
How to Convert Decimal to Octal
Octal (base 8) is a number system that uses only the digits 0 through 7. While less common than binary or hexadecimal in modern computing, octal has one very practical everyday application: Unix and Linux file permissions. The chmod command uses octal numbers to represent the 9-bit permission mask for any file or directory, making octal essential knowledge for system administrators and developers.
The Division-by-8 Method
Octal = remainders of (n÷8) read bottom to top (digits 0–7 only)
Example: Convert decimal 255 to octal.
255 ÷ 8 = 31 remainder 7
31 ÷ 8 = 3 remainder 7
3 ÷ 8 = 0 remainder 3
Read bottom to top: 377
Verify: (3×64) + (7×8) + (7×1) = 192 + 56 + 7 = 255 ✓
255 ÷ 8 = 31 remainder 7
31 ÷ 8 = 3 remainder 7
3 ÷ 8 = 0 remainder 3
Read bottom to top: 377
Verify: (3×64) + (7×8) + (7×1) = 192 + 56 + 7 = 255 ✓
Unix File Permissions — Octal Reference (chmod)
Unix file permissions are stored as a 9-bit value split into three groups of 3 bits: owner (user), group, and others. Each 3-bit group represents read (r=4), write (w=2), execute (x=1). The sum of these values gives one octal digit per group. This is why chmod takes octal arguments like 755 or 644.
| chmod | Octal | Binary | Permissions | Typical Use |
|---|---|---|---|---|
| chmod 777 | 777 | 111 111 111 | rwxrwxrwx | All permissions (insecure) |
| chmod 755 | 755 | 111 101 101 | rwxr-xr-x | Directories, executables |
| chmod 644 | 644 | 110 100 100 | rw-r--r-- | Regular files, web content |
| chmod 600 | 600 | 110 000 000 | rw------- | Private files, SSH keys |
| chmod 444 | 444 | 100 100 100 | r--r--r-- | Read-only for everyone |
| chmod 000 | 000 | 000 000 000 | ---------- | No permissions |
Octal Digit Reference
| Decimal | Octal | Binary (3-bit) | Unix meaning |
|---|---|---|---|
| 0 | 0 | 000 | --- (no access) |
| 1 | 1 | 001 | --x (execute only) |
| 2 | 2 | 010 | -w- (write only) |
| 3 | 3 | 011 | -wx (write + execute) |
| 4 | 4 | 100 | r-- (read only) |
| 5 | 5 | 101 | r-x (read + execute) |
| 6 | 6 | 110 | rw- (read + write) |
| 7 | 7 | 111 | rwx (full access) |
Converting Octal in Programming Languages
- Python:
oct(255)→'0o377'|oct(255)[2:]→'377' - JavaScript:
(255).toString(8)→'377' - Java:
Integer.toOctalString(255)→'377' - C:
printf("%o", 255)→377 - Excel:
=DEC2OCT(255)→377
💡 Octal vs. Binary shortcut: Each octal digit = exactly 3 binary bits. To convert octal 377 to binary: 3=011, 7=111, 7=111 → 011111111 = 11111111 (decimal 255). This makes octal a compact shorthand for binary, with 3 bits per digit instead of hex’s 4 bits per digit.
Frequently Asked Questions
Divide by 8 repeatedly, recording remainders 0–7 until quotient = 0. Read remainders bottom to top. Example: 255÷8=31r7, 31÷8=3r7, 3÷8=0r3 → read bottom to top: 377.
Decimal 8 = octal 10. Since 8 is the base of octal, it is the first two-digit octal number, just as 10 is the first two-digit decimal number. Octal 10 = 1×8 + 0×1 = 8.
Decimal 255 = octal 377. Division: 255÷8=31r7, 31÷8=3r7, 3÷8=0r3 → 377. Verify: (3×64)+(7×8)+(7×1)=192+56+7=255. In binary: 11111111.
Unix permissions are 3 groups of 3 bits each (rwx for owner, group, others). Each 3-bit group maps perfectly to one octal digit (0–7). chmod 755 = 111 101 101 binary = rwxr-xr-x. Octal is the most compact way to express these 9-bit permission masks.
chmod 755: 7=111(rwx), 5=101(r-x), 5=101(r-x). Binary: 111101101. Owner gets full access; group and others get read+execute only (no write). Standard for executable scripts and web directories.
chmod 644: 6=110(rw-), 4=100(r--), 4=100(r--). Binary: 110100100. Owner can read and write; group and others can only read. Standard for regular files, web server content, and configuration files.
Use oct(n): oct(255) returns '0o377'. Without prefix: oct(255)[2:] returns '377'. Formatted: format(255,'o') returns '377'. The 0o prefix signals an octal literal in Python 3 source code.
Octal uses only digits 0, 1, 2, 3, 4, 5, 6, 7. The digits 8 and 9 do not exist in octal. Each octal digit represents exactly 3 binary bits (a triad), compared to hex which uses 4 bits per digit.
Each octal digit = 3 binary bits: 0=000, 1=001, 2=010, 3=011, 4=100, 5=101, 6=110, 7=111. Conversion is direct: octal 377 = binary 011111111 = 11111111. Three binary digits always = one octal digit.
Decimal 64 = octal 100. Division: 64÷8=8r0, 8÷8=1r0, 1÷8=0r1 → 100. Verify: 1×64+0+0=64. Octal 100 = decimal 64 = binary 1000000 = hex 40.
Octal is used in: Unix/Linux file permissions (chmod), string escape sequences (\077 = octal 63), assembly language for older processors (PDP-8), and some network protocols. Modern computing primarily uses hex, but octal remains essential for file permissions.
Octal = 3 bits per digit; Hex = 4 bits per digit. Hex is preferred in modern computing because 4 bits divides cleanly into 8-bit bytes (2 hex = 1 byte). Octal requires 2.67 digits per byte, making it less aligned. But for 3-bit Unix permission groups, octal is the perfect fit.
=DEC2OCT(255) returns 377. =DEC2OCT(255,6) returns 000377 (6 digits padded). Excel supports values from -536,870,912 to 536,870,911. For values outside this range, use Python or JavaScript instead.
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