Binary ↔ Octal Converter
Convert between binary (base 2) and octal (base 8) instantly
Octal was historically important in computing, particularly on early mainframe computers like the PDP-8 and Digital Equipment Corporation systems. While less common today than hex, octal is still used in file permissions (Unix/Linux chmod), some digital electronics, and certain programming contexts.
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How It Works
The Binary-Octal Formula
How to convert between binary and octal
Formula
Octal = 3-bit binary chunks converted | Binary = each octal digit to 3-bit binary
Variables
Grouping Rule
Every octal digit represents exactly 3 binary bits. This relationship (2³ = 8) makes octal a natural human-readable shorthand for binary.
Octal Digits
Octal uses digits 0 through 7. It does not use 8 or 9, or any letters (unlike hexadecimal).
Binary to Octal Conversion
Group binary digits into sets of 3 (from the right), convert each chunk to its octal digit. Add leading zeros to the leftmost group if needed to make 3 bits.
Note: Octal was once used extensively in early computing because 3 bits per digit made it a natural human-readable representation. Today, its primary real-world usage is in Unix/Linux file permissions (chmod 755).
Step-by-Step Example
Converting binary 111010 to octal
Write the binary number
Binary: 111 010
Split into groups of 3 bits from the right
First group: 111 | Second group: 010
Convert first group 111 to octal
111 binary = 4+2+1 = 7 octal
Convert second group 010 to octal
010 binary = 0+2+0 = 2 octal
Combine results
Binary 111010 = Octal 72
Reference Guide
| unit | value | note |
|---|---|---|
| Binary 000 | Octal 0 | 3 bits → 1 octal digit |
| Binary 001 | Octal 1 | 1 decimal |
| Binary 010 | Octal 2 | 2 decimal |
| Binary 011 | Octal 3 | 3 decimal |
| Binary 100 | Octal 4 | 4 decimal |
| Binary 101 | Octal 5 | 5 decimal |
| Binary 110 | Octal 6 | 6 decimal |
| Binary 111 | Octal 7 | 7 decimal |
Understanding Octal
Where octal is used in real computing
The most common modern use of octal is in Unix/Linux file permissions. chmod commands like 755, 644, 777 are octal numbers representing read/write/execute permissions for owner, group, and others.
Best for: chmod 755 means: owner=7 (rwx), group=5 (r-x), others=5 (r-x). chmod 644 means: owner=6 (rw-), group=4 (r--), others=4 (r--).
Octal was the primary human-readable representation on many early computers, including the PDP-8 (from Digital Equipment Corporation), because 12-bit, 18-bit, and 36-bit words aligned perfectly with octal digits.
Best for: A 12-bit word is exactly 4 octal digits. This made octal natural for early machine code programming.
Octal can be useful in digital electronics when working with 3-bit groups, such as multiplexers, encoders, and some binary-coded decimal (BCD) applications.
Best for: 3-to-8 line decoders and 8-to-3 line encoders are naturally described using octal representation.
Where Octal Is Used Today
Octal (base 8) uses the digits 0-7. It was historically important in computing because many early computer architectures used 12-bit, 18-bit, or 36-bit word sizes — all multiples of 3, making octal representation perfect. The famous PDP-8 computer used 12-bit words, represented exactly as 4 octal digits. Today, octal's primary real-world usage is in Unix/Linux file permissions via the chmod command. When you see a permissions string like '-rwxr-xr-x' or a chmod command like '755', that 755 is octal: - 7 (111 binary) = read + write + execute - 5 (101 binary) = read + execute (no write) Octal is also used in: - Digital electronics (3-bit group encoding) - Some programming utilities (print formatting with octal escapes) - Historical computer documentation and legacy systems - Postgresql database permissions (GRANT statements use octal) While hex (16) has largely replaced octal as the standard human-readable binary representation (because 4 bits per digit aligns perfectly with the 8-bit byte), octal remains important in specific contexts, particularly system administration.
Key Features
💡 Pro Tips
- →Memorise the 8 binary-to-octal pairs: 000=0, 001=1, 010=2, 011=3, 100=4, 101=5, 110=6, 111=7
- →Unix file permissions are octal: 7=rwx, 6=rw-, 5=r-x, 4=r--, 3=-wx, 2=-w-, 1=--x, 0=---
- →Common chmod examples: 755 = rwxr-xr-x (executable), 644 = rw-r--r-- (normal file), 777 = all permissions (caution)
- →Binary to octal grouping is always done in groups of 3 bits from the RIGHT (least significant bit)
- →Add leading zeros to the leftmost binary group if needed to make 3 bits
- →Some programming languages use a leading 0 to indicate octal (e.g., 0755 in C/JavaScript)
Common Mistakes
Entering digits 8 or 9 in octal input
Octal only uses digits 0-7. 8 and 9 are not valid octal digits and will trigger a validation error.
Grouping binary from the left instead of the right
Always group binary digits from the RIGHT (least significant bit) when converting to octal, not from the left. Add leading zeros to the leftmost group if needed.
Confusing octal with decimal
Octal numbers look like decimal numbers (0-7 only) but represent different values. 10 octal = 8 decimal, not 10 decimal. Our converter handles this correctly.
Research & Citations
All factual claims on this page are sourced from peer-reviewed research
- [1]
The Open Group (2024). chmod — change file modes. POSIX Base Specifications.
Official documentation for octal notation in Unix/Linux file permissions
View source
This calculator is a reference tool and does not constitute medical advice. For personalised sleep health guidance, consult a qualified healthcare provider.
Last updated: January 10, 2025
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Frequently Asked Questions
How do I convert binary to octal?
Group binary digits into sets of 3 from the right, then convert each 3-bit group to its octal digit (0-7). For example, binary 111010 → groups 111 (7) and 010 (2) → octal 72.
How do I convert octal to binary?
Convert each octal digit to its 3-bit binary equivalent and concatenate. For example, octal 72 → 7 = 111, 2 = 010 → binary 111010.
What are file permissions in octal?
Unix/Linux file permissions are expressed in octal. chmod 755 means: owner (7 = rwx), group (5 = r-x), others (5 = r-x). 644 means owner read/write, group and others read-only.
Why do some programming languages use leading 0 for octal?
In C, C++, Java, JavaScript (strict mode), Python (version 2), a leading zero indicates octal. For example, 0755 = 755 octal = 493 decimal. However, modern JavaScript and Python 3 require 0o prefix (0o755).
Is octal still used today?
Yes, primarily for Unix/Linux file permissions (chmod). It is also used in some digital electronics contexts, PostgreSQL permissions, and legacy systems.