Octal To Binary Converter
Enter the octal value to convert to binary or Binary to Octal.
Octal:
Octal (Base-8) digits each correspond to exactly 3 binary digits.
Binary:
Binary (Base-2) is the fundamental representation used by all digital systems.
How to Convert Octal to Binary — Formula:
Replace each octal digit with its 3-bit binary equivalent: 0=000, 1=001, 2=010, 3=011, 4=100, 5=101, 6=110, 7=111.
Example: Octal 377 → 3=011, 7=111, 7=111 → Binary 011111111 = 11111111.
Technical Details:
Direct digit-by-digit expansion. Used in: analyzing Unix permissions (755 → 111 101 101 = rwxr-xr-x), legacy computing systems, and PDP-11 assembly.
Octal To Binary Converter:
Convert octal to binary by expanding each octal digit into 3 binary digits.
Binary ↔ Hex ↔ Octal: Grouping Relationship
Binary → Hex (group by 4 bits):
Binary → Octal (group by 3 bits):
Frequently Asked Questions
How do I convert Octal to Binary?
Replace each octal digit with its 3-bit binary equivalent: 0=000, 1=001, 2=010, 3=011, 4=100, 5=101, 6=110, 7=111.
What is the Octal number system?
Octal (Base-8) digits each correspond to exactly 3 binary digits.
What is the Binary number system?
Binary (Base-2) is the fundamental representation used by all digital systems.
Where is Octal to Binary conversion used?
Direct digit-by-digit expansion. Used in: analyzing Unix permissions (755 → 111 101 101 = rwxr-xr-x), legacy computing systems, and PDP-11 assembly.
Can I convert large octal numbers?
Yes. This converter handles numbers of any practical size. For very large numbers, the conversion is performed using arbitrary-precision arithmetic to ensure accuracy.
How to Convert Octal to Binary (Base-8 to Base-2)
Octal-to-binary conversion is the reverse of binary-to-octal: each octal digit expands to exactly 3 binary bits. This direct mapping makes the conversion trivial — simply replace each digit with its 3-bit pattern. This is used when interpreting Unix permissions at the bit level or expanding octal machine code to binary.
- Take each octal digit individually.
- Convert each digit to its 3-bit binary equivalent (0=000 through 7=111).
- Concatenate all 3-bit groups in order.
- Optionally remove leading zeros from the result.
- Example: 752₈ → 111 101 010 → 111101010₂.
Octal to Binary: Common Patterns
Standard octal digits and permission values with their binary expansions:
| Input | Output |
|---|---|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 755 | 111101101 |
| 644 | 110100100 |
| 377 | 011111111 |
| 777 | 111111111 |
Solved Examples: Octal to Binary
Question 1: Expand octal 644 to binary to see the permission bits.
Solution:
6 = 110 (read + write)
4 = 100 (read only)
4 = 100 (read only)
Combined: 110100100
Answer: 644₈ = 110100100₂ → rw-r--r-- (owner can read/write, others can only read).
Question 2: Convert octal 2750 to binary (setgid permission with 750 base).
Solution:
2 = 010 (setgid bit)
7 = 111 (owner: rwx)
5 = 101 (group: r-x)
0 = 000 (others: ---)
Combined: 010111101000
Answer: 2750₈ = 010111101000₂ — the setgid bit (position 10) plus rwxr-x--- permissions.
Question 3: Convert octal 177 to binary.
Solution:
1 = 001
7 = 111
7 = 111
Combined: 001111111
Answer: 177₈ = 001111111₂ = 127₁₀ — the maximum signed 7-bit value (0x7F).
Question 4: Convert octal 4567 to binary.
Solution:
4 = 100
5 = 101
6 = 110
7 = 111
Combined: 100101110111
Answer: 4567₈ = 100101110111₂ = 2423₁₀.
Practice: Octal to Binary
Try solving these on your own to test your understanding:
- Convert 77 (octal) to binary. (Answer: 111111)
- Convert 123 (octal) to binary. (Answer: 001010011)
- Convert 400 (octal) to binary. (Answer: 100000000)
- Convert 1750 (octal) to binary. (Answer: 001111101000)
- Convert 52 (octal) to binary. (Answer: 101010)
- Convert 7777 (octal) to binary. (Answer: 111111111111)
Interpreting Unix Special Permission Bits
Beyond the basic 9 permission bits, Unix has 3 special bits represented by a fourth octal digit: setuid (4), setgid (2), and sticky (1). Octal 4755 in binary is 100 111 101 101 — the setuid bit (bit 11) is set, meaning the program runs as its owner regardless of who executes it. The /usr/bin/passwd command uses setuid (4755) to allow regular users to change their own password by temporarily running as root.
PDP-11 and the Origin of Octal in Computing
The PDP-11 minicomputer used 16-bit words, and its instruction set was designed to be read in octal. Instructions split into fields of 3 bits each, making octal the natural representation. For example, MOV instruction 012700 in octal breaks down as 01(MOV) 27(autoincrement PC) 00(register 0). This heritage is why Unix (born on PDP-11) uses octal for permissions and why the C language supports octal literals.
Key Takeaways
- Replace each octal digit with its 3-bit binary equivalent.
- No arithmetic needed — pure pattern substitution.
- Permission interpretation: 7=111=rwx, 5=101=r-x, 4=100=r--, 0=000=---.
- Special bits (setuid/setgid/sticky) add a fourth octal digit (0-7).
- Octal 377 = binary 11111111 = hex FF = decimal 255 (one byte).
- Historical: PDP-11 instructions were designed for octal readability.