Octal Number System (Base 8)
The octal number system (base 8) uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7. Place values are powers of 8. Octal was historically used in computing because it groups binary digits into sets of three, making binary much easier to read.
Octal Place Values
| Place (from right) | Power of 8 | Decimal Value |
|---|---|---|
| 1st | 8⁰ | 1 |
| 2nd | 8¹ | 8 |
| 3rd | 8² | 64 |
| 4th | 8³ | 512 |
| 5th | 8⁴ | 4,096 |
Converting Octal to Decimal
Multiply each digit by its place value and add the results.
More Examples
| Octal | Working | Decimal |
|---|---|---|
| 17 | (1×8) + (7×1) | 15 |
| 40 | (4×8) + 0 | 32 |
| 100 | (1×64) | 64 |
| 777 | (7×64)+(7×8)+(7×1) | 511 |
Converting Decimal to Octal
Divide repeatedly by 8; read remainders from bottom to top.
Octal and Binary
Each octal digit maps exactly to a 3-bit binary group:
| Octal | Binary (3-bit) |
|---|---|
| 0 | 000 |
| 3 | 011 |
| 5 | 101 |
| 7 | 111 |
- Octal uses digits 0–7; there is no digit 8 or 9 in octal.
- Place values are powers of 8: 1, 8, 64, 512, …
- Octal is often used in Unix file permissions.
- Converting octal ↔ binary: group binary digits in threes from the right.
Quick Practice
- Convert octal 25 to decimal.
- Convert octal 107 to decimal.
- Convert decimal 56 to octal.
- Convert binary 110101 to octal (group into 3-bit sets).
- What is the highest value a 2-digit octal number can hold?
Summary
The octal number system extends place value thinking to a base-8 framework. Because one octal digit equals exactly three binary digits, octal provides a compact shorthand for binary. Although hexadecimal has largely replaced octal in modern computing, octal remains important in Unix/Linux file permission notation and certain programming contexts.