Octal Calculator

Convert between octal, decimal, binary, and hexadecimal. Perform octal arithmetic with step-by-step results.

Enter a valid octal number (digits 0-7, optional decimal point, optional sign).

Results will be shown in decimal, binary, and hexadecimal.

777 (max 3-digit) 12.34 (fractional) -10 (negative) 0 1234567
345 + 12 777 - 123 56 × 7 100 ÷ 4

Understanding Octal Numbers

The octal numeral system (base‑8) uses digits from 0 to 7. It is often used in computing as a compact representation of binary because each octal digit corresponds to exactly three binary digits (bits).

Octal Digits and Their Binary/Decimal Equivalents
Octal Binary Decimal
0 000 0
1 001 1
2 010 2
3 011 3
4 100 4
5 101 5
6 110 6
7 111 7

Place values: In octal, each position represents a power of 8.

… 8³ (512) , 8² (64) , 8¹ (8) , 8⁰ (1) , 8⁻¹ (0.125) , 8⁻² (0.015625) …

Conversion Methods

1

Octal → Decimal: Multiply each digit by its place value and sum. For fractional part, use negative powers.

345₈ = 3×8² + 4×8¹ + 5×8⁰ = 192 + 32 + 5 = 229₁₀
2

Decimal → Octal: Repeatedly divide by 8, reading remainders from last to first. For fractions, multiply by 8 repeatedly.

3

Octal ↔ Binary: Each octal digit expands to exactly 3 binary bits (group from right).

345₈ = 011 100 101₂ = 11100101₂
Because 8 = 2³, you can group binary digits in threes starting from the point.
4

Octal ↔ Hexadecimal: Usually convert via binary or decimal as intermediate.

Octal Arithmetic

Addition and subtraction work similarly to decimal, but you carry/borrow when a digit exceeds 7. For example:

3 4 5₈
+ 1 2 3₈
-------
4 7 0₈
5+3=8 → 0 with carry 1; 4+2+carry=7; 3+1=4
5 6₈
× 7₈
-----
5 1 2₈
7×6=42₁₀ = 52₈ (write 2, carry 5); 7×5+5=40₁₀ = 50₈ → 512₈

Another addition example with carries:

7 7 7₈
+ 1 1 1₈
-------
1 1 1 0₈
7+1=8 → 0 carry 1; 7+1+carry=9 → 1 carry 1; 7+1+carry=9 → 1 carry 1 → final 1110₈

Historical Note

The word “octal” comes from Latin octo (eight). Octal was widely used in early computers like the PDP-8 (1960s) because it simplified binary display. Today, its most common application is in Unix/Linux file permissions (e.g., chmod 755), where each octal digit represents read/write/execute permissions for owner, group, and others.

Common Uses

  • Unix/Linux file permissions (e.g., chmod 755).
  • Digital systems where groups of three bits are convenient.
  • Microprocessor programming (some older architectures).
  • Character encodings (e.g., ASCII values in octal).

Calculator features:

  • Supports positive/negative integers and fractional octal numbers.
  • Validates input (only digits 0-7, one optional decimal point, optional sign).
  • Shows conversion steps and arithmetic results.
  • Real‑time error checking and clear feedback.

Frequently Asked Questions

Octal remains in use for file permissions on Unix-like systems, and sometimes in programming when dealing with bit groups of three. It's also an easy way to represent binary in a more compact form.

Typically a minus sign is placed before the number (e.g., -777₈). The calculator handles signed octal by converting to decimal, applying the sign, then converting back if needed.

Yes! Our parser accepts a decimal point. The fractional part is converted using negative powers of 8. For example, 0.1₈ = 1×8⁻¹ = 0.125₁₀.

The calculator uses JavaScript numbers (double precision), so very large integers may lose precision. For typical educational use, this is sufficient.