Octal Calculator

Octal is a base-8 number system that uses only the digits 0 through 7. Each position in an octal number represents a power of 8, similar to how each position in a decimal number represents a power of 10. For example, the octal number 17 equals 1×8¹ + 7×8⁰ = 15 in decimal.

Octal became popular in early computing because it maps cleanly to groups of 3 binary digits — three binary bits can represent exactly one octal digit (000–111 = 0–7). This made it much easier for engineers to read and write binary data without long strings of 1s and 0s. Unix/Linux file permissions (e.g., 755, 644) still use octal today.

Octal addition works like decimal addition but you carry when the sum reaches 8 (not 10). For example, 7 + 1 in octal equals 10 (octal), not 8. Add digit by digit from right to left, writing the remainder and carrying 1 whenever the digit sum is 8 or more.

Multiply each octal digit by 8 raised to the power of its position (starting from 0 on the right), then sum the results. For example, octal 17 = 1×8¹ + 7×8⁰ = 8 + 7 = 15 in decimal.

Replace each octal digit with its 3-bit binary equivalent. For example, octal 7 = binary 111, octal 5 = binary 101, so octal 75 = binary 111101. This direct mapping is one of the main reasons octal was favored in early computing.

Only the digits 0, 1, 2, 3, 4, 5, 6, and 7 are valid in octal. The digits 8 and 9 do not exist in base-8. If you enter an 8 or 9, the calculator will flag it as an invalid octal input.

Octal division follows the same long division process as decimal, but all operations are performed in base 8. This calculator converts both operands to decimal, performs the division, and converts the result back to octal. Note that results are shown as integers — any fractional remainder is truncated.

Octal 100 equals 1×8² + 0×8¹ + 0×8⁰ = 64 in decimal. In binary that is 1000000, and in hexadecimal it is 40.