Octal to Decimal Converter
Instantly convert octal decimal with our free converter. See conversion tables, formulas, and step-by-step explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Octal to Decimal Converter
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Formula: Decimal = Sum of (octal_digit x 8^position)
Worked example โ Octal 644 = 420 decimal (rw-r--r-- permission)
Formula
Decimal = Sum of (octal_digit x 8^position)
Each octal digit is multiplied by 8 raised to the power of its position, counting from the right starting at 0. The sum of all these products gives the decimal equivalent. For the reverse, repeated division by 8 with remainders read bottom-to-top produces the octal representation.
Worked Examples
Example 1: Unix File Permission
Problem:Convert octal permission 644 to decimal to understand the permission bits.
Solution:6 x 8^2 = 6 x 64 = 384 4 x 8^1 = 4 x 8 = 32 4 x 8^0 = 4 x 1 = 4 Sum = 384 + 32 + 4 = 420
Result:Octal 644 = 420 decimal (rw-r--r-- permission)
Example 2: Octal to Decimal Conversion
Problem:Convert octal 1750 to decimal.
Solution:1 x 8^3 = 1 x 512 = 512 7 x 8^2 = 7 x 64 = 448 5 x 8^1 = 5 x 8 = 40 0 x 8^0 = 0 x 1 = 0 Sum = 512 + 448 + 40 + 0 = 1000
Result:Octal 1750 = 1000 decimal
Frequently Asked Questions
How do you convert octal to decimal manually?
To convert an octal number to decimal, assign each digit a positional power of 8, starting from the rightmost digit at position 0. Multiply each digit by 8 raised to its position, then add all the products together. For example, octal 745 becomes: 7 times 64 (448) plus 4 times 8 (32) plus 5 times 1 (5), totaling 485 in decimal.
What are some real-world applications of octal numbers?
Octal numbers are primarily used in computing contexts. The most common application is Unix and Linux file permissions, where three octal digits represent read (4), write (2), and execute (1) permissions for owner, group, and others. Octal was also historically used in older computer systems like the PDP-8 and early IBM mainframes. Some programming languages use an octal prefix (like leading zero in C) to denote octal literals.
Why does each octal digit correspond to exactly three binary digits?
Since 8 equals 2 raised to the third power, each octal digit represents exactly three binary digits (bits). The octal digit 0 maps to 000, 1 to 001, up through 7 which maps to 111. This direct correspondence makes octal a convenient shorthand for binary, since you can convert between them by simply grouping or expanding three bits at a time without any arithmetic.
How is octal different from hexadecimal?
Octal is base-8 and uses digits 0 through 7, while hexadecimal is base-16 and uses digits 0 through 9 plus letters A through F. Each octal digit represents 3 binary digits, whereas each hex digit represents 4 binary digits. Hexadecimal is more commonly used in modern computing because it aligns with 8-bit bytes (two hex digits per byte), while octal is mainly seen in Unix file permissions and some legacy systems.
Can octal numbers contain the digits 8 or 9?
No, valid octal numbers can only contain digits 0 through 7. The digit 8 or 9 would be invalid in the octal system because base-8 only needs eight symbols. If you see 8 or 9 in a number that you expect to be octal, it is either an error or the number is actually in a different base such as decimal. Many programming languages will throw a syntax error if you use 8 or 9 in an octal literal.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator ยท Editorial policy
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