Decimalto Octal Converter
Convert between decimal (base-10) and octal (base-8) number systems with step-by-step division method and place value breakdown.
Decimalto Octal Converter
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Formula: Octal = Repeated division by 8, read remainders bottom-to-top | Decimal = Sum of (digit x 8^position)
Worked example โ 493 in decimal = 755 in octal (rwxr-xr-x permission)
Formula
Octal = Repeated division by 8, read remainders bottom-to-top | Decimal = Sum of (digit x 8^position)
Decimal to octal conversion uses repeated division by 8. Each remainder becomes a digit in the octal result, read from bottom to top. Octal to decimal reverses this by multiplying each octal digit by 8 raised to its positional power and summing the results.
Worked Examples
Example 1: File Permission Conversion
Problem:Convert the decimal number 493 to octal to understand a Unix file permission.
Solution:493 / 8 = 61 remainder 5 61 / 8 = 7 remainder 5 7 / 8 = 0 remainder 7 Read remainders bottom to top: 755
Result:493 in decimal = 755 in octal (rwxr-xr-x permission)
Example 2: Large Number Conversion
Problem:Convert 1000 in decimal to octal.
Solution:1000 / 8 = 125 remainder 0 125 / 8 = 15 remainder 5 15 / 8 = 1 remainder 7 1 / 8 = 0 remainder 1 Read remainders bottom to top: 1750
Result:1000 in decimal = 1750 in octal
Frequently Asked Questions
How do you convert a decimal number to octal?
To convert a decimal number to octal, repeatedly divide the number by 8 and record each remainder. Continue dividing the quotient by 8 until the quotient reaches zero. Then read the remainders from bottom to top to form the octal number. For example, converting 255 to octal: 255 divided by 8 gives 31 remainder 7, then 31 divided by 8 gives 3 remainder 7, then 3 divided by 8 gives 0 remainder 3, so 255 in decimal equals 377 in octal.
What is the octal number system used for?
The octal (base-8) number system is commonly used in computing and digital electronics. It provides a compact representation of binary numbers since each octal digit corresponds exactly to three binary digits. Unix and Linux file permissions use octal notation extensively, where values like 755 or 644 define read, write, and execute permissions for owner, group, and others.
What digits are valid in the octal system?
The octal number system uses only the digits 0 through 7. Any number containing the digits 8 or 9 is not a valid octal number. This is because octal is a base-8 system, so it only needs eight unique symbols to represent all values. If you encounter an 8 or 9 in what appears to be an octal number, it is likely a decimal or another base.
How do I convert octal directly to binary?
Replace each octal digit with its three-bit binary equivalent and join them, because 8 is 2 cubed and one octal digit holds exactly three bits. The digit 7 is 111 and 5 is 101, so octal 755 is 111101101. To reverse it, split the binary string into three-bit groups from the right, padding the leftmost group with zeros, and convert each group to one octal digit. No division is required in either direction, unlike converting to or from decimal.
How do Unix file permission modes map onto octal digits?
Each of the three permission digits is the sum of read as 4, write as 2 and execute as 1, and the three digits apply to owner, group and others in that order. Mode 755 means 7 for the owner, which is 4 plus 2 plus 1, or read, write and execute; and 5 for group and others, which is 4 plus 1, or read and execute. Mode 644 gives the owner read and write and everyone else read only. Octal is used because each digit is exactly three permission bits.
What is the difference between octal and hexadecimal?
Octal is base 8 and each digit covers three bits; hexadecimal is base 16 and each digit covers four bits. Because a byte is eight bits, hex divides it evenly into two digits while octal does not, which is why hex dominates modern programming. Octal remains standard where the natural grouping is three bits, most visibly in Unix permission modes. Octal also needs only the digits 0 to 7, whereas hex borrows the letters A to F for the values 10 to 15.
How are octal literals written in programming languages?
C and older languages mark octal with a leading zero, so 0755 is 493 in decimal while 755 is seven hundred and fifty-five. That silent difference is a well-known source of bugs, particularly in zero-padded numeric input. Python 3, modern JavaScript and several other languages replaced it with an explicit 0o prefix, and Python 3 rejects a bare leading zero outright. When reading older code, treat any leading zero on an integer literal as a signal to check the base.
Why did early computer systems favour octal?
Machines with word lengths divisible by three - 12, 18, 24 and 36 bits were all common in the 1950s and 1960s, including the 12-bit PDP-8 and the 36-bit PDP-10 - split cleanly into octal digits with no wasted bits. Front panel switches and lights were grouped in threes to match. As the 8-bit byte and word sizes of 16, 32 and 64 bits became standard, four-bit hexadecimal digits aligned better, and hex displaced octal for most purposes.
What is the largest value a given number of octal digits can hold?
With n octal digits the range runs from 0 to 8 to the power of n minus 1. One digit reaches 7, two digits reach 63, three reach 511, and four reach 4,095. Equivalently, n octal digits carry 3n bits, so four digits cover 12 bits. This is worth checking when a field in a file format or protocol is defined as a fixed-width octal string, because a value that overflows the field will be silently truncated rather than flagged.
What does the fourth digit mean in a four-digit permission mode?
It carries the special bits that sit above the owner, group and other triplets. Reading it the same way as the others, 4 is setuid, 2 is setgid and 1 is the sticky bit, so mode 4755 adds setuid to an otherwise ordinary 755. Setgid on a directory makes new entries inherit its group, and the sticky bit on a shared directory stops users deleting files they do not own. When the leading digit is omitted it is taken as zero.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator ยท Editorial policy
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