Bidirectional conversion between octal and decimal with detailed steps
Input Value
Result
Conversion Result
Step-by-Step Derivation
Base Conversion Principle
Octal → Decimal: Place value expansion (each digit × 8ⁿ) and sum
Decimal → Octal: Repeated division by 8 (reverse remainders)
Digit range: Octal uses only 0~7
Octal is one of the commonly used number systems in computer science. Each octal digit corresponds to 3 binary digits and is frequently used for file permission representation.
⚠Octal numbers only allow digits 0~7. If the input contains 8 or 9, it is not a valid octal number. Results default to integers.
What Are Octal and Decimal?
Octal is a base-8 number system using digits 0~7, with carries every 8. Decimal is a base-10 number system using digits 0~9, with carries every 10 — the most commonly used number system by humans. Converting between octal and decimal is a fundamental skill in computer science.
Octal Characteristics
Octal uses digits 0~7, carrying at 8. Each octal digit corresponds to 3 binary digits. For example, 7₈=111₂, 10₈=001000₂.
Place Value Expansion
Octal to decimal: from right to left, multiply each digit by 8^position. For example, 173₈ = 1×64+7×8+3×1 = 123₁₀.
Repeated Division by 8
Decimal to octal: repeatedly divide by 8, collect remainders, and read in reverse. For example, 123÷8=15 r3, 15÷8=1 r7, 1÷8=0 r1 → 173₈.
Practical Use
Linux file permissions use three octal digits (e.g., 755), where rwx=7, r-x=5, r-x=5. Each position represents the permissions for owner, group, and others.
💡 Teaching Example: Convert octal 173₈ to decimal. Place value expansion: 1×8²+7×8¹+3×8⁰ = 1×64+7×8+3×1 = 64+56+3 = 123₁₀. Conversely, 123₁₀: 123÷8=15 r3, 15÷8=1 r7, 1÷8=0 r1 → 173₈.
What is the difference between octal and decimal?▼
Decimal uses digits 0~9, with carries every 10. It is the everyday counting system for humans. Octal uses digits 0~7, with carries every 8. It is commonly used in computer file permissions and Unix/POSIX systems.
How do you convert octal to decimal?▼
Place value expansion: multiply each octal digit by its power of 8 (starting from 8⁰), then sum. For example, 173₈ = 1×8²+7×8¹+3×8⁰ = 64+56+3 = 123₁₀.
How do you convert decimal to octal?▼
Repeated division by 8: repeatedly divide by 8, collecting remainders. Read the remainders in reverse order. For example, 123÷8=15 remainder 3, 15÷8=1 remainder 7, 1÷8=0 remainder 1 → 173₈.
What practical uses does octal have in computing?▼
Octal is commonly used for: Linux/Unix file permissions (chmod 755 = rwxr-xr-x), C language constants starting with 0 (e.g., 0755), early computer systems (e.g., PDP-8), and octal IPv4 address representation.
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