🔢 Octal to Decimal Converter
Professional Octal ⇄ Decimal Converter | Base-8 to Base-10 Calculator
🔢 Octal to Decimal Conversion Reference
| Octal | Decimal | Octal | Decimal | Octal | Decimal |
|---|---|---|---|---|---|
| 0 | 0 | 10 | 8 | 100 | 64 |
| 1 | 1 | 11 | 9 | 144 | 100 |
| 2 | 2 | 12 | 10 | 200 | 128 |
| 3 | 3 | 20 | 16 | 377 | 255 |
| 4 | 4 | 30 | 24 | 400 | 256 |
| 5 | 5 | 40 | 32 | 755 | 493 |
| 6 | 6 | 50 | 40 | 1000 | 512 |
| 7 | 7 | 77 | 63 | 7777 | 4095 |
📚 Complete Guide to Octal-Decimal Conversion
Understanding Octal and Decimal Number Systems
Decimal (base-10) is the standard human counting system we use daily. Uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each position represents a power of 10: ones \( (10^0=1) \), tens \( (10^1=10) \), hundreds \( (10^2=100) \), thousands \( (10^3=1000) \). Example: 345 = \( 3 \times 100 + 4 \times 10 + 5 \times 1 = 345 \). Natural for humans because we have ten fingers. Universal for commerce, science, mathematics, everyday communication. Octal (base-8) uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7. Note: Digits 8 and 9 do not exist in octal. Each position represents a power of 8: ones \( (8^0=1) \), eights \( (8^1=8) \), sixty-fours \( (8^2=64) \), five-hundred-twelves \( (8^3=512) \). Example: 157₈ = \( 1 \times 64 + 5 \times 8 + 7 \times 1 = 111₁₀ \). Historical significance: Octal was the dominant number system in early computing (1960s-1970s). DEC PDP-8, PDP-11 computers used octal extensively for machine code and memory addresses. IBM mainframes employed octal in debugging and documentation. Assembly language programmers were fluent in octal notation. Scientific calculators featured octal mode alongside decimal and hexadecimal. Why octal was popular in early computing: Compact binary representation: Each octal digit = exactly 3 binary bits (8 = 2³). Easier for humans than long binary strings. Simple conversion: Mental arithmetic for octal-binary straightforward. Historical computer word sizes aligned with octal (12-bit, 24-bit, 36-bit systems divisible by 3). Why octal declined: 8-bit byte became industry standard in 1970s-80s. Hexadecimal (base-16) aligns perfectly with bytes: 1 byte = 2 hex digits. Octal awkward for bytes: 1 byte = 2⅔ octal digits (not clean). Modern systems (x86, ARM) use byte-oriented addressing favoring hex. Modern octal applications: Unix/Linux file permissions: chmod 755 (rwxr-xr-x). Aviation transponder codes: 4-digit octal (0000-7777). Legacy system maintenance: Old mainframe code, PDP-8/PDP-11 emulation. Programming languages: C octal literals (0755), Python (0o755). Digital electronics education: Teaching positional notation simpler than hex.
Octal to Decimal Conversion Method
Positional multiplication algorithm: Multiply each octal digit by its power of 8, sum all products. Formula: For octal number \( d_n d_{n-1} ... d_2 d_1 d_0 \), decimal value = \( d_n \times 8^n + d_{n-1} \times 8^{n-1} + ... + d_2 \times 8^2 + d_1 \times 8^1 + d_0 \times 8^0 \). Powers of 8 reference: \( 8^0 = 1 \); \( 8^1 = 8 \); \( 8^2 = 64 \); \( 8^3 = 512 \); \( 8^4 = 4096 \); \( 8^5 = 32768 \). Step-by-step procedure: (1) Write octal number with position indices (right-to-left, starting 0). (2) Multiply each digit by its positional power of 8. (3) Sum all products to get decimal result. Detailed Example 1: Convert 377₈ to decimal. Positions: 377₈ = digit 3 at position 2, digit 7 at position 1, digit 7 at position 0. Position 2: \( 3 \times 8^2 = 3 \times 64 = 192 \). Position 1: \( 7 \times 8^1 = 7 \times 8 = 56 \). Position 0: \( 7 \times 8^0 = 7 \times 1 = 7 \). Sum: \( 192 + 56 + 7 = 255_{10} \). Result: 377₈ = 255₁₀. Significance: Maximum 8-bit value (255₁₀ = FF₁₆ = 11111111₂). Detailed Example 2: Convert 144₈ to decimal. Position 2: \( 1 \times 64 = 64 \). Position 1: \( 4 \times 8 = 32 \). Position 0: \( 4 \times 1 = 4 \). Sum: \( 64 + 32 + 4 = 100_{10} \). Result: 144₈ = 100₁₀. Detailed Example 3: Convert 755₈ to decimal. Position 2: \( 7 \times 64 = 448 \). Position 1: \( 5 \times 8 = 40 \). Position 0: \( 5 \times 1 = 5 \). Sum: \( 448 + 40 + 5 = 493_{10} \). Unix permission: rwxr-xr-x (7=rwx, 5=r-x, 5=r-x). Detailed Example 4: Convert 52₈ to decimal. Position 1: \( 5 \times 8 = 40 \). Position 0: \( 2 \times 1 = 2 \). Sum: \( 40 + 2 = 42_{10} \). Detailed Example 5: Convert 1234₈ to decimal. Position 3: \( 1 \times 512 = 512 \). Position 2: \( 2 \times 64 = 128 \). Position 1: \( 3 \times 8 = 24 \). Position 0: \( 4 \times 1 = 4 \). Sum: \( 512 + 128 + 24 + 4 = 668_{10} \). Why this method works: Positional notation fundamental to all number systems. Each position has value = digit × base^position. Octal base=8, so multiply by powers of 8. Conversion to decimal (base-10) requires evaluating polynomial in base-8.
Decimal to Octal Conversion Method
Division-remainder algorithm: Repeatedly divide by 8, recording remainders. Read remainders bottom-to-top. Step-by-step procedure: (1) Divide decimal number by 8. (2) Record quotient and remainder (remainder = 0-7 = one octal digit). (3) Use quotient as input for next division. (4) Repeat until quotient = 0. (5) Read remainders from bottom to top for octal result. Detailed Example 1: Convert 255₁₀ to octal. Division 1: 255 ÷ 8 = 31 quotient, remainder 7. Division 2: 31 ÷ 8 = 3 quotient, remainder 7. Division 3: 3 ÷ 8 = 0 quotient, remainder 3. Remainders (bottom-up): 3, 7, 7. Result: 377₈. Verification: \( 3 \times 64 + 7 \times 8 + 7 \times 1 = 255_{10} \) ✓. Detailed Example 2: Convert 100₁₀ to octal. Division 1: 100 ÷ 8 = 12 r 4. Division 2: 12 ÷ 8 = 1 r 4. Division 3: 1 ÷ 8 = 0 r 1. Remainders: 1, 4, 4. Result: 144₈. Detailed Example 3: Convert 493₁₀ to octal. Division 1: 493 ÷ 8 = 61 r 5. Division 2: 61 ÷ 8 = 7 r 5. Division 3: 7 ÷ 8 = 0 r 7. Result: 755₈ (Unix rwxr-xr-x permission). Detailed Example 4: Convert 42₁₀ to octal. Division 1: 42 ÷ 8 = 5 r 2. Division 2: 5 ÷ 8 = 0 r 5. Result: 52₈. Detailed Example 5: Convert 1000₁₀ to octal. Division 1: 1000 ÷ 8 = 125 r 0. Division 2: 125 ÷ 8 = 15 r 5. Division 3: 15 ÷ 8 = 1 r 7. Division 4: 1 ÷ 8 = 0 r 1. Result: 1750₈. Why this method works: Division extracts digits right-to-left in target base. Remainder = digit value in that position (0-7 for octal). Quotient becomes input for next higher position. Process terminates when quotient=0 (no more significant digits). Reading remainders backwards reconstructs number in octal.
Practical Conversion Examples
Example 1: Unix file permission 644 to decimal. Octal 644₈: Position 2: \( 6 \times 64 = 384 \). Position 1: \( 4 \times 8 = 32 \). Position 0: \( 4 \times 1 = 4 \). Decimal: \( 384 + 32 + 4 = 420_{10} \). Permission meaning: rw-r--r-- (owner read/write, others read-only). Binary: 110100100₂. Example 2: Aviation code 7500₈ to decimal. Position 3: \( 7 \times 512 = 3584 \). Position 2: \( 5 \times 64 = 320 \). Position 1: \( 0 \times 8 = 0 \). Position 0: \( 0 \times 1 = 0 \). Decimal: \( 3584 + 320 = 3904_{10} \). Significance: Hijacking emergency code in aviation transponders. Example 3: PDP-8 address 7777₈ to decimal. Position 3: \( 7 \times 512 = 3584 \). Position 2: \( 7 \times 64 = 448 \). Position 1: \( 7 \times 8 = 56 \). Position 0: \( 7 \times 1 = 7 \). Decimal: \( 3584 + 448 + 56 + 7 = 4095_{10} \). Maximum 12-bit address in DEC PDP-8 computer (4K memory). Example 4: Decimal 512 to octal. 512 ÷ 8 = 64 r 0; 64 ÷ 8 = 8 r 0; 8 ÷ 8 = 1 r 0; 1 ÷ 8 = 0 r 1. Result: 1000₈. Significance: 512₁₀ = 8³ = power of 8 (clean octal: 1000₈). Example 5: Decimal 128 to octal. 128 ÷ 8 = 16 r 0; 16 ÷ 8 = 2 r 0; 2 ÷ 8 = 0 r 2. Result: 200₈. Significance: 128₁₀ = 2⁷ = high bit of byte. Example 6: Octal 10₈ to decimal. Position 1: \( 1 \times 8 = 8 \). Position 0: \( 0 \times 1 = 0 \). Result: 8₁₀. Important: Octal 10₈ = decimal 8₁₀ (not 10). Common confusion for beginners.
Common Octal-Decimal Conversion Table
| Octal | Decimal | Description | Common Use |
|---|---|---|---|
| 0-7 | 0-7 | Single digits same in both | Basic digit equivalence |
| 10 | 8 | One "eight" | Base value (8¹) |
| 100 | 64 | One "sixty-four" | Base squared (8²) |
| 144 | 100 | Common round number | Frequently used example |
| 200 | 128 | Two sixty-fours | High byte bit (2⁷) |
| 377 | 255 | Max 8-bit value | Byte maximum (FF hex) |
| 400 | 256 | One byte + 1 | 9-bit minimum |
| 644 | 420 | Unix file permission | rw-r--r-- (common file) |
| 755 | 493 | Unix executable perm | rwxr-xr-x (scripts) |
| 777 | 511 | Full Unix permissions | rwxrwxrwx (dangerous) |
| 1000 | 512 | Cube of 8 | Base cubed (8³) |
| 7777 | 4095 | Max 12-bit | PDP-8 max address (4K) |
Unix File Permissions Using Octal
Most common modern application of octal: Unix/Linux file permissions. Three-digit octal represents owner, group, and others permissions. Each digit encodes three permission bits: read (4), write (2), execute (1). Permission calculation: Add values: read(4) + write(2) + execute(1) = 7 (full access). Read-only: 4 + 0 + 0 = 4. Read/execute: 4 + 0 + 1 = 5. Read/write: 4 + 2 + 0 = 6. Common permission patterns: 755₈ (rwxr-xr-x): Decimal 493₁₀. Owner: 7 = rwx (read, write, execute). Group: 5 = r-x (read, execute). Others: 5 = r-x. Use: Executable scripts, programs. Command: chmod 755 script.sh. 644₈ (rw-r--r--): Decimal 420₁₀. Owner: 6 = rw- (read, write). Group: 4 = r-- (read only). Others: 4 = r--. Use: Regular files, documents. Command: chmod 644 file.txt. 600₈ (rw-------): Decimal 384₁₀. Owner: 6 = rw-. Group: 0 = ---. Others: 0 = ---. Use: Private files, SSH keys. Command: chmod 600 ~/.ssh/id_rsa. 777₈ (rwxrwxrwx): Decimal 511₁₀. Everyone: 7 = rwx (full access). Use: Dangerous (security risk). Avoid except temp directories. 700₈ (rwx------): Decimal 448₁₀. Owner: 7 = rwx. Group/Others: 0 = ---. Use: Private executables. Octal-to-permission conversion: 755₈ = 111101101₂ (binary). Split: 111|101|101 (3 bits each). 111₂ = 7₈ = rwx. 101₂ = 5₈ = r-x. Result: rwxr-xr-x. Why octal for permissions: Each permission group = 3 bits (rwx). 3 bits = 1 octal digit (8 = 2³). Perfect alignment: 755 immediately shows owner=7, group=5, others=5. More intuitive than decimal 493 or hex 1ED.
Why Choose RevisionTown's Octal-Decimal Converter?
RevisionTown's professional converter provides: (1) Bidirectional Conversion—Convert octal→decimal and decimal→octal seamlessly with instant results; (2) Step-by-Step Calculation—Shows complete positional notation breakdown with powers of 8 for educational understanding; (3) Input Validation—Automatically detects invalid octal digits (8-9), provides clear error messages; (4) Large Number Support—Handles values up to JavaScript safe integer limit for comprehensive applications; (5) Copy to Clipboard—One-click copy for immediate use in terminal commands (chmod), code, documentation; (6) Comprehensive Reference Table—Quick lookup for common octal-decimal conversions including Unix permissions; (7) Mobile Optimized—Responsive design works perfectly on smartphones, tablets, desktops; (8) Zero Cost—Completely free with no ads, registration, or limitations; (9) Professional Accuracy—Trusted by computer science students, Unix system administrators, computer historians, programmers, and IT professionals worldwide for Unix administration (converting chmod 755₈=493₁₀, understanding rwx permissions), programming (C octal literals 0755, Python 0o755 in code), computer science education (learning positional notation, base conversion algorithms), legacy system maintenance (PDP-8 addresses 7777₈=4095₁₀, IBM mainframe code), digital electronics (understanding base-8 encoding), aviation (transponder codes 7500₈=3904₁₀ conversion), file permission troubleshooting (calculating numeric values from symbolic permissions), computer history research (analyzing historical documentation, assembly code), and all applications requiring accurate octal-decimal conversions with professional-grade tools for system administration, software development, computer engineering education, and comprehensive understanding of positional number systems worldwide.
❓ Frequently Asked Questions
Multiply each octal digit by its power of 8, sum all products. Formula: For octal \( d_n...d_1d_0 \), decimal = \( d_n \times 8^n + ... + d_1 \times 8^1 + d_0 \times 8^0 \). Example: 377₈ to decimal. Position 2: \( 3 \times 8^2 = 3 \times 64 = 192 \). Position 1: \( 7 \times 8^1 = 7 \times 8 = 56 \). Position 0: \( 7 \times 8^0 = 7 \times 1 = 7 \). Sum: \( 192 + 56 + 7 = 255_{10} \). Example: 144₈ to decimal. \( 1 \times 64 + 4 \times 8 + 4 \times 1 = 64 + 32 + 4 = 100_{10} \). Steps: (1) Identify position of each digit (right-to-left, start 0). (2) Multiply digit × 8^position. (3) Sum all products. Powers of 8: 8⁰=1, 8¹=8, 8²=64, 8³=512, 8⁴=4096. Quick check: Single octal digit (0-7) same in decimal. Two digits minimum 10₈=8₁₀.
Divide by 8 repeatedly, record remainders, read bottom-to-top. Steps: (1) Divide decimal by 8. (2) Record quotient and remainder. (3) Use quotient for next division. (4) Repeat until quotient = 0. (5) Read remainders upward. Example: 255₁₀ to octal. 255÷8 = 31 r7. 31÷8 = 3 r7. 3÷8 = 0 r3. Remainders up: 3,7,7. Result: 377₈. Example: 100₁₀ to octal. 100÷8 = 12 r4; 12÷8 = 1 r4; 1÷8 = 0 r1. Result: 144₈. Verification: Convert result back: 144₈ → 1×64+4×8+4×1 = 100₁₀ ✓. Why this works: Division extracts digits right-to-left in target base. Remainder (0-7) = one octal digit. Alternative: Convert to binary first (if familiar), then binary→octal (group by 3 bits). Example: 255₁₀ → 11111111₂ → 011|111|111 → 377₈.
Octal = base-8 number system using digits 0-7 (no 8 or 9). Each position represents power of 8: 8⁰=1, 8¹=8, 8²=64, 8³=512. Example: 157₈ = \( 1 \times 64 + 5 \times 8 + 7 \times 1 = 111_{10} \). Historical importance: Dominant in 1960s-70s computing (DEC PDP-8, IBM mainframes). Assembly language used octal for machine code, memory addresses. Compact binary representation: Each octal digit = 3 binary bits (8=2³). Easier than long binary strings for humans. Why declined: 8-bit byte became standard (1970s-80s). Hexadecimal better for bytes: 1 byte=2 hex digits vs 2⅔ octal digits. Modern systems byte-oriented favoring hex. Modern uses: (1) Unix/Linux file permissions: chmod 755 (rwxr-xr-x). (2) Aviation transponder codes: 4-digit octal (0000-7777). (3) Legacy systems: PDP-8/PDP-11 emulation, old mainframe code. (4) Programming: C octal literals (0755), Python (0o755). Common confusion: Octal 10₈ = decimal 8₁₀ (not 10). No digit 8 exists in octal.
chmod 755₈ = 493₁₀ (decimal) = rwxr-xr-x permissions. Conversion: Position 2: \( 7 \times 64 = 448 \). Position 1: \( 5 \times 8 = 40 \). Position 0: \( 5 \times 1 = 5 \). Sum: \( 448 + 40 + 5 = 493_{10} \). Permission breakdown: First digit (7): Owner permissions. 7 = read(4) + write(2) + execute(1) = rwx. Second digit (5): Group permissions. 5 = read(4) + execute(1) = r-x. Third digit (5): Others permissions. 5 = read(4) + execute(1) = r-x. Binary representation: 755₈ = 111101101₂ (9 bits). Split: 111|101|101 (3 bits each = rwx|r-x|r-x). Common use: Executable scripts, programs. Owner can read/write/execute; others can read/execute. Command: chmod 755 script.sh. ls -l shows: -rwxr-xr-x. Other permissions: 644₈=420₁₀ (rw-r--r-- files). 600₈=384₁₀ (rw------- private). 777₈=511₁₀ (rwxrwxrwx dangerous). Why octal: 3 permission bits (rwx) = 1 octal digit. Perfect alignment for chmod.
Octal system uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7. Digits 8 and 9 do not exist in octal. Reason: Base-8 system means 8 unique symbols. Range: 0-7 (8 values total). Common error: Writing 8 or 9 in octal = invalid. Example: 389₈ invalid (contains 8 and 9). Must be 0-7 only. Digit values: 0₈=0₁₀, 1₈=1₁₀, 2₈=2₁₀, 3₈=3₁₀, 4₈=4₁₀, 5₈=5₁₀, 6₈=6₁₀, 7₈=7₁₀ (single digits same in both). 10₈=8₁₀ (one "eight" not ten). 17₈=15₁₀ (one eight + seven = 8+7). 20₈=16₁₀ (two eights = 16). Comparison: Binary (base-2): 2 digits (0,1). Octal (base-8): 8 digits (0-7). Decimal (base-10): 10 digits (0-9). Hexadecimal (base-16): 16 symbols (0-9,A-F). Counting in octal: 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21... When reach 7, next is 10₈ (not 8). When reach 77₈, next is 100₈. Maximum values: 1 octal digit: 7₈=7₁₀. 2 digits: 77₈=63₁₀. 3 digits: 777₈=511₁₀. 4 digits: 7777₈=4095₁₀.
377₈ = 255₁₀ (two hundred fifty-five decimal). Calculation: Position 2 (leftmost): \( 3 \times 8^2 = 3 \times 64 = 192 \). Position 1 (middle): \( 7 \times 8^1 = 7 \times 8 = 56 \). Position 0 (rightmost): \( 7 \times 8^0 = 7 \times 1 = 7 \). Sum: \( 192 + 56 + 7 = 255_{10} \). Significance: 255₁₀ = maximum 8-bit value (1 byte). Binary: 11111111₂ (all 8 bits set). Hexadecimal: FF₁₆ (two F's = max byte). Common uses: (1) IP addresses: 255.255.255.255 (broadcast address). Each octet max value. (2) RGB colors: RGB(255,255,255) = white (max intensity). (3) Byte boundaries: Data structures, file formats. (4) Unix permissions: Not 377 typically (would be unusual permission). Related values: 377₈ = 255₁₀ = FF₁₆ = 11111111₂ (all representations of same value). 376₈ = 254₁₀ = FE₁₆ (one less). 400₈ = 256₁₀ = 100₁₆ (one more, 9 bits needed). Memory aid: 377₈ = max octal with 3 sevens (highest digit repeated) = 255₁₀ = max byte.
Octal perfectly aligns with Unix 3-bit permission groups (read, write, execute). Key reasons: (1) 3-bit encoding: Each permission group has 3 bits: read (r), write (w), execute (x). 3 bits = 2³ = 8 combinations (0-7) = 1 octal digit. Perfect match: owner/group/others = 3 groups = 3 octal digits. Example: 755₈ = owner(7)|group(5)|others(5). (2) Range alignment: Permissions range 0-7: 0=--- (none), 1=--x, 2=-w-, 3=-wx, 4=r--, 5=r-x, 6=rw-, 7=rwx. Exactly matches octal digit range 0-7. (3) Historical context: Unix created 1969-70 when octal was computing standard. PDP-7/PDP-11 (where Unix developed) used octal extensively. Natural choice given era's conventions. (4) Intuitive calculation: Mental arithmetic easy: 7=4+2+1 (rwx), 5=4+1 (r-x), 6=4+2 (rw-). Single digit represents one permission group. 755 immediately shows all three groups. Why not other bases: Decimal: 3 bits = 0-7, but base-10 confusing (why stop at 7?). Hexadecimal: Overkill (4 bits but only need 3). Binary: 111101101 too long (9 digits vs 3 octal). Modern alternative: Symbolic: chmod u+rwx,go+rx (same as 755). More verbose but explicit. Octal remains standard for conciseness.
Different languages use different octal notation prefixes. C/C++: Leading zero indicates octal. int perm = 0755; // octal 755 = decimal 493. printf("%o", 493); // outputs "755" (octal). Caution: 0755 looks like typo of 755 but different value! Python 3: 0o prefix (oh-zero). perm = 0o755 # octal 755. os.chmod('file', 0o644). oct(493) returns '0o755' (string). Python 2 (legacy): 0755 worked, but Python 3 requires 0o755 for clarity. JavaScript: 0o prefix (strict mode). const perm = 0o755; // octal 755. Legacy: 0755 worked (non-strict), now discouraged. parseInt('755', 8) parses string as octal. Java: Leading zero. int perm = 0755; // octal 755. System.out.printf("%o", 493); // octal output. Ruby: 0o prefix or leading zero. perm = 0o755 or perm = 0755. Go: 0o prefix. perm := 0o755. Rust: 0o prefix. let perm = 0o755; Shell scripting: Direct octal in chmod. chmod 755 file.sh (no prefix needed). Best practice: Use explicit prefix (0o) in modern code to avoid confusion. Leading-zero notation (0755) considered legacy/dangerous (easy to mistake for decimal). Output formatting: printf family: %o format specifier for octal output. Example: printf("Octal: %o\n", 493); outputs "Octal: 755".
