🔄 Octal to Hex Converter
Professional Octal ⇄ Hexadecimal Converter | Base-8 to Base-16 Calculator
🔢 Octal-Hex Conversion Reference
| Octal | Hex | Decimal | Octal | Hex | Decimal |
|---|---|---|---|---|---|
| 10 | 8 | 8 | 100 | 40 | 64 |
| 20 | 10 | 16 | 144 | 64 | 100 |
| 40 | 20 | 32 | 200 | 80 | 128 |
| 77 | 3F | 63 | 377 | FF | 255 |
| 177 | 7F | 127 | 400 | 100 | 256 |
| 644 | 1A4 | 420 | 755 | 1ED | 493 |
| 1000 | 200 | 512 | 7777 | FFF | 4095 |
📚 Complete Guide to Octal-Hexadecimal Conversion
Understanding Octal and Hexadecimal Number Systems
Octal (base-8) uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7. Each position represents a power of 8: \( 8^0=1, 8^1=8, 8^2=64, 8^3=512 \). Example: 144₈ = \( 1 \times 64 + 4 \times 8 + 4 \times 1 = 100_{10} \). Historical computing standard from 1960s-70s (DEC PDP-8, IBM mainframes). Each octal digit = exactly 3 binary bits (8 = 2³). Modern uses: Unix file permissions (chmod 755), aviation transponder codes, legacy systems. Hexadecimal (base-16) uses sixteen symbols: 0-9 and A-F. Digits 0-9 have usual values; letters A=10, B=11, C=12, D=13, E=14, F=15. Each position represents power of 16: \( 16^0=1, 16^1=16, 16^2=256, 16^3=4096 \). Example: FF₁₆ = \( 15 \times 16 + 15 \times 1 = 255_{10} \). Each hex digit = exactly 4 binary bits (16 = 2⁴). Modern computing standard: memory addresses (0x7FFFFFFF), RGB colors (#FF5733), protocols, data formats. Key relationship through binary: Octal digit = 3 bits (range 0-7 = 000₂-111₂). Hex digit = 4 bits (range 0-F = 0000₂-1111₂). No direct digit correspondence (3 and 4 have no common factor except 1). Conversion requires intermediate step: binary or decimal. Why hex replaced octal in modern computing: (1) Byte alignment: 8-bit byte = 2 hex digits exactly (FF₁₆ = 11111111₂). 8-bit byte = 2⅔ octal digits awkwardly (377₈ = 11111111₂). Modern systems byte-oriented. (2) Memory addressing: 32-bit address = 8 hex digits cleanly (0x00000000-0xFFFFFFFF). 32-bit address = 10⅔ octal digits messily. (3) RGB colors: 24-bit color = 6 hex digits (#RRGGBB natural). Octal representation awkward for color values. (4) Nibble concept: Half-byte (4 bits) = 1 hex digit. No clean octal equivalent for nibble. Historical transition: 1960s-70s: Octal dominant (word sizes 12/24/36 bits divisible by 3). 1980s onward: Hex dominant (8-bit byte standardization). IBM PC, Intel x86, Motorola 68000 drove hex adoption. Modern programmers fluent in hex, less familiar with octal.
Octal to Hexadecimal Conversion Methods
Method 1: Binary Bridge Method (Most Common)—Convert octal→binary→hex using direct bit patterns. Step-by-step procedure: (1) Convert each octal digit to 3-bit binary equivalent. (2) Concatenate all 3-bit groups into complete binary number. (3) Group binary from right in 4-bit chunks (pad left with zeros if needed). (4) Convert each 4-bit group to one hex digit. Detailed Example 1: Convert 377₈ to hex. Step 1—Octal to binary (3 bits each): 3₈ = 011₂; 7₈ = 111₂; 7₈ = 111₂. Binary: 011111111₂. Step 2—Regroup for hex (4 bits each from right): 0|1111|1111. Pad left: 0000|1111|1111 (not needed, leading zeros). Groups: 1111|1111. Step 3—Binary to hex: 1111₂ = F₁₆; 1111₂ = F₁₆. Result: FF₁₆. Verification: 377₈ = 255₁₀ = FF₁₆ ✓. Detailed Example 2: Convert 144₈ to hex. Octal to binary: 1₈=001₂, 4₈=100₂, 4₈=100₂. Binary: 001100100₂. Regroup (4 bits): 0|0110|0100. Pad: 0000|0110|0100. Groups: 0110|0100. Binary to hex: 0110₂=6₁₆, 0100₂=4₁₆. Result: 64₁₆. Verification: 144₈ = 100₁₀ = 64₁₆ ✓. Detailed Example 3: Convert 755₈ to hex. Octal to binary: 7₈=111₂, 5₈=101₂, 5₈=101₂. Binary: 111101101₂. Regroup: 1|1110|1101. Pad: 0001|1110|1101. Binary to hex: 0001₂=1₁₆, 1110₂=E₁₆, 1101₂=D₁₆. Result: 1ED₁₆. Decimal check: 755₈ = 493₁₀ = 1ED₁₆ ✓. Why this method preferred: Direct pattern substitution, no arithmetic. Fast for manual conversion (memorize 8 octal + 16 hex patterns). Shows binary relationship clearly. Method 2: Decimal Bridge Method—Convert octal→decimal→hex using arithmetic. Step-by-step procedure: (1) Convert octal to decimal using positional notation (multiply by powers of 8). (2) Convert decimal to hex using repeated division by 16. Detailed Example: Convert 377₈ to hex via decimal. Octal to decimal: \( 3 \times 64 + 7 \times 8 + 7 \times 1 = 192 + 56 + 7 = 255_{10} \). Decimal to hex: 255 ÷ 16 = 15 r15 (F); 15 ÷ 16 = 0 r15 (F). Read remainders up: FF₁₆. Result: 377₈ = 255₁₀ = FF₁₆. When to use decimal method: Calculator available. Less familiar with binary patterns. Need decimal value anyway.
Hexadecimal to Octal Conversion Methods
Method 1: Binary Bridge Method—Convert hex→binary→octal. Step-by-step procedure: (1) Convert each hex digit to 4-bit binary. (2) Concatenate all 4-bit groups. (3) Regroup binary from right in 3-bit chunks (pad left if needed). (4) Convert each 3-bit group to one octal digit. Detailed Example 1: Convert FF₁₆ to octal. Hex to binary (4 bits each): F₁₆=1111₂, F₁₆=1111₂. Binary: 11111111₂. Regroup (3 bits from right): 11|111|111. Pad: 011|111|111. Binary to octal: 011₂=3₈, 111₂=7₈, 111₂=7₈. Result: 377₈. Detailed Example 2: Convert 64₁₆ to octal. Hex to binary: 6₁₆=0110₂, 4₁₆=0100₂. Binary: 01100100₂. Regroup: 001|100|100. Pad: 001|100|100 (already 3-bit). Binary to octal: 001₂=1₈, 100₂=4₈, 100₂=4₈. Result: 144₈. Detailed Example 3: Convert 1ED₁₆ to octal. Hex to binary: 1₁₆=0001₂, E₁₆=1110₂, D₁₆=1101₂. Binary: 000111101101₂. Regroup: 111|101|101. Pad: 111|101|101 (already aligned). Binary to octal: 111₂=7₈, 101₂=5₈, 101₂=5₈. Result: 755₈. Method 2: Decimal Bridge—Hex→decimal→octal. Example: FF₁₆ → \( 15 \times 16 + 15 = 255_{10} \) → 255÷8 repeatedly → 377₈. Slower but works without binary knowledge.
Common Octal-Hex Conversion Table
| Octal | Hexadecimal | Decimal | Binary | Common Use |
|---|---|---|---|---|
| 10 | 8 | 8 | 1000 | Base value (8¹) |
| 20 | 10 | 16 | 10000 | Hex base value (16¹) |
| 77 | 3F | 63 | 111111 | 6-bit maximum |
| 100 | 40 | 64 | 1000000 | Octal "hundred" (8²) |
| 144 | 64 | 100 | 1100100 | Decimal 100 |
| 177 | 7F | 127 | 1111111 | Max signed 7-bit |
| 200 | 80 | 128 | 10000000 | High byte bit (2⁷) |
| 377 | FF | 255 | 11111111 | Max 8-bit byte |
| 400 | 100 | 256 | 100000000 | Hex "hundred" (16²) |
| 644 | 1A4 | 420 | 110100100 | Unix rw-r--r-- |
| 755 | 1ED | 493 | 111101101 | Unix rwxr-xr-x |
| 1000 | 200 | 512 | 1000000000 | Octal "thousand" (8³) |
| 7777 | FFF | 4095 | 111111111111 | Max 12-bit (PDP-8) |
Practical Conversion Applications
Unix File Permissions: Octal to Hex—Though octal standard for chmod, hex occasionally seen in system programming. Example: chmod 755 to hex. Octal 755₈ via binary: 7₈=111₂, 5₈=101₂, 5₈=101₂ → 111101101₂. Regroup for hex: 0001|1110|1101 → 1ED₁₆. Result: 755₈ = 1ED₁₆ = 493₁₀. Use case: System calls using integer permission values in hex notation. Example: 644₈ (rw-r--r--) to hex. Binary: 110100100₂. Hex: 1A4₁₆. Decimal: 420₁₀. Memory Addresses: Converting Historical Octal—Legacy documentation may show octal addresses needing hex translation. Example: PDP-8 address 7777₈ to modern hex. Octal 7777₈: Each 7₈=111₂ → 111111111111₂ (12 bits). Regroup: 1111|1111|1111 → FFF₁₆. Result: 7777₈ = FFF₁₆ = 4095₁₀ (max 12-bit, 4K address). Data Format Conversion—When reading legacy data files, octal values may need hex for modern tools. Example: Octal dump 377 200 144 to hex. 377₈=FF₁₆, 200₈=80₁₆, 144₈=64₁₆. Hex sequence: FF 80 64 (bytes in modern format). Color Code Translation—Though colors typically hex, understanding octal conversion helps with unusual formats. Example: If color specified as octal RGB values (rare): R=377₈, G=200₈, B=144₈. Convert to hex: #FF8064. Decimal: RGB(255, 128, 100). Assembly Language—Historical assembly used octal, modern uses hex. Converting old PDP-11 code: Instruction 062706 (octal) → Convert to hex for modern assembler. Binary: 000110010111000110₂ → Hex: 0CAE₆ (easier for x86 programmers). Network Protocols—Some legacy network protocols documented in octal. Example: Protocol field 011₈ → Hex: 09₁₆ → Decimal: 9₁₀ (IGP protocol number).
Conversion Verification Techniques
Three-way check: Octal ↔ Decimal ↔ Hex. Verify conversion by checking all three representations match same value. Example: Verify 755₈ = 1ED₁₆. Forward (octal→hex via binary): 755₈ → 111101101₂ → 1ED₁₆ ✓. Via decimal: 755₈ → \( 7 \times 64 + 5 \times 8 + 5 = 493_{10} \). 493₁₀ → 1ED₁₆ (493÷16=30 r13(D), 30÷16=1 r14(E), 1÷16=0 r1) ✓. Reverse (hex→octal): 1ED₁₆ → 000111101101₂ → 755₈ ✓. All match: 755₈ = 1ED₁₆ = 493₁₀ confirmed. Binary digit count check. Octal digits × 3 = total bits. Hex digits × 4 = total bits. Must equal (allowing leading zeros). Example: 755₈ (3 digits) = 3×3 = 9 bits minimum. 1ED₁₆ (3 digits) = 3×4 = 12 bits maximum. Binary 111101101₂ = 9 bits. Fits: 9 bits for octal, requires padding to 12 for full hex representation ✓. Range check. Max n-digit octal = \( 8^n - 1 \). Max m-digit hex = \( 16^m - 1 \). Example: 3-digit octal max = 777₈ = 511₁₀. 3-digit hex max = FFF₁₆ = 4095₁₀. 755₈ (3 digits, 493₁₀) within 3-digit octal range ✓. 1ED₁₆ (3 digits, 493₁₀) within 3-digit hex range ✓. Calculator verification. Use programming calculator (Windows Calc, macOS Calculator in Programmer mode). Input 755 in OCT mode. Switch to HEX mode → displays 1ED. Switch to DEC mode → displays 493. All match ✓.
Why Choose RevisionTown's Octal-Hex Converter?
RevisionTown's professional converter provides: (1) Bidirectional Conversion—Convert octal→hex and hex→octal seamlessly with instant results; (2) Dual Method Support—Shows both binary bridge and decimal bridge conversion paths for educational understanding; (3) Step-by-Step Breakdown—Displays intermediate binary conversion, bit grouping, and final result with full explanation; (4) Flexible Output Options—Choose uppercase/lowercase hex (FF vs ff), optional 0x prefix for programming contexts; (5) Input Validation—Automatically detects invalid characters (8-9 in octal, G-Z in hex), provides clear error messages; (6) Large Number Support—Handles values up to JavaScript safe integer limit for comprehensive applications; (7) Copy to Clipboard—One-click copy for immediate use in code, terminal commands, documentation; (8) Comprehensive Reference Table—Quick lookup for common octal-hex conversions including Unix permissions, memory boundaries; (9) Mobile Optimized—Responsive design works perfectly on smartphones, tablets, desktops; (10) Zero Cost—Completely free with no ads, registration, or limitations; (11) Professional Accuracy—Trusted by computer science students, Unix system administrators, embedded systems engineers, computer historians, legacy system maintainers, reverse engineers, and IT professionals worldwide for Unix administration (converting chmod permissions between octal and hex notation), programming (translating octal literals to hex in modern code), computer science education (learning base conversion algorithms, understanding positional notation), legacy system maintenance (PDP-8/PDP-11 address translation, IBM mainframe code modernization), digital electronics (format conversion for hardware specifications), data format conversion (reading octal dumps in hex-native tools), assembly language (translating historical octal opcodes to hex for modern assemblers), reverse engineering (analyzing legacy binaries with mixed notation), file format analysis (hex editors displaying octal-documented structures), embedded systems (register configuration using preferred notation), network protocol analysis (converting octal protocol specifications to hex), and all applications requiring accurate octal-hexadecimal conversions with professional-grade tools for system administration, software development, computer engineering education, and comprehensive understanding of positional number systems worldwide.
❓ Frequently Asked Questions
Two methods: (1) Binary bridge—Octal→Binary→Hex. (2) Decimal bridge—Octal→Decimal→Hex. Binary method (faster): Step 1: Convert each octal digit to 3-bit binary. Step 2: Regroup binary into 4-bit chunks from right. Step 3: Convert each 4-bit group to hex digit. Example: 377₈ to hex. Octal→Binary: 3=011, 7=111, 7=111 → 011111111₂. Regroup (4 bits): 1111|1111. Binary→Hex: 1111=F, 1111=F. Result: FF₁₆. Example: 144₈ to hex. Octal→Binary: 1=001, 4=100, 4=100 → 001100100₂. Regroup: 0110|0100. Binary→Hex: 0110=6, 0100=4. Result: 64₁₆. Decimal method: Convert 377₈ → 255₁₀ (3×64+7×8+7×1) → FF₁₆ (255÷16=15rF, 15÷16=0rF). Why binary method preferred: Direct pattern substitution, no arithmetic. Shows bit-level relationship. Faster for manual conversion.
Binary bridge method: Hex→Binary (4 bits per digit)→Octal (3 bits per digit). Steps: (1) Convert each hex digit to 4-bit binary. (2) Concatenate into complete binary. (3) Regroup from right in 3-bit chunks (pad left if needed). (4) Convert each 3-bit group to octal digit. Example: FF₁₆ to octal. Hex→Binary: F=1111, F=1111 → 11111111₂. Regroup (3 bits): 011|111|111. Pad: 011|111|111. Binary→Octal: 011=3, 111=7, 111=7. Result: 377₈. Example: 64₁₆ to octal. Hex→Binary: 6=0110, 4=0100 → 01100100₂. Regroup: 001|100|100. Binary→Octal: 001=1, 100=4, 100=4. Result: 144₈. Example: 1ED₁₆ to octal. Hex→Binary: 1=0001, E=1110, D=1101 → 000111101101₂. Regroup: 111|101|101. Binary→Octal: 7, 5, 5. Result: 755₈. Verification: Convert back: 755₈ → 1ED₁₆ ✓. Alternative: Via decimal (slower): FF₁₆ → 255₁₀ → 377₈.
Base and digits differ; both compact binary representations. Octal (base-8): Uses 8 digits: 0-7. Each digit = 3 binary bits (8=2³). Example: 7₈=111₂ (max 3-bit value). Historical standard (1960s-70s computing). Modern use: Unix permissions (chmod 755), aviation codes. Awkward for bytes: 8 bits = 2⅔ octal digits. Hexadecimal (base-16): Uses 16 symbols: 0-9, A-F (A=10, F=15). Each digit = 4 binary bits (16=2⁴). Example: F₁₆=1111₂ (max 4-bit value). Modern computing standard. Perfect for bytes: 8 bits = 2 hex digits exactly (FF₁₆=255₁₀). Comparison: Octal 377₈ = Hex FF₁₆ = Decimal 255₁₀ = Binary 11111111₂ (same value, different notation). Octal 3 digits = 9 bits max. Hex 3 digits = 12 bits max. No direct digit mapping (must convert via binary or decimal). Why hex dominant today: Byte-aligned (8=2×4 bits). Memory addresses natural in hex (0x00000000-0xFFFFFFFF). RGB colors (#RRGGBB). Protocols, file formats. When octal used: chmod 755 (rwxr-xr-x). Legacy PDP-8/PDP-11 systems. C octal literals (0755).
377₈ = FF₁₆ (hexadecimal). Conversion via binary: Octal→Binary (3 bits each): 3₈ = 011₂. 7₈ = 111₂. 7₈ = 111₂. Complete binary: 011111111₂ = 11111111₂ (8 bits). Binary→Hex (4 bits each): 1111₂ = F₁₆. 1111₂ = F₁₆. Result: FF₁₆. Conversion via decimal: Octal→Decimal: \( 3 \times 64 + 7 \times 8 + 7 \times 1 = 192 + 56 + 7 = 255_{10} \). Decimal→Hex: 255 ÷ 16 = 15 remainder 15 (F). 15 ÷ 16 = 0 remainder 15 (F). Read up: FF₁₆. Significance: 377₈ = FF₁₆ = 255₁₀ = maximum 8-bit byte value. All 8 bits set (11111111₂). Common values: Max RGB channel (255 white), IP broadcast (255.255.255.255), byte boundary. Verification: All representations equal 255₁₀ ✓. Binary: 8 ones ✓. Octal: 3×3=9 bits (with leading 0). Hex: 2×4=8 bits exactly ✓.
755₈ = 1ED₁₆ (hexadecimal) = 493₁₀ (decimal). Conversion via binary: Octal→Binary: 7₈=111₂, 5₈=101₂, 5₈=101₂ → 111101101₂ (9 bits). Regroup for hex (4 bits): 1|1110|1101 → Pad: 0001|1110|1101. Binary→Hex: 0001₂=1₁₆, 1110₂=E₁₆, 1101₂=D₁₆. Result: 1ED₁₆. Conversion via decimal: 755₈ → \( 7 \times 64 + 5 \times 8 + 5 = 448 + 40 + 5 = 493_{10} \). 493₁₀ → Hex: 493÷16=30 r13(D), 30÷16=1 r14(E), 1÷16=0 r1. Read up: 1ED₁₆. Significance (Unix permissions): 755₈ = chmod rwxr-xr-x (common executable permission). Owner: 7=rwx (read/write/execute). Group: 5=r-x (read/execute). Others: 5=r-x. Hex 1ED₁₆ = same permission value in hex notation. Decimal 493₁₀ = integer permission value in system calls. Verification: 755₈ = 1ED₁₆ = 493₁₀ = 111101101₂ (all equal) ✓.
No direct digit-to-digit mapping because 3 and 4 (bits per digit) have no common factor. Mathematical reason: Octal digit = 3 binary bits (8=2³). Hex digit = 4 binary bits (16=2⁴). 3 and 4 coprime (GCD=1, LCM=12). Would need 12-bit grouping for direct mapping (impractical). Example issue: Octal 7₈=111₂ (3 bits) doesn't align with hex 4-bit boundaries. Why binary bridge works: Binary common denominator for both. Octal→Binary: Expand to bits (×3 per digit). Binary→Hex: Compress from bits (÷4 per digit). Example: 10₈ → 001|000₂ (6 bits) → 00|1000 → 08₁₆. Contrast with hex-binary: Hex↔Binary direct (each hex digit = exactly 4 bits, no regrouping). F₁₆ ↔ 1111₂ instant. Octal↔Binary direct (each octal digit = exactly 3 bits). 7₈ ↔ 111₂ instant. Hex↔Octal requires bridging. LCM approach (theoretical): Could use 12-bit chunks (LCM of 3 and 4). 12 bits = 4 octal digits = 3 hex digits. Example: 7777₈ (4 digits) = FFF₁₆ (3 digits) = 111111111111₂ (12 bits). Still requires binary intermediate for calculation. Practical solution: Always convert via binary (fast) or decimal (calculator-friendly). No shortcut exists for direct octal-hex mapping.
Legacy system migration, cross-notation documentation, modernizing historical code. Use case 1: Legacy System Migration. Old DEC PDP-8/PDP-11 systems used octal (1960s-70s). Modern systems use hex. Migrating code/data requires conversion. Example: PDP-8 instruction 062706₈ → Hex 0CAE₁₆ for modern assembler. Use case 2: Unix Permission Calculation. chmod uses octal (755₈), but system programming APIs may use hex. Converting 755₈ → 1ED₁₆ for integer permission constants. Example: C code: #define PERM 0x1ED /* rwxr-xr-x = 755 octal */. Use case 3: Historical Documentation. Vintage computer manuals specify addresses in octal. Modern hex editors need hex addresses. Example: Manual says "See address 7777₈" → Convert to FFF₁₆ for hex editor. Use case 4: Data Format Translation. Legacy data files with octal dumps. Modern tools expect hex. Example: Octal dump 377 200 144 → Hex FF 80 64 for analysis. Use case 5: Assembly Language Modernization. Old assembly code with octal opcodes. Translating to hex for modern assemblers/disassemblers. Example: PDP-11 MOV @#377,R0 (octal immediate) → 0xFF in modern notation. Use case 6: Cross-Platform Debugging. Some embedded systems use octal (historical reasons). Debugging on hex-native tools requires conversion. Use case 7: Network Protocol Analysis. Legacy protocols documented in octal. Modern analyzers display hex. Converting specifications for accurate decoding.
Three methods: (1) Convert back (round-trip), (2) Use decimal as intermediate check, (3) Verify binary bit counts. Method 1: Round-trip conversion. Convert forward then backward, should equal original. Example: 755₈ → 1ED₁₆ → 755₈. If matches, correct ✓. Method 2: Decimal verification. Convert both to decimal independently, compare. Example: 755₈ → Decimal: 7×64+5×8+5 = 493₁₀. 1ED₁₆ → Decimal: 1×256+14×16+13 = 256+224+13 = 493₁₀. Both equal 493₁₀ ✓. Method 3: Binary bit count. Octal digits × 3 = total bits (minimum). Hex digits × 4 = total bits (maximum, with padding). Example: 755₈ (3 digits) → 3×3 = 9 bits minimum. 1ED₁₆ (3 digits) → 3×4 = 12 bits with padding. Binary 111101101₂ = 9 bits actual. Fits: 9-12 bit range ✓. Method 4: Calculator verification. Programmer calculator (Windows/Mac): Enter 755 in OCT mode → Switch to HEX → Shows 1ED → Switch to DEC → Shows 493. All match ✓. Method 5: Online converter cross-check. Use multiple converters (RevisionTown, others), compare results. Agreement indicates correctness. Common errors to catch: Wrong bit grouping (regrouping from left instead of right). Digit typos (O vs 0, 5 vs S). Base confusion (treating octal 10 as decimal 10).
