🔢 Octal to Binary Converter
Professional Octal ⇄ Binary Converter | Base-8 to Base-2 Calculator
🔢 Octal to Binary Conversion Reference (0-7)
| Octal | Binary (3-bit) | Decimal | Octal | Binary (3-bit) | Decimal |
|---|---|---|---|---|---|
| 0 | 000 | 0 | 4 | 100 | 4 |
| 1 | 001 | 1 | 5 | 101 | 5 |
| 2 | 010 | 2 | 6 | 110 | 6 |
| 3 | 011 | 3 | 7 | 111 | 7 |
📚 Complete Guide to Octal-Binary Conversion
Understanding Octal and Binary Number Systems
Binary (base-2) is the fundamental language of digital electronics and computing. Uses only two digits: 0 and 1 (bits). Each bit position represents a power of 2: \( 2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128 \). Example: 1101₂ = \( 1×8 + 1×4 + 0×2 + 1×1 = 13₁₀ \). Digital circuits implement binary naturally through two voltage states (high/low), making it the universal language of processors, memory, and data transmission. Challenges with binary for humans: Long sequences difficult to read/memorize. 8-bit value: 11010110 (8 digits). 32-bit value: 32 digits (extremely impractical). Error-prone: 1011 vs 1101 easily confused. Manual calculations tedious. Octal (base-8) emerged as compact binary notation in early computing era. Uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7. Each position represents power of 8: \( 8^0=1, 8^1=8, 8^2=64, 8^3=512 \). Example: 157₈ = \( 1×64 + 5×8 + 7×1 = 111₁₀ \). Perfect mathematical relationship: 8 = 2³. Each octal digit = exactly 3 binary bits. Range: 3 bits represent 0-7 values = one octal digit. Direct conversion through simple substitution (no arithmetic required). Example: 5₈ = 101₂ (memorize 8 patterns for instant conversion). Three octal digits = 9 binary bits (more compact than writing 9 separate bits). Historical importance of octal: Dominant in 1960s-1970s computing (DEC PDP-8, IBM mainframes). Assembly language listings used octal for memory addresses, machine code. Scientific calculators featured octal mode. Octal-based programming common before hexadecimal gained preference. Why octal declined in modern computing: 8-bit byte became standard (8 not evenly divisible by 3). Hexadecimal (base-16) aligns perfectly with bytes: 1 byte = 2 hex digits vs 2⅔ octal digits. Hex became dominant for memory addresses, RGB colors, modern protocols. Modern octal applications: Unix/Linux file permissions (chmod 755 = rwxr-xr-x). Aviation transponder codes (4-digit octal codes for aircraft identification). Digital electronics education (simpler than hex for teaching base conversion). Legacy system maintenance (older hardware, assembly code). Some programming contexts (C/Python octal literals: 0o755).
Octal to Binary Conversion Method
Conversion algorithm: Replace each octal digit with its 3-bit binary equivalent. Direct substitution using 8 memorized patterns—no arithmetic calculation needed. 8 fundamental patterns to memorize: 0₈=000₂, 1₈=001₂, 2₈=010₂, 3₈=011₂, 4₈=100₂, 5₈=101₂, 6₈=110₂, 7₈=111₂. Memory aids: 0₈: All zeros (000₂). 1₈: Rightmost bit only (001₂). 7₈: All ones (111₂, maximum 3-bit value). 4₈: Middle bit only (100₂, represents 2²=4). Pattern recognition: Even digits (0,2,4,6) end in 0; odd digits (1,3,5,7) end in 1. Step-by-step procedure: (1) Write octal number, separate digits. (2) Replace each digit with 3-bit binary equivalent. (3) Concatenate all 3-bit groups. (4) Remove leading zeros if desired (optional). Detailed Example 1: Convert 377₈ to binary. Separate: 3 | 7 | 7. Convert: 3₈ = 011₂ (3 decimal = 2+1 = 011); 7₈ = 111₂ (7 decimal = 4+2+1 = 111, all bits set); 7₈ = 111₂. Concatenate: 011 111 111. Result: 011111111₂ = 11111111₂ (drop leading zero). Verification: 11111111₂ = 128+64+32+16+8+4+2+1 = 255₁₀ = 377₈ ✓. Significance: 377₈ = max 8-bit value (255₁₀ = FF₁₆). Detailed Example 2: Convert 144₈ to binary. Separate: 1 | 4 | 4. Convert: 1₈ = 001₂; 4₈ = 100₂ (middle bit set, represents 4); 4₈ = 100₂. Concatenate: 001 100 100. Result: 001100100₂ = 1100100₂. Verification: Binary 1100100₂ = 64+32+4 = 100₁₀ = 144₈ ✓. Detailed Example 3: Convert 755₈ to binary. 7₈=111₂, 5₈=101₂, 5₈=101₂. Result: 111101101₂ (9 bits). Decimal: 256+128+64+32+8+4+1 = 493₁₀. Unix permissions: rwxr-xr-x. Detailed Example 4: Convert 52₈ to binary. 5₈=101₂, 2₈=010₂. Result: 101010₂ (6 bits). Decimal: 32+8+2 = 42₁₀. Detailed Example 5: Convert 1234₈ to binary. 1₈=001₂, 2₈=010₂, 3₈=011₂, 4₈=100₂. Result: 001010011100₂ = 1010011100₂ (10 bits). Decimal: 512+128+16+8+4 = 668₁₀. Why this method works: Octal base 8 = 2³, so each octal digit position = 3 binary bit positions. Each octal digit encodes exactly 3 bits with no overlap or gaps. Direct mapping eliminates need for division/multiplication algorithms used in other base conversions.
Binary to Octal Conversion Method
Reverse process: Group binary into 3-bit chunks (from right), convert each to octal digit. Step-by-step procedure: (1) Write binary number. (2) Group bits into sets of 3, starting from rightmost bit (LSB). (3) Pad leftmost group with leading zeros if needed (make it 3 bits). (4) Convert each 3-bit group to single octal digit. (5) Concatenate octal digits. Detailed Example 1: Convert 11111111₂ to octal. Group from right: 11 | 111 | 111 (leftmost group needs padding). Pad: 011 | 111 | 111 (add one leading zero). Convert: 011₂=3₈, 111₂=7₈, 111₂=7₈. Result: 377₈. Decimal check: 255₁₀ ✓. Detailed Example 2: Convert 1100100₂ to octal. Group from right: 1 | 100 | 100 (leftmost needs padding). Pad: 001 | 100 | 100. Convert: 001₂=1₈, 100₂=4₈, 100₂=4₈. Result: 144₈. Decimal: 100₁₀ ✓. Detailed Example 3: Convert 111101101₂ to octal. Group: 111 | 101 | 101 (already aligned). Convert: 111₂=7₈, 101₂=5₈, 101₂=5₈. Result: 755₈. Unix permission: rwxr-xr-x ✓. Detailed Example 4: Convert 10101₂ to octal. Group: 10 | 101 (leftmost needs padding). Pad: 010 | 101. Convert: 010₂=2₈, 101₂=5₈. Result: 25₈. Decimal: 21₁₀ ✓. Detailed Example 5: Convert 1010011100₂ to octal. Group: 1 | 010 | 011 | 100 (leftmost needs padding). Pad: 001 | 010 | 011 | 100. Convert: 1₈, 2₈, 3₈, 4₈. Result: 1234₈. Decimal: 668₁₀ ✓. Why group from right (LSB): Rightmost bit = 2⁰=1 (least significant). Middle bits maintain proper positional values. Grouping from left would misalign powers of 2. Example: 10101₂ grouped wrong (from left): 101|01 → incorrect positions. Correct: 010|101 → 25₈. Common mistakes to avoid: Grouping from left instead of right (incorrect). Forgetting to pad leftmost group to 3 bits. Confusing octal digit 8 (doesn't exist—max is 7). Not recognizing that 8₈ is invalid (octal uses 0-7 only).
Practical Conversion Examples
Example 1: Unix file permission 755 octal to binary. Octal: 755. Convert: 7₈=111₂ (owner: rwx), 5₈=101₂ (group: r-x), 5₈=101₂ (others: r-x). Result: 111101101₂. Permission interpretation: Owner: 111₂ = read(4)+write(2)+execute(1) = rwx. Group: 101₂ = read(4)+execute(1) = r-x. Others: 101₂ = read(4)+execute(1) = r-x. Command: chmod 755 file.txt sets these exact binary permission bits. Example 2: Unix permission 644 to binary. 6₈=110₂ (rw-), 4₈=100₂ (r--), 4₈=100₂ (r--). Result: 110100100₂. Meaning: rw-r--r-- (owner can read/write, others read-only). Example 3: Aviation transponder code 7500 to binary. 7₈=111₂, 5₈=101₂, 0₈=000₂, 0₈=000₂. Result: 111101000000₂ (12 bits). Significance: 7500₈ = hijacking code (aviation emergency). Example 4: Binary subnet mask 11111111111111111111111100000000₂ to octal. Group: 011|111|111|111|111|111|111|111|111|111|100|000|000. Convert: 3|7|7|7|7|7|7|7|7|7|4|0|0. Result: 37777774000₈ (clumsy—shows why hex preferred for bytes). Example 5: PDP-8 memory address 7777₈ to binary. 7₈=111₂ repeated. Result: 111111111111₂ (12 bits). Decimal: 4095₁₀. Significance: Max 12-bit address in DEC PDP-8 computer (4K memory). Example 6: Binary 10000000₂ to octal. Group: 010|000|000. Pad: 010|000|000. Convert: 2₈, 0₈, 0₈. Result: 200₈ (128₁₀, high bit of byte). Example 7: Octal 10 to binary. 1₈=001₂, 0₈=000₂. Result: 001000₂ = 1000₂ (4 bits). Decimal: 8₁₀. Note: Octal 10₈ ≠ decimal 10₁₀ (octal 10 = decimal 8).
Historical and Modern Applications
Unix/Linux File Permissions (chmod): Three-digit octal represents owner/group/others permissions. Each octal digit = 3 permission bits (read/write/execute). Calculation: read(4) + write(2) + execute(1). Common permissions: 755₈ = rwxr-xr-x (executable, public readable). 644₈ = rw-r--r-- (file, public readable). 600₈ = rw------- (private file). 777₈ = rwxrwxrwx (full access, dangerous). 700₈ = rwx------ (private executable). Binary relationship: 755₈ → 111|101|101₂ → each octal digit directly maps 3 permission bits. Command examples: chmod 755 script.sh; chmod 644 config.txt; chmod 600 private.key. Aviation Transponder Codes: 4-digit octal codes (0000-7777) identify aircraft. Range: 0000₈ to 7777₈ (0-4095₁₀ = 4096 unique codes). Special codes: 7500₈ = hijacking. 7600₈ = radio failure. 7700₈ = general emergency. Each digit 0-7 allows simple dial-based entry in cockpit. Binary encoding: Each digit = 3 bits, total 12 bits for squawk code. DEC PDP-8 Computer (1960s): 12-bit word architecture naturally suited octal. Memory addresses: 0000₈-7777₈ (4K = 4096 words). Machine instructions displayed in octal (easier than 12-bit binary). Assembly programming used octal for instruction encoding. Example: TAD 1234 (add memory location 1234₈). Programmers fluent in octal-to-binary mental conversion. IBM Mainframes: Early IBM systems used octal for debugging, memory dumps. Octal representation in documentation, error codes. Byte-oriented systems eventually favored hex, but octal persisted in utilities. C Programming Language: Octal literals start with 0: int perm = 0755; (octal 755). Used for Unix permissions in system calls: chmod("file", 0644). Escape sequences: \101 = 'A' (octal ASCII code 101₈ = 65₁₀). Modern style prefers hex (0x...) but octal legacy remains. Python Programming: Octal prefix 0o: perm = 0o755. os.chmod('file', 0o644) for file permissions. Legacy: Python 2 allowed 0755 (leading zero), Python 3 requires 0o755 for clarity. Digital Electronics Education: Simpler than hex for teaching base conversion (8 patterns vs 16). Demonstrates 3-bit grouping concept. Historical significance in computer science curriculum. Practice for understanding positional notation.
Why Choose RevisionTown's Octal-Binary Converter?
RevisionTown's professional converter provides: (1) Bidirectional Conversion—Convert octal→binary and binary→octal seamlessly with instant results; (2) Step-by-Step Breakdown—Shows conversion process digit-by-digit with 3-bit grouping visualization for educational understanding; (3) Flexible Output Options—Choose spaced octal output, optional 0o prefix for Python/modern programming contexts; (4) Large Number Support—Handles values up to JavaScript safe integer limit for comprehensive applications; (5) Input Validation—Automatically detects invalid characters (8-9 in octal, non-binary in binary), provides clear error messages; (6) Automatic Padding—Binary-to-octal conversion automatically pads leftmost group to 3 bits for correct alignment; (7) Copy to Clipboard—One-click copy for immediate use in terminal commands (chmod), code, or documentation; (8) Comprehensive Reference Table—Quick lookup for all 8 octal-binary mappings (0-7 to 000-111); (9) Mobile Optimized—Responsive design works perfectly on smartphones, tablets, and 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, and IT professionals worldwide for Unix administration (converting chmod permissions 755₈=111101101₂ for understanding), programming (octal literals in C/Python code, debugging legacy systems), computer science education (learning base conversion, understanding historical computing), digital electronics (demonstrating 3-bit encoding, truth tables), aviation (understanding transponder codes in binary format), legacy system maintenance (PDP-8, IBM mainframe code analysis), file permission troubleshooting (visualizing rwx bits from octal values), embedded systems (register configuration using octal notation), computer history research (analyzing historical documentation, assembly listings), security auditing (examining Unix permission vulnerabilities in binary), and all applications requiring accurate octal-binary conversions with professional-grade tools for system administration, software development, computer engineering education, and comprehensive understanding of historical and modern computing systems worldwide.
❓ Frequently Asked Questions
Replace each octal digit with its 3-bit binary equivalent. No calculation needed—direct substitution. Memorize 8 patterns: 0=000, 1=001, 2=010, 3=011, 4=100, 5=101, 6=110, 7=111. Example: 377₈ to binary. 3=011, 7=111, 7=111. Result: 011111111₂ = 11111111₂ (drop leading zero). Example: 144₈ to binary. 1=001, 4=100, 4=100. Result: 001100100₂ = 1100100₂. Steps: (1) Separate octal into individual digits. (2) Replace each with 3-bit binary. (3) Concatenate all groups. (4) Optionally remove leading zeros. Why this works: Each octal digit (base-8) = exactly 3 binary bits (base-2) because 8 = 2³. Direct 1-to-1 mapping without arithmetic. Memory tips: 0₈=all zeros (000₂). 7₈=all ones (111₂). Even octal (0,2,4,6) end in 0; odd (1,3,5,7) end in 1. 4₈=middle bit only (100₂).
One octal digit = exactly 3 binary bits. Mathematical reason: 8 = 2³, so base-8 digit represents 3 base-2 digits. Range: 3 bits represent 0-7 values = one octal digit 0-7 = 000-111 binary. Examples: 0₈ = 000₂ (3 bits, all zero). 7₈ = 111₂ (3 bits, all one). 5₈ = 101₂ (3 bits). 4₈ = 100₂ (3 bits). Practical implications: (1) Compact notation: 9 binary bits = 3 octal digits. Example: 111101101₂ (9 bits) = 755₈ (3 digits). (2) Unix permissions: 3 octal digits = 9 permission bits. 755₈ = 111|101|101₂ (owner|group|others, each 3 bits). (3) Grouping: Binary always groups by 3 for octal conversion. 10101₂ → 010|101 → 25₈ (must pad to multiples of 3). (4) Word sizes: 12-bit word = 4 octal digits (PDP-8: 0000-7777₈). 24-bit = 8 octal digits. Calculation: n octal digits = 3n binary bits. 4 octal digits = 12 bits. 8 octal digits = 24 bits. Comparison to hex: Hex digit = 4 bits (16=2⁴). Octal digit = 3 bits (8=2³). Hex better for bytes (8 bits = 2 hex); octal awkward (8 bits = 2⅔ octal).
Group binary into 3-bit chunks from right, convert each group to octal digit. Steps: (1) Write binary number. (2) Starting from rightmost bit (LSB), group into sets of 3. (3) If leftmost group has fewer than 3 bits, pad with leading zeros. (4) Convert each 3-bit group to single octal digit. (5) Concatenate results. Example: 11111111₂ to octal. Group from right: 11|111|111 (leftmost=2 bits). Pad leftmost: 011|111|111 (add one 0). Convert: 011₂=3₈, 111₂=7₈, 111₂=7₈. Result: 377₈. Example: 1100100₂ to octal. Group: 1|100|100. Pad: 001|100|100. Convert: 001₂=1₈, 100₂=4₈, 100₂=4₈. Result: 144₈. Example: 111101101₂ to octal. Group: 111|101|101 (already 3-bit aligned). Convert: 7₈, 5₈, 5₈. Result: 755₈ (Unix rwxr-xr-x). Why group from right: Preserves positional values (LSB=rightmost=2⁰). Grouping from left misaligns bit positions. Common errors: Grouping from left (wrong). Forgetting to pad leftmost. Using digit 8 or 9 (doesn't exist in octal—max is 7).
chmod 755 = 111101101₂ (9 bits) = rwxr-xr-x permissions. Breakdown: Octal 755₈ converts: 7₈=111₂, 5₈=101₂, 5₈=101₂. Result: 111|101|101₂ (three 3-bit groups). Permission interpretation: (1) Owner (7₈=111₂): Bit 1 (read) = 1 = r. Bit 2 (write) = 1 = w. Bit 3 (execute) = 1 = x. Total: rwx (full access). (2) Group (5₈=101₂): Read = 1 = r. Write = 0 = -. Execute = 1 = x. Total: r-x (read/execute only). (3) Others (5₈=101₂): Same as group = r-x. Numeric calculation: Each permission bit has value: read=4 (2²), write=2 (2¹), execute=1 (2⁰). Owner: 4+2+1=7₈. Group: 4+0+1=5₈. Others: 4+0+1=5₈. Common permissions: 644₈ = 110|100|100₂ = rw-r--r-- (files). 600₈ = 110|000|000₂ = rw------- (private). 777₈ = 111|111|111₂ = rwxrwxrwx (dangerous, all access). 700₈ = 111|000|000₂ = rwx------ (private script). Command usage: chmod 755 script.sh (make script executable for owner, readable for all). ls -l shows: -rwxr-xr-x (matches 755₈ = 111101101₂).
Octal perfectly aligns with 3-bit permission groups (read, write, execute). Reasons octal chosen: (1) Natural grouping: 3 permissions (rwx) = 3 bits = 1 octal digit. Owner, group, others = 3 groups = 3 octal digits. Perfect match: 755₈ = 7|5|5 directly maps owner|group|others. Each digit immediately shows that group's permissions. (2) Range match: Permissions range 0-7 (000₂ to 111₂ = no permissions to full). Octal digits 0-7 exactly cover this range. Hex (0-F) would waste values 8-F (never used for 3-bit permissions). (3) Historical timing: Unix created 1969-1970 when octal was standard in computing. PDP-7/PDP-11 computers (where Unix developed) used octal extensively. Natural choice given era's conventions. (4) Simplicity: Single octal digit per permission group easy to remember/calculate. 755 more intuitive than hex 1ED (requires conversion to understand). Mental math: 7=4+2+1 (rwx) instantly recognizable. Why not hex: Hex digit = 4 bits, but permission group = 3 bits. Awkward mapping: Would need hex to represent 9 bits (not byte-aligned). Example: 755₈ = 1ED₁₆ (less intuitive, requires more mental conversion). Modern perspective: Octal one of few areas where octal still preferred over hex. Symbolic mode alternative: chmod u+rwx,go+rx (same as 755) avoids numbers entirely but less concise.
Learn 8 patterns using systematic approach and visual memory. Method 1: Binary counting 0-7. Count in binary from 0 to 7: 000₂=0₈, 001₂=1₈, 010₂=2₈, 011₂=3₈, 100₂=4₈, 101₂=5₈, 110₂=6₈, 111₂=7₈. Practice counting: "zero-zero-zero, zero-zero-one, zero-one-zero..." until automatic. Method 2: Pattern recognition. Extremes: 0₈=000₂ (all off), 7₈=111₂ (all on). Middle: 4₈=100₂ (only high bit, represents 2²=4). Powers: 1₈=001₂ (2⁰), 2₈=010₂ (2¹), 4₈=100₂ (2²). Even/odd: Even digits (0,2,4,6) end in 0; odd digits (1,3,5,7) end in 1. Method 3: Bit positions. Label bits: [4][2][1]. Example: 5₈ = 4+1 = set bits 4 and 1 = 101₂. 6₈ = 4+2 = 110₂. 3₈ = 2+1 = 011₂. Method 4: Flashcards. Create 8 cards (octal on front, binary on back). Daily practice until instant recognition. Method 5: Real-world practice. Convert chmod permissions: 755→111,101,101. File mode 644→110,100,100. Practice on actual Unix commands builds muscle memory. Quick test: Random octal (5, 3, 6) → binary (101, 011, 110). Verify with calculator. Reverse: binary → octal. Goal: instant recognition without counting.
0o prefix indicates octal literal in Python (Python 3+) and modern programming languages. Format: 0o followed by octal digits (0-7). Example: 0o755 = 493₁₀ (octal 755). Why 0o prefix needed: (1) Distinguish bases: 10 = ten (decimal) vs 0o10 = eight (octal) vs 0x10 = sixteen (hex). Without prefix, ambiguous. (2) Historical change: Python 2: Leading zero indicated octal (0755 = octal). Python 3: Requires 0o755 for clarity (leading zero alone = error). Reason: 0755 confusing (looks like typo of 755 decimal). (3) Consistency: Matches hex (0x...) and binary (0b...) prefix style. All bases now explicit: 0b1010 (binary), 0o12 (octal), 0x0A (hex), 10 (decimal). Code examples: Python: perm = 0o755; os.chmod('file', 0o644). C (no 0o, uses leading 0): int perm = 0755; (leading zero = octal in C). JavaScript: No octal prefix (legacy 0755 works but discouraged; use 0o755 in strict mode). Conversion: Python: int('755', 8) = 493₁₀ (parse string as octal). oct(493) = '0o755' (convert decimal to octal string). Common usage: File permissions: os.chmod('script.sh', 0o755). Legacy code compatibility: 0o777 (full permissions). Error prevention: 0o8 = SyntaxError (8 not valid octal digit). 0755 in Python 3 = SyntaxError (must use 0o755).
Hexadecimal aligns perfectly with 8-bit bytes (modern standard), while octal does not. Key reason: Byte alignment. (1) Byte = 8 bits: 8-bit byte became universal in 1970s-80s (IBM System/360 established standard). Hex: 1 byte = 2 hex digits exactly (8 bits = 2×4 bits). FF₁₆ = 11111111₂ = 255₁₀ (max byte). Octal: 1 byte = 2⅔ octal digits (8 bits = 8/3 = 2.67 digits). Awkward: 377₈ = 11111111₂ (max byte uses 3 octal digits, not power of 2). (2) Address representation: 32-bit address = 8 hex digits (0x00000000-0xFFFFFFFF). 32-bit address = 10⅔ octal digits (awkward, not clean). Hex cleanly represents memory boundaries, page sizes (powers of 16). (3) Nibble (half-byte): 4-bit nibble fundamental in modern computing. Hex digit = 1 nibble (0-F). Octal digit = 3 bits (doesn't align with nibble concept). (4) RGB colors: 24-bit color = 3 bytes = 6 hex digits (#RRGGBB). Natural representation: #FF5733. Octal equivalent: 77653463₈ (8 digits, less intuitive). Historical transition: 1960s-70s: Octal dominant (DEC PDP-8, IBM). 1980s onward: Hex dominant (IBM PC, Motorola 68000, Intel x86). 8-bit byte standardization drove hex adoption. Octal remnants: Unix permissions (legacy from 1970s). Aviation codes. Legacy system documentation. C leading-zero octal (historical artifact). Modern code: Hex vastly more common than octal.
