🔢 Decimal to Octal Converter
Professional Decimal to Octal Calculator | Base-10 to Base-8 Conversion Tool
🔄 Common Decimal to Octal Conversions
📚 Complete Guide to Decimal to Octal Conversion
Understanding Octal Number System
Octal (Base-8) Number System: Octal is a positional numeral system using eight distinct digits: 0, 1, 2, 3, 4, 5, 6, 7. Each position represents a power of 8, reading from right to left: ones place \( (8^0 = 1) \), eights place \( (8^1 = 8) \), sixty-fours place \( (8^2 = 64) \), five-hundred-twelves place \( (8^3 = 512) \), and continuing exponentially. Example: The octal number 157 means \( 1 \times 8^2 + 5 \times 8^1 + 7 \times 8^0 = 1 \times 64 + 5 \times 8 + 7 \times 1 = 64 + 40 + 7 = 111 \) in decimal. Octal emerged as a compact representation for binary data because each octal digit corresponds exactly to three binary digits (bits): 0₈=000₂, 1₈=001₂, 2₈=010₂, 3₈=011₂, 4₈=100₂, 5₈=101₂, 6₈=110₂, 7₈=111₂. This clean 3-bit grouping made octal popular in early computing systems (1960s-1970s) for representing binary machine code, memory addresses, and instruction sets more concisely than lengthy binary strings. Historical Computing Context: Early computers like DEC PDP-8 (1965) used 12-bit words naturally grouped as four octal digits (12÷3=4), making octal the preferred notation for programmers. IBM mainframes used octal for memory dumps and debugging. Assembly language programmers routinely worked in octal to read machine code. Example: 12-bit binary 101110011010₂ groups as 101 110 011 010 = 5632₈ (much more readable than 12 binary digits). However, modern computing standardized on 8-bit bytes (powers of 2), making hexadecimal (base-16) more natural than octal for representing 8-bit, 16-bit, 32-bit, and 64-bit values. Despite declining general use, octal persists in specific domains. Modern Applications of Octal: (1) Unix/Linux File Permissions: The chmod command uses 3-digit octal notation representing read (4), write (2), execute (1) permissions for owner, group, and others. Example: chmod 755 means 111 101 101 in binary = rwxr-xr-x (owner: read+write+execute=7; group: read+execute=5; others: read+execute=5). Common permissions: 644 (rw-r--r--), 755 (rwxr-xr-x), 777 (rwxrwxrwx full access). (2) Digital Electronics: Octal occasionally used in circuit design where 3-bit groupings natural (3-bit ADC, RGB 3-bit color encoding legacy systems). (3) Aviation/Aerospace: Some avionics systems use octal for equipment codes, transponder codes (Mode A uses 4-digit octal 0000-7777 = 4,096 unique codes). (4) Subnetting: Network engineers sometimes convert subnet masks to octal for rapid binary visualization (255.255.255.0 = 377.377.377.0 octal = 11111111.11111111.11111111.00000000 binary). Notation Conventions: Subscript 8 indicates octal base: 12₈ means "12 in octal" = 10 decimal. Programming languages use prefix "0o" (Python, modern) or leading "0" (C, Unix traditional—leading zero means octal: 0755 = 493 decimal). Without notation, decimal assumed. Octal digits never include 8 or 9 (only 0-7 valid in base-8).
Division-by-8 Conversion Method
Standard Algorithm: Repeated Division by 8. This systematic approach converts any positive decimal integer to octal by successively dividing by 8 and recording remainders. The remainders, read in reverse order (bottom-to-top), form the octal representation. Step-by-step procedure: (1) Divide decimal number by 8. (2) Record the remainder (0-7)—this becomes one octal digit. (3) Replace the decimal number with the quotient (result of division, ignoring remainder). (4) Repeat steps 1-3 until quotient equals 0. (5) Read all remainders from bottom to top (last remainder to first)—this sequence is the octal equivalent. Detailed Example 1: Convert 10 decimal to octal. Division 1: \( 10 \div 8 = 1 \) quotient, remainder \( 2 \). Division 2: \( 1 \div 8 = 0 \) quotient, remainder \( 1 \) (quotient 0 means stop). Reading remainders bottom-to-top: 1, 2 → Octal: 12₈. Verification: \( 1 \times 8^1 + 2 \times 8^0 = 1 \times 8 + 2 \times 1 = 8 + 2 = 10 \) decimal ✓. Detailed Example 2: Convert 64 decimal to octal. Step 1: \( 64 \div 8 = 8 \) remainder \( 0 \). Step 2: \( 8 \div 8 = 1 \) remainder \( 0 \). Step 3: \( 1 \div 8 = 0 \) remainder \( 1 \). Bottom-to-top: 1, 0, 0 → Octal: 100₈. Verify: \( 1 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 1 \times 64 + 0 + 0 = 64 \) decimal ✓. Note: 64 is a power of 8 (\( 8^2 = 64 \)), so octal representation is 100₈ (similar to how 100 decimal = \( 10^2 \)). Detailed Example 3: Convert 255 decimal to octal. Step 1: \( 255 \div 8 = 31 \) remainder \( 7 \). Step 2: \( 31 \div 8 = 3 \) remainder \( 7 \). Step 3: \( 3 \div 8 = 0 \) remainder \( 3 \). Bottom-to-top: 3, 7, 7 → Octal: 377₈. Verify: \( 3 \times 8^2 + 7 \times 8^1 + 7 \times 8^0 = 3 \times 64 + 7 \times 8 + 7 \times 1 = 192 + 56 + 7 = 255 \) decimal ✓. Significance: 377₈ is maximum 3-digit octal (like 999 is max 3-digit decimal, 377₈ is max before needing 4th octal digit). Detailed Example 4: Convert 512 decimal to octal. \( 512 \div 8 = 64 \) r \( 0 \); \( 64 \div 8 = 8 \) r \( 0 \); \( 8 \div 8 = 1 \) r \( 0 \); \( 1 \div 8 = 0 \) r \( 1 \). Remainders: 1, 0, 0, 0 → Octal: 1000₈. This represents \( 8^3 = 512 \) decimal (first 4-digit octal, like 1000 decimal = \( 10^3 \)). Mathematical justification: Any decimal N can be expressed \( N = q \times 8 + r \) where q = quotient, r = remainder (0-7). The remainder r represents the rightmost octal digit (\( 8^0 \) position). Repeatedly dividing quotient extracts successive octal digits from right to left, decomposing the number into its base-8 place values through successive modulo-8 operations.
Octal to Decimal Conversion
Positional Notation Method: Multiply and Sum. Converting octal to decimal reverses the process by multiplying each octal digit by its positional power of 8, then summing all products. Formula: For octal number \( o_n o_{n-1} \ldots o_2 o_1 o_0 \), decimal value = \( o_n \times 8^n + o_{n-1} \times 8^{n-1} + \ldots + o_2 \times 8^2 + o_1 \times 8^1 + o_0 \times 8^0 \). Example 1: Convert 12₈ to decimal. Positions (right-to-left): digit 0 (rightmost), digit 1 (leftmost). Octal: 1 2. Position powers: \( 8^1 = 8, 8^0 = 1 \). Calculation: \( 1 \times 8 + 2 \times 1 = 8 + 2 = 10 \) decimal. Result: 12₈ = 10₁₀. Example 2: Convert 100₈ to decimal. Octal: 1 0 0 (3 digits). Powers: \( 8^2 = 64, 8^1 = 8, 8^0 = 1 \). Calculation: \( 1 \times 64 + 0 \times 8 + 0 \times 1 = 64 + 0 + 0 = 64 \) decimal. Example 3: Convert 377₈ to decimal. Octal: 3 7 7. Powers: \( 8^2 = 64, 8^1 = 8, 8^0 = 1 \). Calculation: \( 3 \times 64 + 7 \times 8 + 7 \times 1 = 192 + 56 + 7 = 255 \) decimal. Example 4: Convert 1000₈ to decimal. Four digits: 1 0 0 0. Calculation: \( 1 \times 8^3 + 0 \times 8^2 + 0 \times 8^1 + 0 \times 8^0 = 1 \times 512 + 0 + 0 + 0 = 512 \) decimal. Quick patterns: Single octal digit 0-7 equals same decimal value (3₈ = 3₁₀). Powers of 8 in octal are 10₈, 100₈, 1000₈ (decimal 8, 64, 512). Maximum n-digit octal: all 7s = \( 8^n - 1 \) (like 77₈ = 63₁₀ = \( 8^2-1 \); 777₈ = 511₁₀ = \( 8^3-1 \)). Shortcut: Ignore digits that are 0 (contribute nothing), only sum positions with non-zero octal digits. Example: 3047₈ has 3 in position 3, 0 in position 2 (skip), 4 in position 1, 7 in position 0. Calculate: \( 3 \times 512 + 4 \times 8 + 7 \times 1 = 1536 + 32 + 7 = 1575 \) decimal.
Common Decimal to Octal Conversion Table
| Decimal | Octal | Calculation (Powers of 8) | Significance |
|---|---|---|---|
| 0 | 0 | 0 | Zero (minimum value) |
| 1 | 1 | \( 8^0 = 1 \) | One (single digit same) |
| 7 | 7 | \( 7 \times 8^0 = 7 \) | Maximum single octal digit |
| 8 | 10 | \( 1 \times 8^1 + 0 = 8 \) | First power of 8 (like 10 decimal) |
| 10 | 12 | \( 1 \times 8 + 2 = 10 \) | Common decimal value |
| 16 | 20 | \( 2 \times 8 + 0 = 16 \) | Two eights (2×8) |
| 63 | 77 | \( 7 \times 8 + 7 = 63 \) | Maximum 2-digit octal (8²-1) |
| 64 | 100 | \( 1 \times 8^2 = 64 \) | Second power of 8 (8²) |
| 100 | 144 | \( 1×64 + 4×8 + 4×1 = 100 \) | Hundred in decimal |
| 255 | 377 | \( 3×64 + 7×8 + 7×1 = 255 \) | 8-bit max (byte max, FF hex) |
| 256 | 400 | \( 4 \times 64 = 256 \) | 256 = 2⁸ (1 byte = 256 values) |
| 511 | 777 | \( 7×64 + 7×8 + 7 = 511 \) | Maximum 3-digit octal (8³-1) |
| 512 | 1000 | \( 1 \times 8^3 = 512 \) | Third power of 8 (8³) |
| 4096 | 10000 | \( 1 \times 8^4 = 4096 \) | Fourth power (transponder codes max) |
Unix File Permissions Using Octal
File Permission System in Unix/Linux: The most prevalent modern use of octal notation is representing file and directory permissions in Unix-based operating systems (Linux, macOS, BSD, Solaris). Each file has three permission sets for three user classes: owner (u), group (g), and others (o). Each permission set contains three bits: read (r), write (w), and execute (x). Permission bit values: Read = 4 (binary 100), Write = 2 (binary 010), Execute = 1 (binary 001). These values are summed to create octal digit 0-7 representing any combination: 0 = --- (no permissions, 000 binary); 1 = --x (execute only, 001); 2 = -w- (write only, 010); 3 = -wx (write+execute, 011); 4 = r-- (read only, 100); 5 = r-x (read+execute, 101); 6 = rw- (read+write, 110); 7 = rwx (all permissions, 111). Three-digit octal notation: First digit = owner permissions, second digit = group permissions, third digit = others permissions. Common permission examples: 755 (rwxr-xr-x): Owner 7 (read+write+execute), Group 5 (read+execute), Others 5 (read+execute). Binary: 111 101 101. Used for executable scripts, programs, and directories that everyone can enter. Example: /usr/bin/ls executable, website directories. 644 (rw-r--r--): Owner 6 (read+write), Group 4 (read-only), Others 4 (read-only). Binary: 110 100 100. Standard for regular text files, documents. Owner can edit, everyone can read. Example: README.md, configuration files. 600 (rw-------): Owner 6 (read+write), Group 0 (no access), Others 0 (no access). Binary: 110 000 000. Private files, SSH private keys (~/.ssh/id_rsa must be 600 for security), password files. Owner-only access prevents unauthorized reading. 777 (rwxrwxrwx): Everyone 7 (full permissions). Binary: 111 111 111. Security risk—avoid unless specific need (world-writable directories like /tmp use special sticky bit). Allows anyone to read, modify, execute. 700 (rwx------): Owner 7 (full), Group 0, Others 0. Binary: 111 000 000. Private directories, personal scripts. Owner-only access. Example: ~/.gnupg directory. chmod command syntax: chmod 755 filename (sets permissions using octal). chmod u+x filename (adds execute for user using symbolic notation). ls -l shows permissions: -rwxr-xr-x 1 user group 4096 Dec 16 file.txt (first character = file type; next 9 = permissions rwx|r-x|r-x = 755). Directory permissions differ: Read permission (4) allows listing directory contents (ls). Write permission (2) allows creating/deleting files within directory. Execute permission (1) allows entering directory (cd) and accessing files inside. Common directory permission: 755 (drwxr-xr-x) allows everyone to browse, only owner to modify contents. Special permission bits: Setuid (4000 octal), Setgid (2000), Sticky bit (1000) add fourth octal digit. Example: chmod 4755 sets setuid + 755 permissions (runs with owner's privileges). Sticky bit on /tmp (1777) prevents users from deleting others' files despite world-writable directory.
Octal-Binary Relationship
Direct 3-bit Binary Grouping: Octal's most significant advantage is the perfect correspondence between each octal digit and exactly three binary digits, enabling rapid mental conversion without arithmetic. This 3-bit grouping made octal historically popular for reading binary computer output, though hexadecimal (4-bit grouping) has largely supplanted it in modern computing. Conversion table octal ↔ binary: 0₈ = 000₂; 1₈ = 001₂; 2₈ = 010₂; 3₈ = 011₂; 4₈ = 100₂; 5₈ = 101₂; 6₈ = 110₂; 7₈ = 111₂. Memorizing these eight patterns enables instant conversion in both directions. Binary to Octal conversion: Group binary digits in sets of three starting from the rightmost bit (least significant), then convert each triplet to its octal digit. If leftmost group has fewer than 3 bits, pad with leading zeros. Example 1: Convert 101110₂ to octal. Group: 101 | 110 (already 3-bit aligned). Convert: 101₂ = 5₈, 110₂ = 6₈. Result: 56₈. Example 2: Convert 11010101₂ to octal. Group: 011 | 010 | 101 (pad left with 0 to make 3 bits). Convert: 011₂ = 3₈, 010₂ = 2₈, 101₂ = 5₈. Result: 325₈. Verify: 3×64 + 2×8 + 5×1 = 192+16+5 = 213₁₀. Binary verify: 128+64+16+4+1 = 213₁₀ ✓. Octal to Binary conversion: Replace each octal digit with its 3-bit binary equivalent. Ensure each digit produces exactly 3 bits (pad with leading zeros if necessary). Example 1: Convert 77₈ to binary. 7₈ = 111₂, 7₈ = 111₂. Result: 111111₂ (six bits). Decimal verify: 32+16+8+4+2+1 = 63₁₀ = 7×8+7×1 ✓. Example 2: Convert 144₈ to binary. 1₈ = 001₂, 4₈ = 100₂, 4₈ = 100₂. Result: 001100100₂ = 1100100₂ (leading zeros optional). Decimal: 64+32+4 = 100₁₀ = 1×64+4×8+4×1 ✓. Why 3-bit grouping works mathematically: \( 2^3 = 8 \), so three binary positions represent values 0-7, exactly matching octal digit range. Each octal position represents three binary positions. Historical computing used 12-bit, 18-bit, 24-bit, 36-bit word sizes (multiples of 3), making octal natural. Modern 8-bit bytes (not divisible by 3) favor hexadecimal (16 = \( 2^4 \), perfect for 8-bit = two hex digits). Comparison octal vs hexadecimal: Octal: base-8, digits 0-7, groups 3 bits, efficient for 12-bit systems, used Unix permissions. Hexadecimal: base-16, digits 0-9 A-F, groups 4 bits, perfect for 8/16/32/64-bit systems, used memory addresses, RGB colors (#FF00FF), debugging. Modern preference: Hexadecimal dominates because 8-bit bytes = 2 hex digits (00-FF = 0-255), making hex more compact and aligned with hardware than octal.
Why Choose RevisionTown's Decimal to Octal Converter?
RevisionTown's professional converter provides: (1) Bidirectional Conversion—Instantly convert decimal↔octal with accurate division-by-8 algorithm; (2) Step-by-Step Division Method—Shows complete division process with remainders for educational understanding; (3) Bulk Processing—Convert multiple values simultaneously for file permissions, programming, and system administration; (4) Large Number Support—Handles up to 9,999,999 decimal for comprehensive applications; (5) Formula Display—Demonstrates octal-to-decimal calculation with power-of-8 expansion verification; (6) Comprehensive Reference—Quick lookup table including Unix permission values 0-777; (7) Mobile Optimized—Responsive design works perfectly smartphones, tablets, desktops; (8) Zero Cost—Completely free with no ads, registration, or usage limitations; (9) Professional Accuracy—Trusted by computer science students, Unix/Linux system administrators, programmers, digital electronics students, and IT professionals worldwide for homework assignments (converting decimal 10 to octal 12 with division steps), Unix/Linux administration (understanding chmod 755 = 111 101 101 = rwxr-xr-x permissions structure), shell scripting (setting proper file permissions chmod 644, chmod 700), programming (legacy code maintenance requiring octal literals like 0755 in C, Python 0o755), network engineering (understanding historical octal subnetting, transponder codes), digital electronics (3-bit circuit design, ADC/DAC octal display panels), educational purposes (learning number systems, base conversion algorithms, positional notation), embedded systems (legacy firmware using octal notation for register configuration), debugging (reading octal dumps from old mainframe systems, PDP-8 emulation), and all applications requiring accurate decimal-octal conversions with clear step-by-step educational explanations for professional computer science, system administration, software development, and comprehensive computational mathematics worldwide.
❓ Frequently Asked Questions
10 (decimal) = 12 (octal). Division method: Step 1: 10 ÷ 8 = 1 remainder 2. Step 2: 1 ÷ 8 = 0 remainder 1. Read remainders bottom-to-top: 1, 2 → Octal: 12₈. Verification: \( 1 \times 8^1 + 2 \times 8^0 = 1 \times 8 + 2 \times 1 = 8 + 2 = 10 \) ✓. Meaning: 12₈ means 1 eight plus 2 ones = 10 decimal. Common confusion: 12 octal ≠ 12 decimal (12₁₀ = 14₈). Context matters—without subscript notation, decimal assumed. Binary relationship: 12₈ = 001 010₂ (each octal digit = 3 binary bits).
Division-by-8 method: (1) Divide decimal by 8. (2) Write remainder (0-7). (3) Replace number with quotient. (4) Repeat until quotient = 0. (5) Read remainders bottom-to-top = octal. Example: Convert 64 to octal. 64÷8=8 r0; 8÷8=1 r0; 1÷8=0 r1. Remainders up: 1,0,0 → 100₈. Verify: 1×64+0×8+0×1=64 ✓. Alternative (powers of 8): Find largest power of 8 ≤ number: 64 = 1×64 (8²) + 0×8 + 0×1. Digits: 1,0,0 → 100₈. Quick check: Result valid only if digits 0-7 (no 8 or 9 in octal). Binary shortcut: Convert decimal→binary, then group 3 bits→octal faster for some values.
Octal = base-8 number system using eight digits: 0, 1, 2, 3, 4, 5, 6, 7. No digit 8 or 9 exists in octal (like no digit 10 exists in decimal—next position needed). Each position represents power of 8 (right-to-left): \( 8^0=1, 8^1=8, 8^2=64, 8^3=512, 8^4=4096 \). Example: 157₈ = \( 1×64 + 5×8 + 7×1 = 64+40+7 = 111₁₀ \). Why computers used octal: Each octal digit = exactly 3 binary bits (0₈=000₂, 7₈=111₂). Early computers (1960s-70s) had 12-bit, 18-bit, 24-bit word sizes (multiples of 3), making octal compact notation for binary machine code. DEC PDP-8 (famous 12-bit computer) used octal extensively. Modern uses: Unix/Linux file permissions (chmod 755 = rwxr-xr-x); aviation transponder codes (4-digit octal 0000-7777); legacy systems. Hexadecimal largely replaced octal for modern 8-bit byte systems.
Multiply each octal digit by its positional power of 8, then sum. Formula: For octal \( o_2 o_1 o_0 \), decimal = \( o_2×8^2 + o_1×8^1 + o_0×8^0 \). Example: Convert 144₈ to decimal. Positions (right-to-left): 0,1,2. Octal: 1 4 4. Calculation: \( 1×8^2 + 4×8^1 + 4×8^0 = 1×64 + 4×8 + 4×1 = 64+32+4 = 100 \). Result: 144₈ = 100₁₀. Shortcut: Only calculate non-zero digits. 205₈: \( 2×64 + 0×8 + 5×1 = 128+5 = 133₁₀ \) (skip 0 middle digit). Quick mental: Memorize powers: 8¹=8, 8²=64, 8³=512, 8⁴=4096. Single octal digit (0-7) = same decimal value (5₈ = 5₁₀). Practice: 77₈ = 7×8+7 = 63₁₀; 100₈ = 64₁₀; 377₈ = 255₁₀.
chmod 755 = rwxr-xr-x permissions in Unix/Linux. Octal 755 breaks down: Owner=7, Group=5, Others=5. Owner permissions (7): 7 = 4+2+1 = read(4) + write(2) + execute(1) = rwx. Binary: 111. Full control—owner can read, modify, and execute file/enter directory. Group permissions (5): 5 = 4+0+1 = read(4) + no write + execute(1) = r-x. Binary: 101. Group members can read and execute but not modify. Others permissions (5): Same as group = r-x. Binary: 101. Everyone else can read and execute, not modify. Use cases: Executable scripts, programs (755 allows everyone to run but only owner to edit); website directories (755 lets web server read/list files; owner maintains); shared scripts (multiple users run, one owner manages). Security: Less restrictive than 700 (owner-only) but safer than 777 (world-writable). View permissions: ls -l shows drwxr-xr-x (directory) or -rwxr-xr-x (file). Set permissions: chmod 755 script.sh makes executable.
64 (decimal) = 100 (octal). Conversion: 64÷8=8 r0; 8÷8=1 r0; 1÷8=0 r1. Remainders up: 1,0,0 → 100₈. Verification: \( 1×8^2 + 0×8^1 + 0×8^0 = 1×64 + 0 + 0 = 64 \) ✓. Significance: 64 = \( 8^2 \) (exact power of 8, second power). Pattern: Powers of 8 in octal always single 1 followed by zeros: 8=10₈, 64=100₈, 512=1000₈, 4096=10000₈ (similar to powers of 10 in decimal: 10, 100, 1000). Number of zeros = exponent. Computing relevance: 64 = common computer value (64-bit architecture, base64 encoding, 8×8=64 chessboard squares, IPv6 prefix length /64). Binary: 64₁₀ = 1000000₂ (7 bits, single 1 in position 6). Octal grouping: 001 000 000₂ = 100₈ demonstrates 3-bit octal-binary relationship. Memory: 64 KB = 65,536 bytes (64×1024 = \( 2^{16} \)).
Group binary digits into sets of three (from right), convert each triplet to octal digit. Method: (1) Start from rightmost bit (least significant). (2) Group into 3-bit sections. (3) Pad leftmost group with zeros if needed to make 3 bits. (4) Convert each 3-bit group: 000₂=0₈, 001₂=1₈, 010₂=2₈, 011₂=3₈, 100₂=4₈, 101₂=5₈, 110₂=6₈, 111₂=7₈. (5) Concatenate octal digits. Example 1: 11010101₂ to octal. Group: 011 | 010 | 101 (pad left). Convert: 011₂=3₈, 010₂=2₈, 101₂=5₈. Result: 325₈. Verify: 3×64+2×8+5=213₁₀; binary: 128+64+16+4+1=213₁₀ ✓. Example 2: 111111₂ to octal. Group: 111 | 111. Convert: 7₈, 7₈. Result: 77₈ (decimal 63). Reverse (octal→binary): Replace each octal digit with 3-bit binary: 144₈ = 001 100 100₂ = 1100100₂. Fast bidirectional conversion without decimal intermediate step.
Octal provided compact notation for binary data in early computers with word sizes divisible by 3. (1) 3-bit grouping: Each octal digit = exactly 3 binary bits (8 = \( 2^3 \)). Enables instant mental conversion: 5₈ = 101₂ without arithmetic. Much shorter than writing full binary: 12-bit 101110011010₂ = 5632₈ (4 octal vs 12 binary digits). (2) Historical word sizes: Early computers used 12-bit (DEC PDP-8), 18-bit (PDP-1, PDP-15), 24-bit, 36-bit architectures—all divisible by 3. 12 bits = 4 octal digits perfect fit; 18 bits = 6 octal digits. Programmers read machine code, memory dumps in octal rather than unwieldy binary. (3) Assembly language: 1960s-70s programmers routinely worked in octal for instruction encoding, memory addresses, debugging. Example: PDP-8 instruction 7402₈ = specific opcode. (4) Modern decline: 8-bit byte (not divisible by 3) became standard. Hexadecimal (base-16 = \( 2^4 \)) maps perfectly: 8 bits = 2 hex digits (00-FF = 0-255). Hex replaced octal for modern systems. (5) Surviving uses: Unix permissions (chmod 755); legacy system emulation; niche applications where 3-bit grouping natural.
