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<p>Created page with &quot;= Octal: Reading &amp; Conversion = This page introduces the octal numeral system, explains how it relates to binary, and shows how to convert between octal, decimal, and binary. == Introduction == Octal uses base 8. Digits are 0–7. The number written as 10₈ means 8 in decimal. Octal was more common in early computers, especially before hexadecimal became dominant. It is still useful for compactly representing binary numbers grouped in 3 bits. == Positional va...&quot;</p> <p><b>New page</b></p><div>= Octal: Reading &amp; Conversion =<br /> This page introduces the octal numeral system, explains how it relates to binary, and shows how to convert between octal, decimal, and binary.<br /> <br /> == Introduction ==<br /> Octal uses base 8. <br /> Digits are 0–7. <br /> The number written as 10₈ means 8 in decimal. <br /> <br /> Octal was more common in early computers, especially before hexadecimal became dominant. <br /> It is still useful for compactly representing binary numbers grouped in 3 bits.<br /> <br /> == Positional values in octal ==<br /> Each column has a value 8 times the column to its right. <br /> This corresponds to powers of 8.<br /> <br /> From right to left:<br /> * 8⁰ = 1<br /> * 8¹ = 8<br /> * 8² = 64<br /> * 8³ = 512<br /> * 8⁴ = 4096<br /> * 8⁵ = 32 768<br /> * 8⁶ = 262 144<br /> * 8⁷ = 2 097 152<br /> <br /> Example: 76225₈ = <br /> 7 × 8⁴ + 6 × 8³ + 2 × 8² + 2 × 8¹ + 5 × 8⁰ <br /> = 7 × 4096 + 6 × 512 + 2 × 64 + 2 × 8 + 5 × 1 <br /> = 28 672 + 3072 + 128 + 16 + 5 <br /> = 31 893₁₀.<br /> <br /> == From decimal to octal ==<br /> Method: Division by 8<br /> 1. Divide the decimal number by 8.<br /> 2. Record the remainder.<br /> 3. Repeat with the quotient until 0.<br /> 4. Read the remainders backwards.<br /> <br /> Example: Convert 83₁₀ to octal <br /> 83 ÷ 8 = 10 remainder 3 <br /> 10 ÷ 8 = 1 remainder 2 <br /> 1 ÷ 8 = 0 remainder 1 <br /> Result: 123₈.<br /> <br /> == Binary to octal ==<br /> Binary can be grouped directly into octal digits.<br /> * Each octal digit = 3 binary digits (bits).<br /> * Group binary digits into blocks of 3, starting from the right.<br /> * Convert each block to a single octal digit.<br /> <br /> Example: 101110₂ → group as 101 110 → 5 6 → 56₈. <br /> Check: 101110₂ = 46₁₀, and 56₈ = 46₁₀.<br /> <br /> == Octal to binary ==<br /> Each octal digit expands to 3 binary digits.<br /> <br /> Example: 725₈ → 7 = 111, 2 = 010, 5 = 101 → 111010101₂.<br /> <br /> == Programming notes ==<br /> In the C programming language, a number starting with 0 is interpreted as octal. <br /> * Example: 012 means 10 decimal, not twelve. <br /> * This can lead to confusion. <br /> * In modern languages, octal literals are often written with a prefix like 0o (Python, Rust).<br /> <br /> == Arithmetic in octal ==<br /> Arithmetic is similar to decimal but carries occur after 7.<br /> <br /> Addition example: 57₈ + 25₈ <br /> 57₈ = 47₁₀, 25₈ = 21₁₀ <br /> 47 + 21 = 68 <br /> 68 in octal: 104₈. <br /> So 57₈ + 25₈ = 104₈.<br /> <br /> Subtraction and multiplication follow the same principles, always working with base 8.<br /> <br /> [[Category:Numeral Systems]]</div>

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