View source for Octal: Reading & Conversion

= Octal: Reading & 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 values in octal ==
Each column has a value 8 times the column to its right.  
This corresponds to powers of 8.

From right to left:
* 8⁰ = 1
* 8¹ = 8
* 8² = 64
* 8³ = 512
* 8⁴ = 4096
* 8⁵ = 32 768
* 8⁶ = 262 144
* 8⁷ = 2 097 152

Example: 76225₈ =  
7 × 8⁴ + 6 × 8³ + 2 × 8² + 2 × 8¹ + 5 × 8⁰  
= 7 × 4096 + 6 × 512 + 2 × 64 + 2 × 8 + 5 × 1  
= 28 672 + 3072 + 128 + 16 + 5  
= 31 893₁₀.

== From decimal to octal ==
Method: Division by 8
1. Divide the decimal number by 8.
2. Record the remainder.
3. Repeat with the quotient until 0.
4. Read the remainders backwards.

Example: Convert 83₁₀ to octal  
83 ÷ 8 = 10 remainder 3  
10 ÷ 8 = 1 remainder 2  
1 ÷ 8 = 0 remainder 1  
Result: 123₈.

== Binary to octal ==
Binary can be grouped directly into octal digits.
* Each octal digit = 3 binary digits (bits).
* Group binary digits into blocks of 3, starting from the right.
* Convert each block to a single octal digit.

Example: 101110₂ → group as 101 110 → 5 6 → 56₈.  
Check: 101110₂ = 46₁₀, and 56₈ = 46₁₀.

== Octal to binary ==
Each octal digit expands to 3 binary digits.

Example: 725₈ → 7 = 111, 2 = 010, 5 = 101 → 111010101₂.

== Programming notes ==
In the C programming language, a number starting with 0 is interpreted as octal.  
* Example: 012 means 10 decimal, not twelve.  
* This can lead to confusion.  
* In modern languages, octal literals are often written with a prefix like 0o (Python, Rust).

== Arithmetic in octal ==
Arithmetic is similar to decimal but carries occur after 7.

Addition example: 57₈ + 25₈  
57₈ = 47₁₀, 25₈ = 21₁₀  
47 + 21 = 68  
68 in octal: 104₈.  
So 57₈ + 25₈ = 104₈.

Subtraction and multiplication follow the same principles, always working with base 8.

[[Category:Numeral Systems]]