Number SystemFree Interactive Tool

Octal to Binary

Convert octal numbers into binary by expanding each octal digit into its 3-bit binary triplet. Designed for software developers, electrical engineers, students, and computer architecture researchers requiring deterministic client-side accuracy.

Engine: Client-Side Verified (0ms Latency)

Octal String (Base 8)

Enter valid octal digits (0 to 7).

Calculated Output

Primary Representation

Calculating...

Step-by-Step Mathematical Proof

Active Derivation

Ready. Enter input above to generate derivation.

Concept

What Is Octal to Binary?

Octal to binary conversion translates base-8 numbers into raw binary bits. Each octal digit maps directly to a 3-bit binary triplet without requiring any intermediate decimal conversion.

Methodology

How Does It Work?

Expand each octal digit into its exact 3-bit binary pattern (0=000, 1=001, 2=010, 3=011, 4=100, 5=101, 6=110, 7=111) and concatenate them together.

Formula & Rules

Mathematical Algorithm

d_k \in \{0..7\} \rightarrow [b_2 b_1 b_0]_2 \quad \text{where } d_k = 4b_2 + 2b_1 + b_0

Worked Problem

Step-by-Step Example

Convert 327_8 to binary: 3 -> 011 2 -> 010 7 -> 111 Concatenating yields: 011010111_2 (or 11010111_2 without leading zero).

Important Rules & Edge Cases

Each octal digit must expand into exactly 3 binary bits (e.g. 2 is 010, not just 10).
Input can only contain digits 0 through 7.

Practical Applications in Engineering

Converting chmod octal permissions (e.g., 755) into binary bitmasks (111 101 101).
Configuring digital hardware multiplexers and bus selectors.
Translating legacy microcode instructions into binary control lines.