Octal to Binary Converter

Step 1 — Identify the symbols

Input (Octal): Each digit is 0–7 (eight possibilities per position).

Output (Binary): Each position is either 0 or 1.

Step 2 — The Two-Step Method (via Decimal)

Part A — Into decimal

Number positions from the right, starting at 0. At each position, multiply that digit by 8 raised to the position index, then add every term. The total is your decimal number.

Digits after the dot: use negative powers of 8 (8⁻¹, 8⁻², …). Each place is still (digit × weight); add the fractional side to the whole side.

Part B — Out of decimal

Repeated division by 2: divide the decimal number by 2, write each remainder, replace the number with the whole quotient, repeat until the quotient is 0. Read the remainders from last division to first to read the binary digits.

Fractional decimal values: convert the whole part with repeated division, then multiply the fractional part by the target base over and over; each integer you get is the next digit after the radix point (same idea the calculator shows in Formulas).

Step 3 — Worked examples

Two practice values in Octal, converted to Binary using the same rules as Step 2. Example 3 uses a fractional part (digits after the radix point); the hub and pair calculators accept a single . on binary, octal, decimal, and hex inputs.

Octal "12"
= 1×8¹ + 2×8⁰
= 8 + 2
= 10  (decimal)

Whole part — repeated division:
10 ÷ 2 = 5  R 0
5 ÷ 2 = 2  R 1
2 ÷ 2 = 1  R 0
1 ÷ 2 = 0  R 1

Read remainders bottom → top → 1010

→ tool: 0b1010

Octal "377"
= 3×8² + 7×8¹ + 7×8⁰
= 192 + 56 + 7
= 255  (decimal)

Whole part — repeated division:
255 ÷ 2 = 127  R 1
127 ÷ 2 = 63  R 1
63 ÷ 2 = 31  R 1
31 ÷ 2 = 15  R 1
15 ÷ 2 = 7  R 1
7 ÷ 2 = 3  R 1
3 ÷ 2 = 1  R 1
1 ÷ 2 = 0  R 1

Read remainders bottom → top → 11111111

→ tool: 0b11111111

Octal "12.4"
Left of . : 1×8¹ + 2×8⁰ = 10
Right of .: 4×8^-1 = 0.5
= 10.5  (decimal)

Whole part — repeated division:
10 ÷ 2 = 5  R 0
5 ÷ 2 = 2  R 1
2 ÷ 2 = 1  R 0
1 ÷ 2 = 0  R 1

Read remainders bottom → top → 1010

Fractional part — multiply by 2, integer of each product = next digit after .
  0.5×2 = 1  →  1
Digits after . (in order): .1

→ tool: 0b1010.1