Octal to binary converter

What is the binary number system?

The binary system is a positional numeral system that uses only two digits: 0 and 1. Each digit in a binary number represents a power of 2, starting from the rightmost bit, which is $2^0$. This system is the foundation of modern computing because it aligns perfectly with the ON/OFF logic of electronic circuits.

For example, the binary number $1011_2$ can be interpreted as:

$$(1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 8 + 0 + 2 + 1 = 11_{10}$$

What is the octal number system?

The octal system (base 8) uses digits from 0 to 7. It is sometimes used in computing as a more compact way to represent binary numbers, since each digit in an octal number corresponds exactly to three binary bits. This makes octal particularly convenient for working with binary-encoded data.

Example: $765_8$ stands for:

$$(7 \times 8^2) + (6 \times 8^1) + (5 \times 8^0) = 448 + 48 + 5 = 501_{10}$$

Formula for conversion

The most straightforward way to convert an octal number to binary is to replace each octal digit with its equivalent 3-bit binary representation.

Here is the conversion chart for each octal digit to binary:

But you can use also 2-step conversion: first convert octal to decimal, then convert decimal to binary.

Example conversion

Let’s convert the octal number 65₈ to binary.

Step 1: Convert each octal digit to its 3-bit binary equivalent

Step 2: Combine the binary groups

$$65_8 = 110101_2$$

Thus, the octal number 65 in binary form is 110101.

Verification

To verify correctness, let’s convert octal number to decimal, then convert decimal number to binary.

Octal to decimal:

$$(6 \times 8^1) + (5 \times 8^0) = 48 + 5 = 53_{10}$$

Decimal to binary:

Reading the remainders from bottom to top gives the binary result:

$$53_{10} = 110101_2$$

Interesting historical fact

Early computers like the PDP-8 (developed by Digital Equipment Corporation) used octal as their primary numeric representation system. This was because its machine words were 12 bits long, easily represented as four octal digits. It simplified the reading and manual entry of binary instructions.

Notes

• Each octal digit corresponds exactly to three binary digits.
• Leading zeroes can be omitted without changing numerical value.
• Always read binary groups from left to right in the same order as the octal digits.

Frequently asked questions

How to convert an octal number 123₈ to binary?

Convert each digit separately: 1 → 001, 2 → 010, 3 → 011
Combine: $001010011_2$ or $1010011_2$ after removing leading zeros.

How many binary bits are needed to represent one octal digit?

Each octal digit corresponds to three binary bits.

123 from octal to binary

Let’s convert the octal number 123₈ to decimal.

Octal to decimal:

$$(1 \times 8^2) + (2 \times 8^1) + (3 \times 8^0) = 64 + 16 + 3 = 83_{10}$$

Decimal to binary:

So, $123_8 = 1010011_2$.

Can binary numbers be converted back to octal easily?

Yes. Group binary digits into sets of three bits from right to left and replace each with its corresponding octal digit.

Why do computers use binary and not octal?

Computers use binary because it directly corresponds to physical states (ON or OFF). Octal is used only as a human-readable shorthand for binary data.