Octal to Binary Converter

What Is the Octal Number System?

The octal number system uses base 8 and only the digits 0 through 7. Each position has a power of 8. This makes octal a compact way to represent binary values. For example, octal $7$ equals binary $111$, while octal $10$ equals binary $1000$.

What Is Binary System?

The binary system uses base 2 and only two digits, 0 and 1. Computers use binary because electronic circuits can represent two clear states. Each binary digit is one bit. For example, octal $5$ becomes binary $101$, and octal $10$ becomes binary $1000$.

How to Convert Octal to Binary Manually

You can convert octal to binary in three useful ways. The indirect method changes octal to decimal first, then changes decimal to binary. The direct method replaces every octal digit with its three bit binary form. You can also calculate the binary value without looking at a conversion table. The direct method usually takes the fewest steps. Always check that every octal digit falls between 0 and 7 before starting. Use a clear example beside your work so you can spot missing bits or reversed remainders.

Indirect Method

Formula

$N_{10} = d_n \times 8^n + ... + d_1 \times 8^1 + d_0 \times 8^0$

Step by step Guide

1. Write the octal digits from right to left.
2. Give the rightmost digit the power $8^0$.
3. Increase the power by one for each position.
4. Multiply each digit by its power of 8.
5. Add the values to get the decimal number.
6. Divide the decimal number by 2 repeatedly.
7. Record each remainder.
8. Read the remainders from bottom to top.

Practical Example

Convert $157_8$ to binary.

$157_8 = 1 \times 8^2 + 5 \times 8^1 + 7 \times 8^0$

$= 64 + 40 + 7 = 111_{10}$

Now convert $111_{10}$ to binary.

$111 \div 2 = 55$ remainder $1$

$55 \div 2 = 27$ remainder $1$

$27 \div 2 = 13$ remainder $1$

$13 \div 2 = 6$ remainder $1$

$6 \div 2 = 3$ remainder $0$

$3 \div 2 = 1$ remainder $1$

$1 \div 2 = 0$ remainder $1$

Read the remainders upward:

$157_8 = 1101111_2$

Direct Method

Formula

$1$ octal digit $=$ $3$ binary bits.

For example, $5_8 = 101_2$.

Step by step Guide

1. Take each octal digit separately.
2. Replace it with its three bit binary form.
3. Keep zeros inside each three bit group.
4. Join the groups together.
5. Remove extra zeros from the beginning if needed.

Practical Example

Convert $157_8$ to binary.

$1_8 = 001_2$

$5_8 = 101_2$

$7_8 = 111_2$

Join the groups:

$157_8 = 001\ 101\ 111_2$

Remove the leading zeros:

$157_8 = 1101111_2$

This method works quickly because every octal digit matches exactly three binary bits.

Octal to Binary Conversion without Using the Conversion Table

Formula

$d = 4b_2 + 2b_1 + 1b_0$

Here, each binary bit uses a place value of $4$, $2$, or $1$.

Step by step Guide

1. Take one octal digit at a time.
2. Check whether the digit contains a $4$.
3. Check whether the remaining value contains a $2$.
4. Check whether the remaining value contains a $1$.
5. Write $1$ when a value fits the place.
6. Write $0$ when it does not fit.
7. Keep three binary positions for every octal digit.
8. Join all three bit groups.

Practical Example

Convert $526_8$ without using a table.

For $5$:

$5 = 4 + 1 = 101_2$

For $2$:

$2 = 2 + 0 + 0 = 010_2$

For $6$:

$6 = 4 + 2 = 110_2$

Join the groups:

$526_8 = 101\ 010\ 110_2$

Therefore:

$526_8 = 101010110_2$

Octal to Binary conversion table

Note: Each octal digit maps to exactly three binary bits. Keep leading zeros inside each group, then remove only extra zeros at the start of the final whole number.

Octal to Binary Conversion Applications

Octal to binary conversion helps students learn number systems, binary coding, and digital electronics. It also appears in programming, computer science, logic circuits, and data representation. Engineers and developers can use base conversion when working with low level values.