What is an octal number?

An octal number is a number written in base 8. It uses eight digits: 0, 1, 2, 3, 4, 5, 6 and 7. Each position represents a power of 8 instead of a power of 10 as in the decimal system.

How do you convert octal to decimal?

To convert octal to decimal, multiply each digit by 8 raised to the power of its position index from right to left, starting at 0, and then add the results. For example, 157₈ = 1×8² + 5×8¹ + 7×8⁰ = 64 + 40 + 7 = 111₁₀.

How do you convert decimal to octal?

To convert decimal to octal, repeatedly divide the decimal number by 8, keeping track of the remainders. The octal representation is the sequence of remainders read from last to first. For example, 111₁₀ ÷ 8 gives remainders 7, 5, 1, so 111₁₀ = 157₈.

How do you convert between octal and binary?

Each octal digit corresponds exactly to three binary bits. To convert octal to binary, replace each octal digit with its 3-bit binary equivalent (e.g., 7₈ = 111₂). To convert binary to octal, group the binary digits into sets of three from the right and convert each group to an octal digit.

Where is the octal number system used?

Octal was historically used in computing on systems where word sizes were multiples of 3 bits, such as 12, 24 or 36 bits. It is still seen in Unix-style file permissions and in some low-level programming and digital electronics contexts.

Octal Number System Converter – Octal to Binary, Decimal, Hex

What is the octal number system?

The octal number system is a positional numeral system with base 8. It uses exactly eight digits: 0, 1, 2, 3, 4, 5, 6, 7. Each position represents a power of 8, just as each position in decimal represents a power of 10.

For an octal number like \( (a_n a_{n-1} \dots a_1 a_0)_8 \), its decimal value is

\[ (a_n a_{n-1} \dots a_1 a_0)_8 = \sum_{k=0}^{n} a_k \cdot 8^k,\quad a_k \in \{0,\dots,7\} \]

Place values in octal

From right to left, the place values are:

\(8^0 = 1\)
\(8^1 = 8\)
\(8^2 = 64\)
\(8^3 = 512\)
\(8^4 = 4096\)

So the octal number \(157_8\) means:

\[ 157_8 = 1\cdot 8^2 + 5\cdot 8^1 + 7\cdot 8^0 = 64 + 40 + 7 = 111_{10} \]

How to convert octal to decimal

Write down the octal digits and their positions (starting from 0 on the right).
Multiply each digit by \(8^{\text{position}}\).
Add all the products.

Example: \( 345_8 \) to decimal

\[ 345_8 = 3\cdot 8^2 + 4\cdot 8^1 + 5\cdot 8^0 = 3\cdot 64 + 4\cdot 8 + 5\cdot 1 = 192 + 32 + 5 = 229_{10} \]

How to convert decimal to octal

Use repeated division by 8 and track the remainders.

Divide the decimal number by 8.
Record the remainder (0–7).
Use the quotient as the new number and repeat until the quotient is 0.
The octal digits are the remainders read from last to first.

Example: \( 229_{10} \) to octal

229 ÷ 8 = 28 remainder 5   → least significant digit
 28 ÷ 8 =  3 remainder 4
  3 ÷ 8 =  0 remainder 3   → most significant digit

Read remainders from bottom to top: 3 4 5 → 345₈

Octal and binary: a perfect match

Octal is especially convenient in computing because each octal digit corresponds exactly to three binary bits:

Octal	Binary (3 bits)
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Octal → binary

Replace each octal digit with its 3-bit binary equivalent.
Optionally remove leading zeros.

Example: \( 57_8 \)

\[ 5_8 = 101_2,\quad 7_8 = 111_2 \Rightarrow 57_8 = 101111_2 \]

Binary → octal

Group the binary digits into sets of 3 from right to left (pad with leading zeros if needed).
Convert each 3-bit group to its octal digit.

Example: \( 101111_2 \)

Group: 101 111
101₂ = 5₈, 111₂ = 7₈ → 57₈

Octal and hexadecimal

Both octal (base 8) and hexadecimal (base 16) are used to represent binary data more compactly:

1 octal digit ↔ 3 binary bits
1 hex digit ↔ 4 binary bits

To convert between octal and hex, it is usually easiest to go through binary: octal → binary → hex or hex → binary → octal. The converter above does this for you automatically.

Where is octal used?

Historical computers with word sizes that are multiples of 3 bits (12, 24, 36 bits).
Unix and Linux file permissions (e.g. 755, 644 are octal codes).
Digital electronics and low-level programming in some legacy systems.

Common questions about octal

Is 8 a valid octal digit?

No. Valid octal digits are only 0–7. If you see 8 or 9, the number is not a valid octal number.

How do I check if a number is octal?

A string is a valid octal literal if it contains only the characters 0–7, optionally with a leading sign (+ or -) and a single decimal point if you allow fractions. The calculator validates this for you and highlights invalid input.

Octal Number System Converter

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