Octal converter

What is the octal number system?

The octal number system, also known as base-8, is a positional numeral system that uses eight digits — 0, 1, 2, 3, 4, 5, 6, and 7. Each digit’s position represents a power of 8, much like the decimal system (base-10) uses powers of 10. For example, in the number $135_8$, the leftmost digit 1 represents $1 \times 8^2$, the middle digit 3 represents $3 \times 8^1$, and the last digit 5 represents $5 \times 8^0$.

Thus, the value of $135_8$ in the decimal system can be calculated as follows:

$$135_8 = (1 × 8^2) + (3 × 8^1) + (5 × 8^0) = 64 + 24 + 5 = 93_{10}$$

This number system was widely used in early computer systems because three binary digits correspond precisely to one octal digit (as $2^3 = 8$). Therefore, converting from binary to octal and vice versa is simple and efficient.

How the calculator converter works

The octal converter allows users to convert numbers from any numeral system (between base 2 and base 36) directly into the octal system. You can input a binary, decimal, hexadecimal, or even alphanumeric base 36 number, and the converter will automatically display its equivalent in base 8.

The process involves two steps:

1. Convert the input number (in its original base) into a decimal number.
2. Convert the resulting decimal number into octal.

Even though this process can be done manually, the converter performs it instantly and with full precision.

Formula

To convert a decimal number $N_{10}$ into its octal equivalent $N_{8}$, the following algorithm is applied:

1. Divide the decimal number $N_{10}$ by 8.
2. Record the remainder — it becomes the least significant digit (rightmost) of the octal number.
3. Use the quotient as the new number and repeat the division by 8 until the quotient equals 0.
4. Write the remainders in reverse order — this forms the octal representation.

Mathematically, this can be represented as:

$$N_8 = \sum_{k=0}^{m} r_k \times 8^k$$

where $r_k$ are the remainders obtained in each division step.

Example 1 — Convert decimal to octal

Let’s convert the decimal number 600 into octal manually.

Now, reading the remainders from bottom to top gives the octal result:

$$600_{10} = 1130_{8}$$

Example 2 — Binary to octal conversion

Let’s convert $101101010_2$ (binary) to octal.

1. Convert the binary number to decimal:

$$101101010_2 = 1×2^8 + 0×2^7 + 1×2^6 + 1×2^5 + 0×2^4 + 1×2^3 + 0×2^2 + 1×2^1 + 0×2^0 = 362_{10}$$

2. Convert the decimal number to octal:

Reading remainders from bottom to top gives the octal result:

$$362_{10} = 552_{8}$$

Example 3 — Hexadecimal to octal conversion

Let’s convert $1A_{16}$ (hexadecimal) to octal.

Step 1: Convert to decimal.
$1A_{16} = 1 \times 16 + 10 = 26_{10}$

Step 2: Convert decimal to octal.

Reading remainders from bottom to top:

$$1A_{16} = 32_{8}$$

Frequently asked questions

How to convert 3 from decimal to octal manually?

To convert the decimal number 3 to octal manually, follow these steps:

1. Divide the number by 8 and note the quotient and remainder:

$$3÷8=0(quotient),remainder=3$$

So $3_{10} = 0_8 + 3_{8}$.

2. Stop when the quotient is 0. The remainders, read from last to first, form the octal equivalent.

3. Read the remainder: The remainder 3 is the only digit needed.

Thus, the octal equivalent of decimal 3₁₀ is 3₈.

How many digits are used in the octal system?

The octal system uses eight digits — 0 through 7 — to represent all numbers.

How to convert an octal number to decimal?

Multiply each octal digit by the corresponding power of 8 and sum the results.
Example: $127_8 = 1×8^2 + 2×8^1 + 7×8^0 = 64 + 16 + 7 = 87_{10}$.

What is the main difference between octal and hexadecimal systems?

The octal system is base-8, using digits 0–7, while the hexadecimal system is base-16, using digits 0–9 and letters A–F. Hexadecimal can represent larger numbers with fewer digits.