Octal to decimal converter

What is the octal number system?

The octal number system is a positional numeral system that uses base 8. This means it uses eight distinct digits — 0, 1, 2, 3, 4, 5, 6, and 7 — to represent all numbers. Each position in an octal number represents a power of 8, just as in the decimal system, each position represents a power of 10. The system is shorter and more compact than decimal for certain computer operations, as it can represent large binary numbers (base 2) more simply by grouping bits in sets of three.

For example, the octal number 345₈ means:

$$345_8 = 3 \times 8^2 + 4 \times 8^1 + 5 \times 8^0$$

which equals $3 \times 64 + 4 \times 8 + 5 \times 1 = 192 + 32 + 5 = 229$ in decimal form.

The main advantage of using octal comes from its close relationship to binary. Since $8 = 2^3$, each octal digit exactly corresponds to three binary digits, simplifying the representation and conversion between these two number systems.

What is the decimal number system?

The decimal (base 10) system is the standard numeral system used in daily life. It uses ten digits — 0 to 9 — where each position denotes a power of 10. The rightmost digit represents units, the next left represents tens, then hundreds, and so on.

For example, the decimal number 347 can be expressed as:

$$347_{10} = 3 \times 10^2 + 4 \times 10^1 + 7 \times 10^0$$ $$= 3 \times 100 + 4 \times 10 + 7 \times 1 = 347$$

How the octal to decimal converter works

The octal to decimal converter on our website automatically converts a number written in base 8 into its decimal (base 10) counterpart. The converter interprets each octal digit, multiplies it by 8 raised to the power of its positional index, and then adds all these values together to produce the equivalent decimal number.

This tool simplifies and speeds up the manual conversion process, minimizing errors and saving time, especially when working with large numbers or programming tasks involving base conversions.

Step-by-step example

Let’s demonstrate the process using a smaller example:

Example: Convert octal number 36 in decimal number system.

Step 1: Expand by powers of 8:

$$36_8 = 3 \times 8^1 + 6 \times 8^0$$

Step 2: Compute each term:

$$3 \times 8 + 6 \times 1 = 24 + 6 = 30$$

Step 3: Add up the results:

$$30$$

Hence, $36_8 = 30_{10}$.

Practical uses of octal numbers

Although the octal system is not commonly used in everyday arithmetic, it played a key historical role in computing. Many early computer systems, such as the PDP series from the 1960s and 1970s, utilized octal notation because their word sizes (12, 24, or 36 bits) were multiples of three bits, which corresponded perfectly with one octal digit.

Even today, octal is occasionally used in programming, particularly in specifying file permissions in Unix and Linux systems. In these operating systems, each permission bit group for owner, group, and others corresponds to an octal digit:

• rwx (read, write, execute) permissions per user type can be expressed succinctly as an octal digit between 0 and 7. For example, the permission chmod 755 translates to: $7 = 111_2 = rwx$, $5 = 101_2 = r-x$, $5 = 101_2 = r-x$.

This correlation between binary and octal digits makes octal a convenient notation for representing low-level binary information.

Detailed examples

Example 1

Convert $542_8$ to decimal.

$$542_8 = (5 \times 8^2) + (4 \times 8^1) + (2 \times 8^0)$$ $$= (5 \times 64) + (4 \times 8) + (2 \times 1)$$ $$= 320 + 32 + 2 = 354$$

So $542_8 = 354_{10}$.

Example 2

Convert decimal number 78 to octal number.

Divide 78 by 8 and get the remainder:

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

$$78_{10} = 116_8$$

Notes

1. Octal representation never includes digits beyond 7. Any number containing 8 or 9 is not a valid octal number.
2. When converting from octal to decimal, the positional value increases by powers of 8 as you move leftward.
3. If the number includes fractional octal parts, the same principle applies to digits after the point — except powers of 8 are negative: $$3.47_8 = (3 \times 8^0) + (4 \times 8^{-1}) + (7 \times 8^{-2})$$ $$= 3 + 0.5 + 0.109375 = 3.609375_{10}$$

Frequently asked questions

How to convert octal number 345 to decimal number?

Break the digits and multiply by powers of 8:

$$3 \times 8^2 + 4 \times 8^1 + 5 \times 8^0 = 192 + 32 + 5 = 229$$

Thus, $345_8 = 229_{10}$.

How to recognize an invalid octal number?

If the number contains digits 8 or 9, it is invalid in octal since the highest allowed digit is 7. For instance, 128₈ is not valid.

How to convert decimal number 110 to octal number?

Divide 110 by 8 and get the remainder:

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

$$110_{10} = 156_8$$