Octal Number System

Octal Number System
Octal Number System

Computer science and digital electronics employ the base-8 octal number system. Our daily decimal system is identical to this positional number system. An Octal digits indicate powers of 8, with the rightmost digit representing units, the next the 8s, and so on. Octal numbers can represent three binary digits, making them more compact.An  Octal numbers are useful in programming and Unix permissions.

Octal Number System Definition

The positional octal number system employs eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each octal digit is a power of 8. Rightmost digit of octal number 347 is units (7), next digit is 8s (4 multiplied by 8), and the leftmost digit is 64s (3 multiplied by 8 squared). Computer scientists utilize octal numbers to express groups of three binary digits (bits) more compactly.

1.2 Octal Number System History
1.2 Octal Number System History

The ancient Egyptians and Mayans utilized an octal number system. The contemporary octal number system was popularized in the mid-20th century by computers. Its direct association with the binary number system, the core of computers, made it important in computer science. Early computer systems and programming languages relied on octal numbers. Octal numerals are still employed in computer science, digital electronics, and Unix-based operating systems.


Wonder what those 0–7 numbers mean? In the octal number system, everything revolves around eights. Here’s where to start with octal. This simple tutorial will bring you up to speed quickly. You’ll master octal conversion and arithmetic by the end.

Introduction to Octal Numbers

The base-8 octal number system employs eight numerals (0-7) to represent numbers. The base-10 decimal system you use every day uses ten digits (0-9).

Decimal to Octal Conversion

To convert a decimal number to octal, repeatedly divide by 8 and record the remainders from right to left. This remaining sequence is the octal number.

For instance, octalize 65:

65/8=8 remaining 1/8 = 1 remaining 0

The octal representation of 65 is 101.

Octal addition and subtraction

Add and subtract in octal is comparable to decimal, but remember you’re in base 8. Add or subtract octal digits, carrying or borrowing when appropriate.

As an example:

Octal 546

Octal 273

821 (noctal)

Also, subtract.

Octal 721

An Octal 354

Octal 367

the Octal Multiplication and Dividing

An Octal multiplication operates like decimal multiplication, but you must carry any product greater than 7.

As an example:

Octal 324 x 15

A Octal 2610

The octal division operates like decimal long division. Divide, multiply, subtract, and bring down remainders until none remain.

Using base 8 may appear unusual at first, but experience will make it familiar. Have more questions? Let me know!

Octal, Decimal, Binary Conversion

Converting octal, decimal, and binary numbers is easy. Just remember the place values for each number system.

Octal numbers are powers of 8, just like decimal numbers are powers of 10. The octal number 738 is equivalent to 448 in decimal form.

Start at the right, multiply each octal digit by the place value, and sum up the results to convert to decimal. For instance, 127 is (1 x 8^2) + (2 x 8^1) + (7 x 8^0) = 87 in decimal.

Changing decimal to octal is the opposite. Divide the decimal by 8 to get the rightmost octal. Divide by 8 and the right-to-left remainders become the octal number. To convert 79 to octal: 9 remains from 79/8 The rightmost octal digit is 7. 9/8 equals 1 (1 is the middle octal digit) Remainder 1 (1 is the leftmost octal digit) So 79 decimal is 117 octal.

To switch between octal and binary, remember that each octal digit is 3 binary digits. So 738 octal equals 111 101 000 binary. So 100101 binary is 155 octal.

Switching number systems can be enjoyable arithmetic challenge once you get used to it. Understanding different bases helps you understand our number system. Mastering octal conversion will impress at all those cool coding parties!

Octal Arithmetic

Calculating with the octal number system employs base-8 instead of base-10, but the rules are the same. Since octal only uses 0-7, you must “carry” more. But octal math can be mastered with practice.

Add two octal numbers by adding the digits in the columns, carrying to the left column if the sum is higher than 7. To add 135 and 246:

135 +246 = 381

Borrow from the left column to subtract. Example: subtract 246 from 531:

531 -246 = 285

Multiplication involves multiplying numerals but carrying more often. Basic steps:

Multiply ones column digits.

If the product exceeds 7, move it to the next column.

Write the last 1.

Repeat in the following column.

To multiply 53×32:

3×3 = 9, carry 1 160+600 = 1766 (5 x 2 = 10, carry 1, write 0).

The lengthy division technique for decimal numbers applies to octal. Divide by octal divisors and the remainders will always be 0–7. Remember that 8 = 10, so “carry” extra. Regular practice will make octal division easy.

The octal number system may seem unfamiliar, yet its base-8 arithmetic follows the same reasoning. Regular use can make octal computations automatic. If you get stuck, think step-by-step, carry, and double-check. Soon, you’ll think octal!

Real-World Octal Number Applications

The octal number system has practical uses. Linux and Unix file permissions use octal numbers, which IT professionals are familiar with.

File Rights

Linux uses octal numbers for file and folder permissions. Each octal digit represents user, group, and other permissions. Zero means no permission, seven means full. For instance, the octal number 755 grants read, write, and execute permission to the user (7), group (5), and others (5). This system simplifies Linux file and folder permissions with octal numbers.

Color Codes RGB

Octal numbers are used to represent RGB colors. Each main color—red, green, and blue—has an octal digit from 0 to 7. Combining the three octal digits lets you define any RGB color. For instance, blue is 067, red 700, and yellow 070. For precision and simplicity, several graphic and online design applications provide RGB color codes in octal.

File Compression (Deflate)

Deflate compression, utilized in ZIP, PNG, and HTTP/2, represents code lengths, literals, and match distances in octal numbers. The Deflate algorithm encodes repeating strings in input data with octal numbers for high compression. Deflate can be implemented in programs using open-source tools like zlib.

Octal numbers are useful in computers and technology. Octal numbers are less popular than decimal and binary number systems, although they can express information efficiently. Understanding the octal system will help you understand these applications.

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