The octal system, also known as base-8, is a numeral system that uses eight symbols to represent values. These symbols are the digits from 0 to 7. In contrast to the decimal system (base-10) that we commonly use, which has ten symbols (0 to 9), the octal system has a smaller set of symbols. Each position in an octal number represents a power of 8, similar to how each position in a decimal number represents a power of 10.

Here’s a breakdown of the octal system:

  1. Digits: The octal system uses the digits 0 to 7 as its symbols.
  2. Positional Notation: Just like the decimal system, each digit’s position in an octal number represents a power of 8. The rightmost digit represents 8^0 (which is 1), the next digit to the left represents 8^1 (which is 8), the next one represents 8^2 (which is 64), and so on.
  3. Conversion: To convert an octal number to its decimal equivalent, you can use the formula:
    decimal_value = d_n * 8^n + d_{n-1} * 8^{n-1} + … + d_1 * 8^1 + d_0 * 8^0 Here, d_n represents the digit at the nth position from the right (starting from 0), and n represents the position.
  4. Usage: The octal system was more commonly used in the past, especially in early computer systems. It’s less frequently used today, with the binary (base-2) and hexadecimal (base-16) systems being more prevalent in computing.
  5. Representation: In programming languages, an octal number is usually represented by placing a “0” (zero) prefix before the number. For example, “075” represents an octal number.
  6. Limitations: The octal system’s smaller symbol set means that octal numbers are more compact compared to their decimal equivalents, but they can quickly become longer than hexadecimal numbers for the same value.

Overall, the octal system is an interesting and historical numeral system that played a role in the early days of computing but has been largely replaced by other systems that better align with modern computing architectures.