Binary, Decimal, Hexadecimal, and Octal Systems

 Binary, Decimal, Hexadecimal, and Octal Systems

Binary, decimal, hexadecimal, and octal are different number systems used for representing and working with numbers in computing and mathematics. Each system has its own base and set of symbols for representing numeric values. Here's an overview of each of these number systems:


1. Binary System (Base-2):

  • Base: Binary is a base-2 number system, meaning it uses only two symbols: 0 and 1.

  • Representation: In binary, each digit represents a power of 2, starting from the rightmost digit (2^0) and increasing by 1 for each digit to the left.

  • Example: The binary number 1101 is equivalent to 1 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 = 8 + 4 + 0 + 1 = 13 in decimal.


2. Decimal System (Base-10):

  • Base: Decimal is the most common number system and is used worldwide. It is a base-10 system, using the digits 0 through 9.

  • Representation: In decimal, each digit represents a power of 10, starting from the rightmost digit (10^0) and increasing by 1 for each digit to the left.

  • Example: The decimal number 426 is equivalent to 4 * 10^2 + 2 * 10^1 + 6 * 10^0 = 400 + 20 + 6 = 426.


3. Hexadecimal System (Base-16):

  • Base: Hexadecimal is a base-16 number system, using the digits 0-9 and the letters A-F (or a-f) to represent values from 0 to 15.

  • Representation: In hexadecimal, each digit represents a power of 16, starting from the rightmost digit (16^0) and increasing by 1 for each digit to the left.

  • Example: The hexadecimal number 1A3 is equivalent to 1 * 16^2 + 10 * 16^1 + 3 * 16^0 = 256 + 160 + 3 = 419 in decimal.


4. Octal System (Base-8):

  • Base: Octal is a base-8 number system, using the digits 0-7.

  • Representation: In octal, each digit represents a power of 8, starting from the rightmost digit (8^0) and increasing by 1 for each digit to the left.

  • Example: The octal number 347 is equivalent to 3 * 8^2 + 4 * 8^1 + 7 * 8^0 = 192 + 32 + 7 = 231 in decimal.


These different number systems are used in various aspects of computing and programming. Binary is fundamental in digital systems and low-level programming. Hexadecimal is commonly used in computer science and programming for representing binary data more concisely. Octal is less commonly used today but has historical significance in computing. Decimal, being the most common number system, is used for most everyday calculations and representation of numeric values in various contexts. Converting between these number systems is a fundamental skill for computer scientists and programmers.

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