Number systems are methods used to represent and manipulate numbers. Here’s an overview of three commonly used number systems:
- Binary Number System:
- The binary number system is a base-2 numbering system, meaning it uses only two symbols: 0 and 1.
- Each digit in a binary number represents a power of 2, starting from the rightmost digit.
- For example, the binary number 1010 represents (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0) = 10 in decimal.
- Binary numbers are commonly used in digital electronics and computer systems because they directly represent the on-off states of electronic switches.
- Octal Number System:
- The octal number system is a base-8 numbering system, meaning it uses eight symbols: 0, 1, 2, 3, 4, 5, 6, and 7.
- Each digit in an octal number represents a power of 8, starting from the rightmost digit.
- For example, the octal number 73 represents (7 × 8^1) + (3 × 8^0) = 59 in decimal.
- Octal numbers are less common in modern computing but were historically used in some computer systems and programming languages.
- Hexadecimal Number System:
- The hexadecimal number system is a base-16 numbering system, meaning it uses sixteen symbols: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, and F=15).
- Each digit in a hexadecimal number represents a power of 16, starting from the rightmost digit.
- For example, the hexadecimal number 2F represents (2 × 16^1) + (F × 16^0) = 47 in decimal.
- Hexadecimal numbers are widely used in computing, especially in programming and digital electronics, due to their compact representation of binary data.
Conversion between Number Systems:
- Binary to Decimal: To convert a binary number to decimal, multiply each digit by 2 raised to the power of its position from the rightmost digit and sum the results.
- Decimal to Binary: To convert a decimal number to binary, repeatedly divide the decimal number by 2 and record the remainders from each division, reading the remainders in reverse order.
- Binary to Octal/Hexadecimal: Group binary digits into sets of three (for octal) or four (for hexadecimal), and then convert each group to the corresponding octal or hexadecimal digit.
- Octal/Hexadecimal to Binary: Convert each octal or hexadecimal digit to its binary equivalent.
- Octal/Hexadecimal to Decimal: To convert octal or hexadecimal to decimal, multiply each digit by the base raised to the power of its position from the rightmost digit and sum the results.
Understanding binary, octal, and hexadecimal number systems is fundamental in computer science, particularly in fields such as digital logic, programming, and computer networking.