Computer Science

Information Representation

# Number Representation

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Number Representation

Number Representation

Binary Representation

- Binary is a
**base-2**number system most commonly used in digital systems, as it can easily be represented by two states: On (1) or Off (0) - A binary digit, or
**bit**, is the smallest unit of data in computing and can represent two discrete states - Eight bits together form one
**byte**, the basic addressable element in many computer architectures - Binary numbers are often needed to be padded with leading zeros to make up a certain bit-length
- Binary can be converted back and forth with other number systems like decimal, octal, and hexadecimal

Decimal Representation

- Decimal is a
**base-10**number system used universally in mathematics and arithmetic - Ten distinct symbols (0-9) are used to represent numbers in the decimal system
- Despite its widespread use, decimal is less computer-friendly and is often converted to binary or other forms for easier processing

Octal Representation

- Octal is a
**base-8**number system, not as widely used as binary or decimal, but sometimes seen in computing - Eight distinct symbols (0-7) are used to represent numbers in the octal system
- Each octal digit corresponds to three binary digits, which can simplify certain binary operations

Hexadecimal Representation

- Hexadecimal is a
**base-16**number system, commonly used in computing to reduce the number of digits needed to represent binary numbers - Sixteen distinct symbols (0-9, A-F) are used to represent numbers in the hexadecimal system
- Each hexadecimal digit corresponds to four binary digits, making this system particularly useful in information representation

Two's Complement

**Two's complement**is a method of representing negative binary numbers in a normalized way- In two's complement notation, the leftmost bit indicates the sign (0 for positive, 1 for negative)
- To obtain a negative number in two's complement, one inverts the binary representation of the positive number and adds 1 to the least significant bit

Floating Point Representation

- The IEEE 754
**Floating Point Representation**allows for the representation of very large or very small numbers by separating a number into its sign, exponent and mantissa - Any number can be represented in scientific notation, which forms the basis of floating point representation
- Floating point representation allows efficient scaling of numbers, but can suffer from rounding errors

Character Representation

- Text and special characters are often represented as binary using standardized codes such as ASCII and Unicode
- The
**ASCII**code standard assigns unique binary numbers to 128 characters, including basic Latin alphabet characters, digits, symbols, and control codes **Unicode**extends this to a wider range of characters, including those from non-Latin scripts, and is used worldwide for its compatibility and flexibility.