A Level Computer Science CAIE

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.