Introduction to Computing - Week 2

Introduction to Computing - Week 2: Data Representation

• Data Representation: Modern computers use binary (base 2) representation, unlike early digital computers which used decimal (base 10).
• Bit: A bit is the fundamental unit of information in computing. It can have one of two values: 0 or 1.
• Binary Digits: A binary digit (bit) can also be represented as logical values (true or false; on or off).
• Punched Cards: Early computers used punched cards to represent data, each position on the card carrying one bit of information.
• Word Size: The number of bits a CPU can process in one operation is known as the word size. This is important for processor design and architecture.
• Memory Addressing: The CPU's ability to directly address memory locations depends on its word size. A 32-bit CPU, for example, can directly address 2³² memory locations.
• Memory Organization: Memory consists of millions of storage cells, each storing one bit of information. Memory can be viewed as a collection of words. Each n-bit group is a word, where 'n' describes the word length.

Number Systems

• Number System Bases: Commonly used number systems include decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16).
• Radix: The base of a number system is also called the radix. The radix determines the number of unique digits used in the system. The decimal system has 10 digits, while the binary system uses 2, and so on
• Hexadecimal: Hexadecimal uses 16 digits (0-9 and A-F). It's more concise than binary for representing data in computer systems (e.g., for colors, memory addresses, and error codes).
• Octal: Octal uses 8 digits (0-7). It was commonly used in the past for expressing binary values in a shorter form.

Conversion Between Number Systems

• Decimal to Binary: Divide the decimal number by 2 repeatedly, noting the remainders until the quotient is 0. Write the remainders in reverse order to get the binary representation.
• Binary to Decimal: Multiply each binary digit by the corresponding power of 2 (starting with 2⁰), and sum the results to get the decimal equivalent.
• Decimal to Octal: Divide the decimal number by 8 repeatedly, noting the remainders until the quotient is 0. Write the remainders in reverse order to get the octal representation.
• Octal to Decimal: Multiply each octal digit by the corresponding power of 8 (starting with 8⁰), and sum the results to get the decimal equivalent.
• Decimal to Hexadecimal: Divide the decimal number by 16 repeatedly, noting the remainders until the quotient is 0. Convert the remainders to their hexadecimal equivalent (0-9 and A-F) to get the hexadecimal representation.
• Hexadecimal to Decimal: Multiply each hexadecimal digit by the corresponding power of 16 (starting with 16⁰), sum the results to get the decimal equivalent.

Applications of Different Number Systems

• Data Representation: Computers use binary to represent data. Other number systems offer shorthand representations used in various applications (e.g., colors, hardware locations).
• Computer Architecture: Computers use binary for all internal data representation and logic operations.
• Memory Locations: Hexadecimal is used to represent memory addresses and to describe memory locations.
• Colors: Hexadecimal is used to represent colors (each primary RGB color being described by two hexadecimal digits).