Introduction to Computer Systems: Data Representation

Introduction to Computer Systems: Data Representation

• Data comes in various forms: numbers, text, audio, images, video, and multimedia (combinations of these).
• All data types must be represented as 0s and 1s for computer storage and processing.
• Encoding is the method of representing data as 0s and 1s.

Number Systems

• Number systems (bases, or radices) use unique symbols (digits) to represent numbers.
• Digits have both a value and a positional value (based on its position). A number in a specific base is written as (number)base.
• Decimal (base 10) uses digits 0-9. A number like 5432 in base 10 can be calculated as 5 * 10^3 + 4 * 10^2 + 3 * 10^1 + 2 * 10^0 = 5432.
• Common number systems (bases) include binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16).
• Binary uses 0 and 1, Octal uses 0-7, Decimal uses 0-9 and Hexadecimal uses 0-9 and A-F.
• The number of possible values in N digits in base R is R^N

Converting Decimal to Other Bases

• Integer conversion: Use integer division by the new base to find remainders.
• Fractional conversion: Multiply the decimal by the new base, retain the integer portion of the product, which becomes the next digit, and repeat.

Converting Between Binary, Octal, and Hexadecimal

• Shortcut methods exist for converting between these bases.
• Octal conversion uses groups of 3 binary digits.
• Hexadecimal conversion uses groups of 4 binary digits.

Binary Arithmetic

• Binary addition, subtraction and other operations are performed using the rules of base-2 arithmetic.
• Key rules are readily available for addition and subtraction.
• Overflow happens when the result of a calculation exceeds the available bits, leading to incorrect representation and potentially incorrect results.

Floating-Point Representation

• Represents real numbers as mantissa x base^exponent.
• Mantissa: The significant digits.
• Exponent: The power to which the base is raised.

IEEE Standard 754

• Standard format for representing floating-point numbers in computers.
• Standardized for single and double precision representations.
• Specific bit layouts define the sign, exponent, and mantissa portions.

Character Coding Schemes

• Codes like ASCII and Unicode represent characters, digits, and symbols as specific bit patterns.
• Each character has a unique bit pattern.
• EBCDIC and ASCII are commonly encountered character coding schemes.
• Unicode is compatible with most other codes allowing text to be exchanged between systems that may use different character codes.