Data Representation

Introduction to Data Representation:

  • Data representation is the process of encoding data for storage and manipulation in a computer system.
  • In computer arithmetic, data representation is crucial for performing numerical calculations.

Number Systems:

  1. Binary Number System:

    • Fundamental to digital computers, consisting of only two symbols, 0 and 1.
    • Each digit is a power of 2 (e.g., 2^0, 2^1, 2^2, ...).
  2. Decimal Number System:

    • The base-10 system used in everyday life.
    • Comprises digits from 0 to 9, with each digit representing a power of 10.

Binary to Decimal Conversion:

  • To convert a binary number to decimal, multiply each digit by the corresponding power of 2 and sum the results.

Decimal to Binary Conversion:

  • To convert a decimal number to binary, repeatedly divide the number by 2, keeping track of the remainders.

Hexadecimal Number System:

  • Base-16 system using digits 0-9 and letters A-F.
  • Convenient for representing binary data in a more compact form.

Bit, Byte, and Word:

  1. Bit (Binary Digit):

    • The smallest unit of data, representing a 0 or 1.
  2. Byte:

    • Consists of 8 bits and is a common unit of data manipulation.
    • Can represent 256 different values (2^8).
  3. Word:

    • A computer's natural unit of data, often defined by the CPU's architecture.
    • Typically 32 or 64 bits in modern systems.

Fixed-Point Representation:

  • Fixed-point representation is used to store integers or fixed precision numbers.
  • The position of the binary point is predetermined.
  • Arithmetic operations can be performed directly.

Floating-Point Representation:

  • Used for real numbers with variable precision.
  • Comprises three components: sign bit, exponent, and mantissa (fraction).
  • Conforms to the IEEE 754 standard for binary floating-point arithmetic.

Overflow and Underflow:

  • In fixed-point and floating-point arithmetic, operations can result in values exceeding the representable range (overflow) or becoming too small to represent (underflow).

Two's Complement:

  • A binary representation of signed integers where the most significant bit (MSB) indicates the sign.
  • Negative numbers are represented by taking the two's complement of their absolute values.

Endianness:

  • Refers to the byte order of multi-byte data types in memory.
  • Big-endian stores the most significant byte at the lowest memory address, while little-endian stores it at the highest address.

Binary Coded Decimal (BCD):

  • A way to represent decimal numbers in binary form.
  • Each decimal digit is represented by a 4-bit binary code.

Conclusion: Data representation in computer arithmetic is fundamental to how computers store and manipulate numbers. Understanding binary, decimal, and hexadecimal number systems, as well as fixed-point and floating-point representations, is essential for working with numerical data in computer science and engineering. Proper handling of data representation is critical to prevent errors like overflow and underflow in computations.