Data Representation in Computers

Data Representation

• Data representation refers to the techniques used to represent data in computers.
• Number systems are fundamental to computer design and organization.
• Computers use four main number systems: Decimal, Binary, Octal, and Hexadecimal.

Decimal Number System

• The decimal system uses ten digits (0-9) and has a base of 10.
• Examples of decimal numbers include 405 and 145.25.

Binary Number System

• The binary system uses only two digits (0 and 1), with a base of 2.
• Computers operate based on two states (on/off) and store information using these states.
• Examples of binary numbers include 101 and 1001.11.

Octal Number System

• The octal system uses eight digits (0-7), with a base of 8.
• Examples of octal numbers include 76 and 55.25.

Hexadecimal Number System

• The hexadecimal system, often called 'Hex', uses 16 symbols (0-9 and A-F) and has a base of 16.
• Examples of hexadecimal numbers include 45 and 11A.

Conversion between Number Systems

• Converting Decimal to Binary:
  • Repeatedly divide the decimal number by 2.
  • Note the remainders.
  • Arrange the remainders in reverse order (the first remainder is the least significant digit, the last is the most significant digit).
• Converting Binary to Decimal:
  • Assign weights (powers of 2) to each bit, starting from the rightmost with 2^0.
  • Multiply each bit by its corresponding weight.
  • Sum the results to get the decimal equivalent.
• Converting Decimal to Octal:
  • Repeatedly divide the decimal number by 8.
  • Note the remainders.
  • Arrange the remainders in reverse order.
• Converting Octal to Decimal:
  • Assign weights (powers of 8) to each digit, starting from the rightmost with 8^0.
  • Multiply each digit by its corresponding weight.
  • Sum the results to get the decimal equivalent.
• Converting Decimal to Hexadecimal:
  • Repeatedly divide the decimal number by 16.
  • Note the remainders.
  • Arrange the remainders in reverse order.
  • Convert any remainders above 9 to their corresponding hexadecimal letters (A-F).
• Converting Binary to Octal:
  • Group the binary digits into groups of 3 (from right to left for integers, left to right for fractions)
  • Convert each group into its decimal equivalent.
  • If a group has less than 3 bits, add leading zeros to complete the group.
• Converting Binary to Hexadecimal:
  • Group the binary digits into groups of 4 (from right to left for integers, left to right for fractions).
  • Convert each group into its decimal equivalent.
  • If a group has less than 4 bits, add leading zeros to complete the group.
• Converting Octal to Binary:
  • Convert each digit of the octal number into its 3-bit binary equivalent.
• Converting Octal to Hexadecimal:
  • Convert the octal number to binary.
  • Group the binary digits into groups of 4.
  • Convert each group to its hexadecimal equivalent.
• Converting Hexadecimal to Decimal:
  • Assign weights (powers of 16) to each digit, starting from the rightmost with 16^0.
  • Multiply each digit by its corresponding weight.
  • Sum the results to get the decimal equivalent.
• Converting Hexadecimal to Binary:
  • Convert each digit of the hexadecimal number to its 4-bit binary equivalent.

Binary representation of integers

• There are three main ways to represent positive and negative integers in binary:
  • Sign magnitude: a sign bit (0 for positive, 1 for negative) is added to the left of the magnitude, represented as a simple binary number.
  • One's Complement: positive numbers are represented as usual in signed magnitude. Negative numbers are represented by inverting (flipping) all the bits of the positive equivalent.
  • Two's Complement: positive numbers are represented as usual in signed magnitude. Negative numbers are obtained by adding 1 to the one's complement representation.