Week-3 Lecture-3_Number Systems

Overview

  • Module Title: CSEC1001 Foundation of Computing and Cyber Security Mathematics for Computing

  • Focus: Understanding number systems and their applications in computing.

Importance of Mathematics in Computing

  • Computing is deeply rooted in mathematical principles.

  • Mathematics provides a systematic framework for:

    • Modeling systems

    • Reasoning about systems

  • Acts as a bridge between theoretical concepts and practical applications.

Number Systems

Types of Number Systems

  • Decimal (Base 10): Uses digits 0-9.

  • Binary (Base 2): Uses digits 0 and 1.

  • Octal (Base 8): Uses digits 0-7.

  • Hexadecimal (Base 16): Uses digits 0-9 and letters A-F.

Conversion Methods

  • Decimal to Binary:

    • Successive division by 2, recording remainders.

  • Binary to Decimal:

    • Weighted multiplication of binary digits based on their positions.

  • Hexadecimal and Octal Conversions:

    • Group binary digits into sets (4 for hex, 3 for octal).

Computer Data Representation

Bits and Bytes

  • Bit: Smallest unit of data (0 or 1).

  • Nibbles: 4 bits.

  • Bytes: 8 bits (can represent 256 values).

  • Word: Typically 16 bits (2 bytes).

  • Double Word: Typically 32 bits (4 bytes).

Number Size and Range

  • Representations in computing vary, with different types falling under:

    • Unsigned: Only positive values.

    • Signed: Positive and negative values.

Signed Numbers Representation

Methods to Represent Signed Numbers

  1. Sign-Magnitude:

    • Uses an extra bit to indicate the sign (0 = positive, 1 = negative).

    • For example, the decimal -5 would be represented as 10101 for 5 in binary, where the first bit is 1.

  2. Two’s Complement:

    • More common representation for signed integers in computers.

    • To find negative numbers, invert the bits of the number and add 1.

    • Example, to represent -5, convert 5 to binary (0101), invert to (1010) and add 1 to get (1011).

ASCII and Character Encoding

  • ASCII uses 7 bits to represent up to 128 characters, allowing for simple text representation.

  • Characters are mapped to numerical values, enabling computers to process text.

Challenges of Binary Numbers

  • Binary representation can be long and complex, making it difficult to evaluate scale or size.

  • Conversions to octal or hexadecimal can simplify expressions:

    • Example: Binary 101011 becomes octal 53 and hexadecimal 2B.

Important Concepts to Remember

  • Conversions are essential for understanding how computers process and store information.

  • Finally, always assume two’s complement for signed numbers unless specified otherwise.