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.