Lec 2 -Spring 2025 - BAS112 - Number Representation and Taylor Theorem (1)

Number Representation and Computer Architecture

Introduction

  • Course: BAS 112 Complex, Special Functions and Numerical Analysis

  • Instructor: Assoc. Prof. Ahmed Farghal

  • Date: 20/02/2025

  • University: Sohag University, Department of Electrical Engineering

  • Focus Areas: Number representation and Taylor theorem.


Numerical Round-off Errors

  • Numerical round-off errors depend on how numbers are stored in computers.

  • Word: A unit of memory composed of binary digits (bits) used for representing data.

  • Number System: A method for representing quantities.

    • Base: The reference number that forms the numbering system.

Base-10 (Decimal) System

  • Uses digits from 0 to 9. Counts from 0 to 9.

  • Larger numbers are formed by combining basic digits, where the position (place value) determines magnitude.

    • Example: 86,409 = 8 × 10^4 + 6 × 10^3 + 4 × 10^2 + 0 × 10^1 + 9 × 10^0.

  • This method is called positional notation.


Binary (Base-2) System

  • Computers use a binary system, where each digit's position represents a power of 2.

    • Example: Binary number 11 = 1 × 2^1 + 1 × 2^0 = 2 + 1 = 3 in decimal.


Integer Representation

  • 2’s Complement Technique: The preferred way to represent signed integers.

    • Advantages:

      • Simplifies arithmetic operations.

      • Eliminates the need for a separate sign bit.

      • Hardware design is simplified (only one circuit is required).

  • Signed Magnitude Method: Uses the first bit for sign (0 for positive, 1 for negative); remaining bits store the number.

    • Less common in modern computers.

    • Example: −173 on a 16-bit system.

16-bit Representation of Integers

  • First bit: sign bit (0 for positive, 1 for negative).

  • Remaining 15 bits: Represent binary numbers.

    • Upper Limit (Positive): 32,767.

    • Range: -32,768 to 32,767. Special 0 representation to avoid 'minus zero.'


Floating-Point Representation

  • Adjusts representation for fractional numbers using a mantissa and an exponent.

  • Format: m • b^e where:

    • m = mantissa,

    • b = base of the number system,

    • e = exponent.

  • Example: 156.78 expressed as 0.15678 × 10^3 in base-10.

Normalization

  • Normalization removes leading zeros from the mantissa to keep values consistent.

  • Example: 1.294 X 10^-1 maintains precision.

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