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.