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 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.
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.
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.
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.
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.'
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 removes leading zeros from the mantissa to keep values consistent.
Example: 1.294 X 10^-1 maintains precision.