Notation and Representation

Module 7

Reading Assignment: Sec. 1-2, 1-3: pp.13-22 Sec. 1-5: pp. 25-28 Sec.4-2, 4-3, 4-4: pp. 151-167 Sec.10-7: pp. 522-527	Schedule of Lectures	Sample Problems
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Notation vs. Representation

• Notations: expressed symbols, e.g.,  octal, hexadecimal
• Representation: meaning

Notations

• Decimal: base 10 (ten digits)
• Two and three digit sequences
• Total  number of such sequences is 10^{(# digits)}, and in general base^{(# digits)}
• Binary: base 2 ( two digits)
• Octal: base 8 (eight digits)
• Hexadecimal: base 16 (sixteen digits)
• Conversion between notations
• Octal à binary, binary à octal
• Hexadecimal à binary, binary à hexadecimal
• Octal à hexadecimal, hexadecimal à octal
• Binary à decimal, decimal à binary

Representations

• Representation -> meaning of symbols
• Unsigned binary (counting sheep)
• Range
• Powers of 2
• Sign + magnitude 
• Range 
• Easy sign detection and negation 
• Problems: 
• special sign bit handling
• two representations for zero
• One's compliment
• Range 
• Easy sign detection, negation
• Problem: two representations for zero
• Two's compliment
• Range 
• Easy sign detection
• One representation for zero
• Fixed point representations
• The binary point and position values
• Algorithm for computing binary representations of fractional components
• Range vs. precision (accuracy)

ASCII Representation  

• American Standard Code for Information Interchange (ASCII)
• Binary encoding of characters
• Example encodings of character strings
• Revisit distinction between notation and representation

Floating Point

• From fixed point to floating point
• binary point, scientific notation, and normalized representation
• IEEE 754 format
• Implicit 1
• Exponent biasing 
• Dynamic range vs. precision 
• Examples
• Conversion to/from decimal
• Representation in binary and hexadecimal notations
• examples

• Finite precision
• How many points are there in [0-1]? how many can be represented?
• The same code should produce the same answers on different machines!
• Non-commutativity and non-associativity
• Rounding modes and the IEEE 754 standard
• Computation of errors due to finite precision
• examples