1. Computer Arithmetic — Number Representation Paolo.Ienne@epfl.ch EPFL – I&C – LAP Vojin.Oklobdzija@epfl.ch EPFL – CSDA and UC Davis – ACSEL

2. © Ienne, Oklobdzjia 2004CompArith — Number Representation2 Why Representation?  Need to map numbers to binary bits  Need a representation anyway…  Representation is consequential  Complexity of arithmetic operations depends heavily on the representation Number representation is the heart of computer arithmetic…

3. © Ienne, Oklobdzjia 2004CompArith — Number Representation3 What Is a Number System?  A number is represented as an ordered n-tuple of symbols (digit vector)  Each symbol is a digit  Digits usually represent integers from a given set—e.g.,

4. © Ienne, Oklobdzjia 2004CompArith — Number Representation4 Rule of Interpretation  Mapping from set of digit vectors to numbers (e.g., integers, reals) “twelve” digit vectorsN, Z, R,…

5. © Ienne, Oklobdzjia 2004CompArith — Number Representation5 Positional Weighted Systems  The rule of interpretation is a scalar product where is the weight vector

6. © Ienne, Oklobdzjia 2004CompArith — Number Representation6 Radix Systems  Weights are not arbitrary but related to a radix vector in the following way

7. © Ienne, Oklobdzjia 2004CompArith — Number Representation7 Mixed-Radix Systems  Fixed-radix if all elements of R are identical  A few mixed-radix systems are very common—e.g., time

8. © Ienne, Oklobdzjia 2004CompArith — Number Representation8 Common Decimal Notation  It is weighted  It is positional  It is fixed-radix

9. © Ienne, Oklobdzjia 2004CompArith — Number Representation9 Common Decimal Notation  It is nonredundant because canonical  10 n possible digit vectors to represent 10 n values Weighted, positional, fixed-radix, nonredundant  also called conventional systems

10. © Ienne, Oklobdzjia 2004CompArith — Number Representation10 Digit Set  Canonical set A canonical system is nonredundant  Any other choice is noncanonical

11. © Ienne, Oklobdzjia 2004CompArith — Number Representation11 Very Large Choice of Weighted Positional Number Systems Source: Parhami, © Oxford 2000

12. © Ienne, Oklobdzjia 2004CompArith — Number Representation12 Sign-and-Magnitude Representation  Some advantages and disadvantages  Familiar for users  Simple naïve multiplication  Adders are not the most efficient  Redundant zero (+0 and –0) may cause some problems (e.g. testing for a variable =0)

13. © Ienne, Oklobdzjia 2004CompArith — Number Representation13 References  M. D. Ercegovac and T. Lang, Digital Arithmetic, Morgan Kaufmann, 2004  I. Koren, Computer Arithmetic Algorithms, Peters, 2002  A. R. Omondi, Computer Arithmetic Systems— Algorithms, Architecture and Implementation, Prentice Hall, 1991  B. Parhami, Computer Arithmetic—Algorithms and Hardware Designs, Oxford, 2000