Computer Arithmetic —

Number Representation

Paolo.Ienne@epfl.chEPFL – I&C – LAP

Vojin.Oklobdzija@epfl.ch EPFL – CSDA and UC Davis – ACSEL

© 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…

© 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.,

01221 ... aaaaa nn

01221 ,,,, aaaaa nn

1,0,19,...,1,01,0 iii aaa or or

© Ienne, Oklobdzjia 2004CompArith — Number Representation4

Rule of Interpretation Mapping from set of digit vectors to numbers

(e.g., integers, reals)

12 “twelve”

digit vectors N, Z, R,…

© Ienne, Oklobdzjia 2004CompArith — Number Representation5

Positional Weighted Systems The rule of interpretation is a scalar product

where

is the weight vector

01221 ,,,..., wwwwwW nn

1

001221 ...

n

iiinn waaaaaaA

© Ienne, Oklobdzjia 2004CompArith — Number Representation6

Radix Systems Weights are not arbitrary but related to a radix

vector

in the following way

01221 ,,,..., rrrrrR nn

1

0110 1

i

jjiii rrwww

© 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

1,60,3600,86400

60,60,24

W

R

in

ii

ii raArw

1

0

© Ienne, Oklobdzjia 2004CompArith — Number Representation8

Common Decimal NotationIt is weighted

It is positional

It is fixed-radixi

i rw

1

001221 ...

n

iiinn waaaaaaA

iwi ononly depends

© Ienne, Oklobdzjia 2004CompArith — Number Representation9

Common Decimal NotationIt is nonredundant because canonical

10n possible digit vectors to represent 10n values

Weighted, positional, fixed-radix, nonredundant

also called conventional systems

1...,,1,0 rai

© Ienne, Oklobdzjia 2004CompArith — Number Representation10

Digit SetCanonical set

A canonical system is nonredundant

Any other choice is noncanonical

2,1,0,1ia

2/,12/,...,0...,),12/( rrrai

1,0,1ia

1...,,1,0 rai

© Ienne, Oklobdzjia 2004CompArith — Number Representation11

Very Large Choice of Weighted Positional Number Systems

Sour

ce: P

arha

mi,

© O

xfor

d 20

00

© Ienne, Oklobdzjia 2004CompArith — Number Representation12

Sign-and-Magnitude Representation

Some advantages and disadvantagesFamiliar for usersSimple naïve multiplicationAdders are not the most efficientRedundant zero (+0 and –0) may cause

some problems (e.g. testing for a variable =0)

© 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