This document discusses different number systems including conventional, redundant, and residue number systems. The conventional system uses a fixed-radix positional notation. Redundant systems like signed-digit code and canonical signed digit code allow for faster addition and subtraction by reducing carry propagation. The residue number system avoids carry propagation by representing numbers as sets of residues from integer divisions by mutually prime moduli.

1. NUMBER SYSTEM
• CONVENTIONAL NUMBER SYSTEM
• REDUNDANT NUMBER SYSTEM
• RESIDUE NUMBER SYSTEM

2. CONVENTIONAL NUMBER SYSTEM
• It is non-redundant,weighted,positonal number
system
• Every number x represented as
wd-word length
wi-weight associated with each digit
Positional number system wi depends on the position of
the digit xi.
Conventional number system wi = xi ,such systems are
called fixed-radix systems.

3. Number
Integer Fractional part.
• A fixed –point number is written from L to R with
MSB at L and LSB at R.
• 2 distinct forms of
fixed-point arithmetic
Integer arithmetic
( RH imprtant) (RH kept RH
arithmetic dropped)
Fractional fixed point
(LH imprtnt)
(LH kept RH dropped)

4. • Fractional fixed point arithmetic- signed mantissa and
exponent.
• Advantage-suppress parasitic oscillation , less chip
area & faster.
Fixed point represented by:
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•
•
•
•
SIGNED –MAGNITUDE REPRESENTATION
COMPLENENT REPRESENTATION
ONE’S COMPLENENT REPRESENTATION
TWO’S COMPLENENT REPRESENTATION
BINARY OFFSET REPRENSATION

5. SIGNED-MAGNITUDE REPRESENTATION
• Magnitude & sign represented separately
• 1st digit-sign,remaining digit-Magnitude
• Representation:
• Zero +0 or -0 , not in the case of 1.
• Adv : Mul , div easy
• Disadv : add,sub – sign-operands

6. COMPLENENT REPRESENTATION
• Pos no. – 1’s ,2’s,binary offset = signed mag rep
• Neg no. -x=R-x
• For y>x
x+(R-y)=R-(y-x)
• For x>y
R-(y-x)=R-(x-y)

7. Radix complement representation
Diminished-radix complement:
complement
Adv: Computation is simpler

8. ONE’S COMPLENENTREPRESENTATION
• Dimnished rep: R=
; r=2 & k=0
• ReP:
• X>0 1’s complement=binary word
• X<0 bit values complement of positive value.
• Arithmetic operations -complicated

9. TWO’S COMPLENENT REPRESENTATION
•
•
•
•
•
Radix complement representation
R= =1 ; r=2 & k=0
Rep:
X>0 complement=binary word
X<0 bit complement +Q

10. BINARY OFFSET REPRENSATION
• Non-redundant representation
• Seq digit = 2’s comp except for the sign bit i.e.
complemented
• Rep:
• Range: -1<=x<=1-Q

11. REDUNDANT NUMBER SYSTEM
• Time consuming , add , sub-without long carry paths.
• Simplify-speed-suitable application-specific
arithmetic units
• 3 such representation are:
• SIGNED-DIGIT CODE
• CANONIC SIGNED DIGIT CODE
• ON-LINE ARITMETIC

12. SIGNED-DIGIT CODE
• SDC –each digit is allowed to have sign.
• Values of (+1,0,-1)
• Rep:
• X range -2+Q<=x<=2-Q
• Quantizing – creates problem-bcoz - sdc not unique.
•
SDC
Conventional representation

13. CANONIC SIGNED DIGIT CODE
• CSDC –Unique representation
• Rep:
• X range (-4/3)+Q<=x<=(4/3)-Q
• Average no. of non-zero digit
Wd/3 + (1+2-wd)/9

14. ON-LINE ARITMETIC
•
•
•
•
•
Compute ith digit.
Uses only ( i + small positive constant)
Favorable - Recursive algorithm
Mul , div -SDC
SDC
2’s complement allows online operation.

15. RESIDUE NUMBER SYSTEM
• RNS-avoids carry propagation
• Rep by a set of residues after an integer division by
mutal prime factor(moduli)
• Relation (m1, m2…mp (moduli) and x ): x=qi mi+ri
• qi, mi,ri are integers x=(r1,r2…rn)
• The Chinese remainder theorem:
A=(a1,a2....ap) & B=(b1,b2…bp)
then,
A B=[(a1 b1)m1,(a2 b2)m2….(ap
bp)mp]

16. • ARITHMETIC OPERATIONS USING RNS:
• Addition and subtraction
• Multiplication
• Division

17. • Adv:
• RNS –Digital computer arithmetic.
Large integer
set of smaller integers
large calculation
series of smaller calculations
hardware implementations

18. • Diadv:
• Restricted to fixed pioint arithmetic
• Comparision of overflow,quantization operation is
difficult in order to round off
RNS
normal no. Rep.
• Not suitable – Recursieve loops.