Machine Number Systems Fixed radix Positional

1
Review - Outline
Dr. Alaaeldin A. Amin Computer Engineering
Department E-mail amin_at_ccse.kfupm.edu.sa http//
www.ccse.kfupm.edu.sa/amin

• Machine Number Systems (Fixed radix Positional)
• Fixed Point Number Representation
• Base Conversion
• Representation of Signed Numbers
• Signed Magnitude (Sign and Magnitude)
• Complement Representation
• Radix Complement (2s Complement)
• Diminished Radix Complement
• Precision Extension
• Arithmetic Shifts

2
Machine Number Systems?

• X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
• Since Numbers Are Stored in Registers of Fixed
  Length
• There is a Finite Number of Distinct Values that
  Can be Represented in a Register
• Let Xmax Xmin Denote The Largest Smallest
  Representable Numbers
• Xmax rk-1 For an Integral of k-Digits
• Xmax 1 - r -m For a Fractional of m-Digits
• Xmax rk - r -m For a of k- Integral Digits
  and m- Fractional Digits
• Xmin 0
• Number of Possible Distinct Values r km

Fractional Part
Integral Part
3
Number Radix Conversion

• Let X be an Integer ? XB in Radix B System
• ? XA in Radix A
  System
• Assumptions
• XB has n digits
• XB (bn-1..b2 b1 b0)B ,
• where bi is a digit in radix B system,
• i.e. bi ? 0, 1, .. , B-1
• XA has m digits
• XA (am-1..a2 a1 a0)A
• where ai is a digit in radix A system,
• i.e. ai ? 0, 1, .., A-1
• Problem Statement
• XB (bn-1..b2 b1 b0)B ? (am-1..a2 a1 a0)A
  

Unknowns
Knowns
4

• XB am-1Am-1 a2A2 a1A1 a0A0
• Where ai ? 0-(A-1)
• Accordingly, dividing XB by A, the remainder will
  be a0.
• XB Q0.Aa0
• Where, Q0 am-1Am-2 a2A1 a1A0
  
• Likewise ? Q0 Q1Aa1 ? Get a1
• Q1 Q2Aa2 ? Get a2
• Qm-3 Qm-2Aam-2 ? Get am-2
• Qm-2 Qm-1Aam-1 ? Get am-1
• Where Qm-1 0 ? Stopping Criteria

Divisible by A
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Representation of Signed Numbers

• Three Main Systems
• 1. Signed Magnitude (Sign and Magnitude)
• 2. Complement Representation
• Radix Complement (2s Complement)
• Diminished Radix Complement
• Signed Magnitude
• Independent Representation of The Sign and The
  Magnitude
• Symmetric Range of Representation
• -(2n-1 -1) (2n-1 -1)
• Two Representations for 0 ? 0 , -0
• ? Nuisance for Implementation
• Addition/Subtraction Harder To Implement
• Multiplication Division Less Problematic

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Representation of Signed Numbers

• Alternatively
• 0, 1, , (R/2 - 1) ? ive Numbers
• R/2, R/21, ., (R-1) ? -ive Numbers ? More
  Complex Sign Detection if r not power of 2

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Representation of Signed Numbers

• Complement Representation
• Positive Numbers (N) Are Represented in Exactly
  the Same Way as in Signed Magnitude System
• Negative Numbers (-N) Are Represented by the
  Complement of N (N)
• Define the Complement of a number N (N) as
• N M -N
• Where M Some Constant
• This representation satisfies the Basic
  Requirement That
• -(-N ) ( N ) M- (M-N) N
• Adding 2 Numbers X (ive) and Y (-ive)
• IF Y gt X
• Result Z is -ive, i.e. Complement Represented
• Z X (M-Y)
• M -(Y-X) Correct Answer
• in Complement Form

ive
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Representation of Signed Numbers

• IF Y lt X
• Result Z is ive
• Z X (M-Y)
• M (X-Y)
• Correct Answer X - Y
• Requirements of M
• 1- Simple Complement Calculations
• 2- Simplify/Eliminate Addition Correction Steps
• Let xi be the Complement of a single Digit xi
• IF X xk-1 xk-2 ... x1 x0 . x-1 x-2 ...
  x-m
• Define X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
  

• IF Y lt X
• Result Z is ive
• Z X (M-Y)
• M (X-Y)
• Correct Answer X - Y
• Requirements of M
• 1- Simple Complement Calculations
• 2- Simplify/Eliminate Addition Correction Steps
• Let xi be the Complement of a single Digit xi
• IF X xk-1 xk-2 ... x1 x0 . x-1 x-2 ...
  x-m
• Define X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
  

M Chosen To Eliminate/Simplify Correction Step
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Representation of Signed Numbers
X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
(r-1) (r-1) .. . . (r-1) ulp
1 1 0 0 0 0 . 0 0 ... 0
r k Thus, X X ulp r k
... (1) where ulp unit in least position
r -m (for s with m Fractional
digits) 1 (for INTEGERS, i.e when m0
) Eqn. (1) Can Be ReWritten As r
k - X X ulp (2)
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Representation of Signed Numbers

• Radix Complement M r k
• N r k - N
• N ulp (Simple Computation)
• Computing N is independent of k
• Computing the Previous Addition Requires No
  Corrections
• Z M (X-Y) r k (X-Y)
• Diminished Radix Complement M r k - ulp
• N r k - ulp - N
• N (Simplest Computation)
• Computing the Previous Addition Requires Simple
  Correction of Adding ulp
• Z M (X-Y) r k (X-Y) - ulp

End Around Carry Discarded Added as ulp
Should Add ulp for Correction
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Representation of Signed Numbers

• 2s Complement
• Asymmetric Range of Representation
• -(2k-1) (2k-1 -ulp)
• - Negation Can Lead to OverFlow

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2s Complement Numbers System

• X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
• X if X is ive
• -2k-1 (2k-1 - X ) X if X is -ive
• Precision/Range Extension of 2s Comp s
• Extending N from n-bits to k-bits (kgt k)
• ive N Pad with 0s to the Left of the
  integral part And/Or to the right of Fractional
  Part.
• -ive N Pad with 0s to the right of
  Fractional Part And/Or Extend Sign Bit to the
  Left of the integral part.
• In General
• Pad with 0s to the right of Fractional Part
  And/Or Extend Sign Bit to the Left of the
  integral part.
• ...xk-1 xk-1 xk-1 xk-2 ... x1 x0 . x-1 x-2 ....
  x-m 000..

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2s Complement System

• Features
• Leftmost Bit indicates Sign
• The Sign Bit is Part of a Truncated -String
• Asymetric Range(-2k-1 (2k-1 -ulp) )
• Overflow may result on Complementing
• Single Zero Representation 0000000 (0)

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Representation of Signed Numbers

• 1s Complement
• Symmetric Range of Representation
• -(2k-1 -ulp) (2k-1 -ulp)
• - 2 Representations of 0

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1s Complement Numbers System

• X xk-1 xk-2 ... x1 x0 . x-1 x-2 ... x-m
• X for ive X
• -(2k-1-ulp) X ) X for
  -ive X
• Precision/Range Extension of 2s Comp s
• Extending N from n-bits to k-bits (kgt k)
• ive N Pad with 0s to the Left of the
  integral part And/Or to the right of Fractional
  Part.
• -ive N Pad with Sign Bit to the right of
  Fractional Part And/Or to the Left of the
  integral part.
• In General
• Pad with Sign Bit to the right of Fractional Part
  And/Or to the Left of the integral part.
• ..xk-1 xk-1 xk-1 xk-2 .. x1 x0 . x-1 x-2 .. x-m
  xk-1 xk-1 ..

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1s Complement Numbers System

• Features
• Leftmost Bit indicates Sign
• The Sign Bit is Part of a Truncated -String
• Symetric Range(-(2k-1 -ulp) (2k-1 -ulp) )
• Two Zero Representation (0 , -0 )

Arithmetic Shifts

• Effect
• Left Shift Multiply Number by radix r
• Right Shift Divide Number by radix r
• (a) Shifting Unsigned or ive Numbers
• Shift-In 0s (for both Left Right Shifts)
• (b) Shifting -ive Numbers
• Depends on the Representation Method
• Signed Magnitude
• Sign Bit is maintained (Unchanged)
• Only The Magnitude is shifted and 0s are
  Shifted-In for both Directions (Left Right)

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Arithmetic Shifts

• 2s Complement
• Left Shifts 0s are Shifted-In
• Right Shifts Sign Bit Extended (Shifted Right)
• 1s Complement
• Left Right Shifts Sign Bit is Shifted-In

Right Shifts
Left Shifts