Fixed-Point Negative Numbers Two Common Forms: 1.Signed-Magnitude Form 2.Complement Forms Signed-Magnitude Numbers First Digit is Sign Digit, Remaining n-1 are the Magnitude Convention (binary) –0 is a Positive Sign bit –1 is a Negative Sign bit Convention (non-binary) –0 is a Positive Sign digit –  -1 is a Negative Sign digit Only 2  n-1 Digit Sequences are Utilized

Signed-Magnitude Example Largest Representable Value is:

Signed-Magnitude Example (cont)

Signed-Magnitude Ternary Example Notice that fractional part is infinite in  =10 but finite in  =3

Signed-Magnitude Ternary Bounds Positive Numbers: Negative Numbers: Range:

Signed-Magnitude Comments Two Representations for zero, +0 and –0 Addition of +K and –K is not zero EXAMPLE Disadvantage since algorithm requires comparison of signs and, if different, comparison of magnitudes Yields a Sum of –20 10 !!!!!

Complement Representations Two Types of Complement Representations 1. Radix Complement (binary – 2’s-complement) 2. Diminished-Radix Complement (binary – 1’s-complement) Positive Values Represented Same Way as Signed Magnitude for Both Types Negative Value, -Y, Represented as R-Y Where R is a Constant Obeys the Identity: Advantage is No Decisions Needed Based on Operand Sign Before Operations are Applied

Complement Representation Example If |Y| > X, Then the Answer is R - (Y - X) If X > |Y|, Then the Answer Should be X - Y –But X + (R - Y) = R + (X - Y), Thus R Must be Discarded! Solution is to Choose the Value of R Carefully X is Positive, Y is Negative, Compute X + Y Using Complement Representation

Requirements for Complementation Value, R Select R to Simplify (or Eliminate) Correction for the X > |Y| Case Calculation of Complement of Y or (R-Y) Should be Simple and Fast Definition of Complement for Single Digit, x i Definition of Digit Complement for a Word, X

Complementation Value, R Add Word and Complement Together: Answer to Addition Now Add 1 ulp Therefore, we see that:

Radix-Complement Form The Radix Complement Form is Defined When: Using  k is Convenient Since Storing Result in Register of Length n Causes MSD of 1 to be Discarded due to Finite Register Length Therefore, it is Easy to Compute the Complement of X by: 1.Take the Digit Complement of X 2.Add 1ulp to Complement

Radix-Complement Form (cont) No Correction is Needed When We have Positive X and Negative Y Such That: Since R=  k And  k is discarded Due to Finite Register Length

Radix-Complement Example Since n = m + k  m = 0 Therefore 1 ulp = 2 0 = 1 Given X, the radix complement (2’s complement) is: Range of Positive Numbers is [0000,0111] 2’s Complement of Largest, 0111: In Radix Complement, There is a Single Representation of Zero (0000) and Each Positive Number has Corresponding Negative Number With MSB=1

Radix-Complement Example In Radix Complement, There is a Single Representation of Zero (0000) and Each Positive Number has Corresponding Negative Number With MSB=1 Accounts for 1(zero)+7(pos.)+7(neg.), But Extra Bit Pattern Left One Additional Negative Number, =-8 10,  X  +7 10

Diminished-Radix Complement In Diminished Radix Complement, the Complementation Process is Easier Since the Addition of 1 ulp is Avoided Range of Positive Numbers is: [0000 2, ]=[0 10,7 10 ] 1’s Complement of Largest is = ’s Complement of Zero is Two Representations of Zero! In All Cases MSB is Sign Bit

Comparison of Two’s Complement, One’s Complement and Signed-Magnitude SequenceTwo’s Complement One’s Complement Signed- Magnitude

Signed-Number Arithmetic Signed Magnitude – Only Use Magnitude Digits Carry-out  Overflow

Radix-Complement Arithmetic Radix Complement; In this case 2’s Complement Carry-out Does NOT Mean Overflow

2’s-Complement Overflow If X, Y have opposite signs overflow never occurs whether carry-out exists or not If X, Y have same sign and result sign differs, overflow occurs No Carry-out Carry-out Carry-out, Overflow No Carry-out, Overflow

1’s-Complement Overflow One’s complement – carry-out indicates a correction is needed If X > Y, then answer should be X-Y however; register contains X-Y-ulp since 2 n is carry-out bit, therefore must “correct” by adding 1 ulp

Example of 1’s-Complement Overflow So-called “end-around” carry Need Correction Since Overflow

“End-Around” Carry Design This is “end-around” carry – always add carry-out to LSD Carry-out

Other Number Systems Binary Number Systems are Most Common In terms of building “fast” systems, we should consider: – Negative Radix – Signed Digit – Log (logarithm) – Signed Log – Complex Radix – Mixed Radix – Residue Number Systems

Negative-Radix Fixed-Position Systems Nega-decimal example:

Nega-Decimal Number System Finite Register Length, n=3 digits: Largest Positive Value, X max : Smallest Value, X min : Asymmetric System!!!: 10 times more positive than negative values represented

Nega-Decimal Number System Finite Register Length, n=4 digits: Nega-decimal System Characteristics: Now more Negative Values than Positive Arithmetic Operations Same Regardless of Sign of Number No Signed Digit/Complement Representation Needed Sign of X Determined by Position of First Non-zero Digit

Nega-Binary Number System Negative Radix: How is this Addition Operation Performed????? Example

Nega-Binary Number System w i Values (5) 10 (1+1=4-2) 10 (0+0=0) 10 (4+4=16-8) 10 (0-8=-8) 10 (5-3=2) -10 Carry-out (-3) 10

Nega-Binary Adder Design Individual Adder Cells Produce Two Carry-out Bits Design a Circuit at Gate Level for a 4-Digit Nega-Binary Adder Hint: Cout Functions Should Look Familiar!