Homework 2

due Tuesday, February 12, 7:20 pm (NO LATE HOMEWORK ACCEPTED)

Reading

Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design

Chapter 1, Numbers and Arithmetic
Chapter 2, Representing Signed Numbers
Chapter 4.1, RNS Representation and Arithmetic
Chapter 17, Floating-Point Representations
Certicom Corp., Polynomial Representation
Certicom Corp., F(2^4) with Polynomial Representation
see errata at http://www.ece.ucsb.edu/~parhami/text_comp_arit.htm#errors

Problems (submit all problems on paper, not WebCT)

Problem 1: 2 points

Parhami, Chapter 1, Problem 1.4a. Hint: "If and only if" means that your proof must show the equivalence in both directions.

Problem 2: 2 points

Parhami, Chapter 1, Problem 1.4b. Hint: "If and only if" means that your proof must show the equivalence in both directions.

Problem 3: 2 points

Parhami, Chapter 2, Problem 2.8

Problem 4: 2 points

Parhami, Chapter 2, Problem 2.4a and 2.4b (i.e. do part a and part b).

Problem 5: 2 points

Two numbers A and B represented in RNS( 8 | 7 | 5 | 3 ) look as follows:

A = ( 7 | 0 | 3 | 0 )_RNS
B = ( 6 | 5 | 4 | 0 )_RNS

Find decimal values of A and B.
Perform addition, subtraction, and multiplication of A and B in the RNS representation and in the decimal representation.
Demonstrate that the results obtained from both computations represent the same values.

Problem 6: 2 points

Determine all bits of the ANSI/IEEE standard single-precision representation of the following numbers:

a. 98765.5419921875₁₀ × 2^-42
b. - 0.0000010001101111011₂ × 2^-122
c. the result of: -5.4766321₁₀ - (-infinity)
d. the result of: 6.87₁₀ / (-infinity)

Problem 7: 2 points

Perform the following computations in the Galois Field GF(2⁴) (note that · is the multiplication sign):

a. "1001" · "0111"
b. "1101" · "0101"

Assume the use of the irreducible polynomial P(x) = x⁴ + x + 1.