Homework 1

Required reading (material covered at the last lecture)

Parhami

Chapter 1, Numbers and Arithmetic, Sections 1.1-1.6, pp. 3-15.

Parhami

Chapter 2, Representing Signed Numbers, Sections 2.1-2.6, pp. 19-31.

Written Assignment 1

Part 1 Due Monday, February 5

Problem 1 (4 points)

a. Determine how many digits are necessary to represent all possible values of the sum of 256 integers in the range from 0 to 999₁₀each, using
- radix-2 conventional system
- radix-3 conventional system

b. Represent the following decimal numbers in binary representation and ternary representation (radix-3).

434445.859375, 223344.78515625

c. Convert the following octal (radix-8) numbers to hexadecimal (radix-16) notation:

367241.426312, 342263.6342321

d. Represent    -101.78125 and -31.703125
    using the following signed number representations with k = 8 and l = 8
    - two's complement
    - one's complement
    - biased with the base B=128
    - signed magnitude.

Part 2 Due Monday, February 12

Please, solve at least four problems from the following list (each problem is worth 2 points; solutions to the fifth and sixth problem will be awarded with bonus points)

Problem 2a

Parhami, Chapter 1, Problem 1.4a (see the text of Problem 1.4 below). Hint: Remember that your proof must show the equivalence in both directions.

Problem 2b

Parhami, Chapter 1, Problem 1.4b-d (see the text of Problem 1.4 below).

Problem 2c

Parhami, Chapter 2, Problem 2.8
Prove a formula for a value of a number in one's complement representation (see Lecture 1, slide "Value of a number in the signed representations").

Problem 2d

Prove a formula for an extension of a signed number in one's complement representation with a k'-bit integer part and an l-bit fractional part to a number with a k'-bit integer part and an l'-bit fractional part, with k' > k and l' > l (see Lecture 1, slide "Extending the number of bits of a signed number").

Problem 2e

Prove a formula for an extension of a signed number in the biased representation with a k-bit integer part and an l-bit fractional part to a number with a k'-bit integer part and an l'-bit fractional part, with k' > k and l' > l (see Lecture 1, slide "Extending the number of bits of a signed number").

Problem 2f

Parhami, Chapter 1, Problem 2.4ab

a. Show conversion procedures from k-bit 2's complement format to k-bit biased representation, with bias = 2^k-1, and vice versa. Pay attention to possible exceptions.

b. Repeat part a. for a bias, B=2^k-1 - 1.

Recommended reading (material covered during the next class)

Parhami

Chapter 17, Floating-point Representations, pp. 279-293.