## Homework 2

Parhami

Chapter 1, Numbers and Arithmetic, Sections 1.1-1.6.

Chapter 2, Representing Signed Numbers, Sections 2.1-2.6.

Chapter 17, Floating-point Representations.

Wikipedia

Endianess

IEEE 754-2008

### Due Friday, March 11, 11:59pm

#### Problem 1 (5 points)

a. Determine how many digits are necessary to represent all possible values of the

A. sum of 1024 integers in the range from 0016 to FF16 each
B. product of 1000
integers in the range from 0 to 99 each

using

b. Represent the following decimal numbers in hexadecimal representation.

85.6328125,   1324.19140625

5667321.434214335267.67423411

d. Represent    -123.7109375 and -83.2890625
using the following binary signed number representations
with
k = 8 and l = 8

- signed magnitude
- one's complement

- two's complement
- biased with the base B=128.

e. Extend all numbers from point d., expressed in the respective signed number representations with k = 8 and l = 8, to the numbers with the same value and the sizes of the integer and fractional part equal respectively to k' = 16 and l' = 16 (see Lecture 4, slide 71, "Extending the number of bits of a signed number").

Problem 2 (3 bonus points)

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 4, slide 71, "Extending the number of bits of a signed number").

#### Problem 3 (3 points)

Determine all bits of the ANSI/IEEE standard single-precision representation of the following numbers (Hint: use default rounding scheme if necessary):

a. -0.11101101010110011112 × 2-121
b. -1001.11110000110010111102 × 2-129
c. (-infinity) / (+infinity)
d. (+infinity) - (-infinity)
e. 0/(infinity)
f. 1111.101111110112 × 2124

Problem 4 (2 bonus points)

Consider the IEEE 32-bit standard floating-point format.

Can floating-point numbers be compared by treating their representations as signed integers in the signed magnitude representation, even when denormals and +/- infinity are considered?