## Homework 3

Parhami

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

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

### Recommended reading (material covered during the next class)

Parhami

Chapter 17, Floating-point Representations.

Wikipedia

IEEE 754-2008

### Due Monday, March 26

#### Problem 2

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

A. sum of 64 integers in the range from 0016 to FF16 each
B. product of 100
integers in the range from 0 to 99 each

using

b. Represent the following decimal numbers in hexadecimal representation.

864.6328125,   1567.19140625

567251.436342347667.6242341

d. Represent    -109.7109375 and -71.2890625
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. (Hint: Apply formulas from Lecture 1, slide 70, "Extending the number of bits of a signed number"). Convert the obtained numbers to the decimal representation and show that they have the same value as numbers you started with in point e.

Problem 3 (bonus)

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 70, "Extending the number of bits of a signed number").

Problem 4 (bonus)

Find the contents of LUT F and LUT G in the implementation of a 5-to-3 parallel counter using a single CLB slice of a Virtex FPGA (see slides from Lecture 4).