ex Practice number representation and computer arithmetic.

• Due: before class Tuesday, 9 February
• Submit: Bring a paper copy to class.
• Relevant Reference:
• Collaboration: individual ex assignment policy, as defined by the syllabus.

These short exercises will help familiarize you with number representation and prepare you for the longer Bit Transfiguration assignment.

## Problem 1

Most people can count to 10 on their fingers; computer scientists can do better. If you regard each finger as one bit, with finger extended as 1 and finger curled as 0, how high can you count in base 2 using ten fingers and starting at zero? With both ten fingers and ten toes?

Now use both hands and both feet, with your left pinky toe as a sign bit for two’s complement numbers. What is the range of expressible numbers?

## Problem 2

Perform the following conversions:

• Show the 8-bit two’s complement representation of -10710 and 10710.
• Show the decimal notation of the signed integers whose 16-bit two’s-complement representations are given in hexadecimal notation as `0x5F8C` and `0xCAFE`.

## Problem 3

Perform the following calculations on the 8-bit representation of unsigned integers. Show the unsigned sums in binary and convert the sums to decimal notation too. Indicate for each whether or not overflow has occurred.

``````  00101101       11111111       00000000
+ 01101111     + 11111111     - 11111111
----------     ----------     ----------
``````

## Problem 4

Repeat the same calculations assuming the 8-bit values represent signed integers. Show the signed sums in binary and convert the sums to decimal notation too. Indicate for each whether overflow has occurred.

## Problem 5

Perform the following calculations on integer representations notated in hexadecimal. Assume that there is no limit on the number of hex digits we can store, i.e., keep your carries.

``````  5480       CAFF       20D
+ 8888     - 1BCD     *  23
------     ------     -----
``````

# Optional Practice Problems

These problems will get you thinking in the style needed for the Bit Transfiguration assignment.

## Problem 6

CSAPP2e Homework Problem 2.76 (p. 126), parts A, B, C, and D.

Thinking about powers of 2 will help.

## Problem 7

CSAPP2e Homework Problem 2.81 (pp. 126-127), parts A, D, and E.

An informal prose description suffices to “describe the underlying mathematical principles”. No proof is required.