# Integer Arithmancy

ex Practice number representation and computer arithmetic.

**Assign:**Monday, 12 September**Due:**before class Monday, 19 September**Submit:**Bring a typed or neatly handwritten 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 just ten toes, with the left pinky toe as a sign bit for two’s (toes) complement numbers. What is the minimum expressible number?
- What is the maximum fingers-and-toes-complement number?

## Problem 2

Perform the following conversions:

- Show the 8-bit two’s complement representation of -107
_{10}and 107_{10}. - 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 in two’s complement representation. Show the sums
in binary and convert the sums to decimal notation too. Indicate for
each whether overflow has occurred.

**Optional** Practice Problems

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

*CSAPP3e*Homework Problem 2.77. Thinking about powers of 2 will help.*CSAPP3e*Homework Problem 2.82. An informal prose description suffices to “describe the underlying mathematical principles”.