# Integer Arithmancy

ex Practice number representation and computer arithmetic.

**Assign:**Thursday, 2 February**Due:**before class Monday, 6 February**Collaboration: individual**ex assignment policy, as defined by the syllabus and honor code.**Submit:**Bring a typed or neatly handwritten paper copy of this standard answer sheet to class.**Relevant Reference:**

**Please download, print (2-sided), and write your answers on
this standard sheet to help us grade and provide
feedback to you faster.**

## Problem 1

Most people can count to 10 on their fingers; computer scientists can do better. Write answers to these expressions as simple arithmetic expressions or exact numbers in base ten.

- 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? (10-bit two’s complement)
- What is the maximum fingers-and-toes-complement (20-bit two’s 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 your work and do the calculuations in
binary. Additionally, convert the sum to decimal notation. Indicate
for each calculation whether or not overflow has occurred.

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

## Problem 4

Repeat all parts of problem 3 assuming the 8-bit values represent
**signed** integers in two’s complement representation.