Assignment 7: Numbers Systems and Primes

**Announcement: There will be a quiz at the beginning of class
on Friday, March 7, on the reading from ZERO: The Biography of a
Dangerous Idea. The reading for Friday is Chapters 3 and 4 of
that book. The quiz might also have a couple of questions from the
earlier chapters.**

We will begin Assignment 7 in class on Wednesday. The assignment
consists of a few short problems from this sheet plus several exercises
from *Problem Solving Through Recreational Mathematics*. You will
work on the problems in groups in class, and might haver more time on
Friday or Monday to do further work.

The assignment is due in class on Wednesday, March 12 (which is the last day of class before Spring Break). Everyone in the class should write up their own individual solutions for the assignment.

We start with a few short questions on number systems and positional
notation. You will need the handout from class with Egyptian and
Babylonian numerals. The book, *ZERO*, might also be helpful.
A better graphic for Greek alphabetic numbers is shown at the right.

**Exercise 1.** Write out the number 61242 in Egyptian, Babylonian,
and Greek numerals. Do the best you can with the artwork (Note: 60^{2} = 3600,
and to translate the number to Babylonian, you will have to figure out how many
3600's there are in 61242.)

Recall that our usual positional number system uses the *base ten*, since powers of 10 are
used for determining the value of a digit, based on its position in the number.
For example, in base 10, 23725 means
2×10^{4} + 3×10^{3} + 7×10^{2} + 2×10^{1} + 5×10^{0}.
(Recall that 10^{0} = 1 and 10^{1} = 10.)
In a *base five* system, 2431 would represent
2×5^{3} + 4×5^{2} + 3×5^{1} + 1×5^{0}.
Written in base 10, that works out to 366. Note that the only valid digits in the base five
are 0, 1, 2, 3, and 4.

**Exercise 2.** Internally, computers use *base two* rather than base ten. In
base two, the only legal digits are 0 and 1, and powers of 2 are used when determining the
meaning of a number. Translate the base-two number 11010 into base 10.
The symbol "11010" is also a legal number in the base five. Translate the base five number
11010 into base 10. Translate the ordinary base 10 number 66 into base 5.

**The remaining exercises** are taken from Chapter 4 of *Problem Solving Through
Recreational Mathematics*, pages 138 and 139. You should do exercises
**4.1**, **4.5**, and **4.6**. Note that these exercises are
based on factoring numbers, and might use the idea of factoring a number into a product
of prime factors. (Two of these problems have answers in the back of the book. Of course, you need more than answers -- you need complete solutions
with explanations!)