Sections whose titles are marked by an asterisk (e.g. section 2.6) are not used later in the notes, and may be omitted. Hovever they contain really neat material, so you will not want to omit them.

In addition to the exercises, there are some questions and statements with the label ``entertainment''. These are for people who find them entertaining. They require more time and thought than the exercises. Some of them are more metaphysical than mathematical, and some of them require the use of a computer or a programmable calculator. If you do not find the entertainments entertaining, you may ignore them. Here is one to start you off.

This problem is very old. The Rhind Papyrus[16, page 92] (c. 1800 B.C.?) contains the following rule for finding the area of a circle:

**RULE I:**
Divide the diameter of the circle into nine equal parts, and
form a square whose side is equal to eight of the parts. Then the
area of the square is equal to the area of the circle.

The early
Babylonians (1800-1600BC) [38, pages 47 and 51]
gave the following rule:

**RULE II:** The area of a circle is 5/60th of the square of the
circumference of the circle.

Archimedes (287-212 B.C.) proved
that the circumference of a circle is
three times the diameter plus a part smaller than one seventh of the diameter,
but
greater than 10/71 of the diameter[3, page 134].
In fact, by using only elementary geometry, he gave a method
by which can be calculated to any degree of accuracy by someone
who can calculate square roots to any degree of accuracy. We do not
know how Archimedes calculated square roots, but people
have tried to figure out what method he used by the form of his
approximations. For example he says with no justification that

and

By using your calculator you can easily verify that these results are correct. Presumably when you calculate you will use a calculator or computer to estimate any square roots you need. This immediately suggests a new problem.

Zu Chongzhi (429-500 A.D.) stated
that is between 3.1415926
and 3.1415927, and gave 355/113 as a good
approximation to .[47, page 82]

Here is a first approximation to . Consider a circle of radius 1 with center at , and inscribe inside of it a square of side with vertices at and Then by the Pythagorean theorem, . But is the area of the square , and since is contained inside of the circle we have

Consider also the circumscribed square with horizontal and vertical sides. This square has side 2, and hence has area 4. Thus, since the circle is contained in ,

It now follows that

A number of extraordinary formulas for are given in a recent paper
on *How to Compute One Billion Digits of Pi*[12].
One amazing formula given in this paper is the following result

which is due to S. Ramanujan(1887-1920)[12, p 201,p 215]. The reciprocal of the zeroth term of this sum i.e.

gives a good approximation to (see exercise 0.4).

and compare this with the correct value of , which is .