Download History of Mathematics–Spring 2015

History of Mathematics–Spring 2015
Homework 2
Due: Friday, January 23, 2015
1.
Finding areas is a very old activity. A square of sides of
length 1 has area 1. Given any polygon, any region whose
boundary is made up of straight line segments, it is easy
to see how squares can be broken up to cover the region.
Polygonal regions are easy to square (finding their area).
But when the region has a curved boundary, things get
harder.
Hippocrates of Chios (ca. 470 BCE-410 BCE), not to be
confused with the semi-mythical father of medicine Hippocrates of Cos, “squared some circular areas.” They are
known as the lunes or moons of Hippocrates. He thought
he was well on the way of squaring the circle. He wasn’t.
One of his lunes is to the right. AOB is a quarter circle.
Using the secant AB as diameter one draws a half circle
as shown.
The lune is the region bounded by the quarter circle and the semicircle, in green in the picture. Prove: The area of
the lune equals the area of the triangle 4AOB.
2.
Another one of Hippocrates’ lunes is to the right. The
quadrilateral ABCD is half of a regular hexagon inscribed
in the circle of diameter AD. A semicircle is drawn Using
CD as diameter. The lune is the region bounded by the
big circle of diameter AD and the semicircle of diameter
CD; in green in the picture.
Prove: Let T be the area of the trapezoid ABCD, S the
area of the semicircle of diameter AB andL the area of the
lune. Prove that T = 3L + S. It may help to know that
the side of a regular inscribed hexagon equals the radius
of the circle.
3. Here is a simple geometric problem. Only a few basic Euclidean principles are needed to solve it. You will, of course,
mention them: Suppose M is the midpoint of a segment AB. Let P be any point not on AB and draw the segments
AP, M P , and BP . Show that triangles AM P and M BP have the same area.
4. Fun with Perfect Numbers. A perfect number is a positive integer that is equal to the sum of its proper divisors.
The proper divisors of a number are just all divisors but the number itself.
2
The number 6 is perfect; its proper divisors are 1, 2, 3 and 1 + 2 + 3 = 6. The number 28 is perfect; its proper divisors
are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. Pythagoreans were very impressed by these numbers. Euclid has a
Proposition that states:
If 1 + 2 + · · · + 2n−1 is prime, then (1 + 2 + · · · + 2n−1 )2n−1 is perfect.
(a) Show that Euclid’s Proposition can be stated in a more compact way as: Assume that 2n − 1 is prime. Then
2n−1 (2n − 1) is perfect.
(b) Show that if 2n − 1 is prime, then n must be prime.
(c) With n = 2 we get 2n−1 (2n − 1) = 6. With n = 3 we get 2n−1 (2n − 1) = 28. The next prime is n = 5 and
25 − 1 = 31 is prime, thus (as can be verified) 24 (25 − 1) = 496 is perfect. Find three more perfect numbers.
Verify they are perfect. Notice by the way that 211 − 1 = 2047 = 23 × 89.
(d) Can you prove Euclid’s Proposition?
Perfect Notes. All of Euclid’s perfect numbers are even. Some 2000 years after Euclid, Euler proved that all
even perfect numbers are in this form; that is, if m is perfect and even, then m = 2n−1 (2n − 1) for some positive
integer n for which 2n − 1 is prime.
It is not known to this date if odd perfect numbers exist.
It is not known if there are an infinite number of perfect numbers.
(e) The Pythagoreans were also quite interested in what is known as the golden section. Given a segment AB the
idea is to find the point C in the segment with the property that the ratio AB/AC equals AC/CB. The segment
AC
AB
=
.
below is cut by the golden section,
AC
AB
√
AB
1+ 5
Show that if C cuts the segment AB into the golden section then
=
.
AC
2
FYI: The ratio AB/AC, known as the golden ratio and usually denoted by the Greek letter φ, appears in a lot of odd places. It appears a lot in the
five pointed star or pentagram, which is obtained
by drawing all the diagonals of a regular pentagon.
It can be drawn without lifting the pen from the
paper. Each side of the pentagram (diagonal of
the pentagon) divides each other side into golden
section. The Pythagoreans thought this was really
awesome. BEWARE The pentagram has magical powers. It has to be treated with great care.
For extra credit: Prove in the picture of the pentagram above that AB/AC = φ.
(f) Utopos is a parallel universe. It is much like our universe except that matter is infinitely divisible, there are no
atoms. Take paper, for example. Here, in our universe, there is a limit on how thin it can be. There, it can be
as thin as you wish. In the big library of the planet Pseudogea in this universe there is a book with an infinite
number of pages. The first page of the book is 1 centimeter thick. Every page is half as thick as the preceding
page.
i. How thick is the whole book?
ii. A library patron decides she wants to read the whole book. She can read at incredible speeds. She starts
leisurely taking five seconds to read the first page. She then reads every consecutive page at three times the
speed at which she read the previous page. Does she ever get to finish the whole book? (Assume only one
side of the page needs to be read.)