Homework 2 from Lecture 6 to Lecture
10
August 31, 2015
Lecture 6
1. The Greek mathematician Menaechnus (350 B.C.) obtained a purely theoretical solution to the duplication problem based on nding the point of intersection of certain
conic sections. In mod
Solutions of Homework (updated weekly)
Shanyu Ji
October 25, 2010
Lecture 8. Eudoxus of Cnidus Who Avoided a Fundamental Conict
1. Use Eudoxuss denition to prove Proposition V-16: If a : b = c : d, then
a : c = b : d.
Solution:
Let us rst review Eudoxus d
Partial Solutions of the Homework
Assignment History of Mathematics
Shanyu Ji
June 17, 2009
Chapter 9. Aristotle and his Profound Inuence
1. Consider the following statement: given a sequence cfw_an , there exists a positive number
M, such that |an | M fo
Homework for Lecture 3
March 21, 2014
Homework:
1. Around the time of Thales, there were two other important thinkers: Anaximander of
Miletus and Heraclitus of Ephesus. Read about these two men. What did they think
about the world?
2. From Figure 3.3 of t
Lecture 15. Mathematics of Medieval China
For a long period, the geographical nature of the country such as mountains, desserts
and seas formed natural boundary which isolated China. As a result, independent of other
civilizations, there was a continuous
Lecture 12. The Alexandrian Greek Period
and Archimedes
The Alexandrian Greek Period
In 352 B.C., Philip II of Macedonians launched
campaigns of conquest on Greece, it led to the destruction of classical Greece. Athens was
defeated in 338 B.C. Alexander t
Lecture 14. Decline of Greek Mathematics
Figure 14.1
Ancient Greece
Summary of the Greek Achievements
mathematicians?
What are the accomplishments of Greek
The Greeks are to be credited with making mathematics abstract
Making mathematics abstract was the
Lecture 16. Mathematics of Medieval India
The second period of Hindu mathematics
Indian mathematics is also known as
Hindu mathematics. After the early Hindu mathematics, the second period of Hindu mathematics may be roughly dated from about A.D. 200 to 1
Lecture 13. Diophantus and Diophantine
equations
Diophantus Diophantus of Alexandria, (about 200 - 284), was a Greek mathematician.
He was sometimes called the father of algebra, a title some claim should be shared by a
Persian mathematician al-Khwrizm (a
Homework for Lecture 24
July 17, 2014
1. Read Rules for the Direction of the Mind in the appendices.
2. In Le Gomtrie, Descartes constructed the positive solutions to the quadratic equation
e e
2
x ax + b2 = 0 where b < a . Given a circle with center (0,
Lecture 17. Mathematics of Medieval Arabs
The Arabs The term Islam means resignation, i.e., resignation to the will of God as
expressed in the Koran, the sacred book, which contains the revelations made to Islams
founder Muhammad (570-632).
Arabs were nom
Lecture 22. Napier and Logarithms
Figure 22.1 John Napier and his birth place: Merchiston Tower, Edinburgh.
The invention of logarithms by Napier is one of very few events in the history of mathematics there seemed to be no visible developments which fore
Lecture 26. Early Stage of Projective
Geometry
Figure 26.1 The woodcut book The Designer of the Lute illustrates how
one uses projection to represent a solid object on a two dimensional canvas.
Projective geometry was rst systematically developed by Desar
Lecture 33. Non-Euclidean Geometry
Figure 33.1. Euclids fth postulate
Euclids fth postulate
In the Elements, Euclid began with a limited number of
assumptions (23 denitions, ve common notions, and ve postulates) and sought to prove
all the other results (
Lecture 34. Complex Numbers
Origin of the complex numbers
Did it come from the equation
Where did the notion of complex numbers came from?
x2 + 1 = 0
(1)
as i is dened today? No. A very long time ago people had no problem accepting the fact
that an equati
Lecture 32. Gauss, the Prince of
Mathematics
Mathematics is called the Queen of sciences and Gauss was called the Prince
of Mathematics.
Gauss and his life
Carl Friedrich Gauss(1777-1855) is a German mathematician who
contributed signicantly to many elds,
Lecture 9. Aristotle and his Profound
Inuence
Figure 9.1
Aristotle (384-322 B.C.)
Aristotle Unlike Socrates and Plato, Aristotle was not originally from Athens. Aristotles
father was a doctor who lived in Macedon north of Greece. When Aristotle was 18, he
Lecture 11. Apollonius and Conic Sections
Figure 11.1
Apollonius of Perga
Apollonius of Perga Apollonius (262 B.C.-190 B.C.) was born in the Greek city of Perga,
close to the southeast coast of Asia Minor. He was a Greek geometer and astronomer. His
major
Lecture 1. The Dawn of Mathematics
The dawn of mathematics
In ancient times, primitive people settled down in one
area by water, built homes, and relied upon agriculture and animal husbandry. At some
point, they began to use simple concepts of numbers suc
Homework for Lecture 25
July 17, 2014
1. Use Fermats method to nd the local maximum of x2 2x3 .
2. Prove: Fermats theorem: For any positive integer a, if p is a prime, then ap a is
divisible by p.
Hint: Use mathematical induction. First prove the case a =
Homework for Lecture 23
July 17, 2014
Homework:
1. Recall Vites theorem: The roots x1 and x2 of the quadratic equation ax2 +bx+c = 0,
e
where a = 0, satisfy
c
b
x1 + x2 = , x1 x2 = .
a
a
What are the generalized formulas for the roots x1 , x2 , x3 of cubi
Homework for Lecture 26
July 17, 2014
1. The language of axiomatic projective geometry is a language consisting of two parts:
point and line, and one binary predicate I(p, L) we call incidence between points
and lines. There are various ways to call I(p,
Homework for Lecture 27
July 17, 2014
1. Find 1 + 2 + . + n by summing the identity (m + 1)2 m2 = 2m + 1 from m = 1 to
n. Similarly nd 12 + 22 + . + n2 using the identity
(m + 1)3 m3 = 3m2 + 3m + 1
together with the previous result.
2. In 1671, the Scotti
Homework for Lecture 29
April 14, 2014
1. Show the result found by Jacob Bernoulli: 1 +
1
22
+
1
32
+ . +
1
n2
+ . 2.
2. In the Ars Conjecandi, Bernoulli obtained the sum of the series
1
1
1
1
+
+
+ . +
12 23 34
n(n + 1)
1
1
by considering 12 = 1 2 ,
the
Homework for Lecture 30
July 17, 2014
Homework:
1. Take a look at one of Eulers papers, in which he proved the famous formula: 1 + 212 +
2
1
+ 412 + . = 6 , in the Appendices.
32
2. Understand Eulers solution for the problem of the seven bridges of Konigs
Homework for Lecture 31
July 23, 2014
1. What is the Lagrange mean value theorem in Calculus ? Prove: |cos x cos y|
|x y|, x, y R by this theorem.
2. Lagrange asserted that any function can be expressed as a power series. Could you
give an example to sho
Homework for Lecture 28
July 17, 2014
1. What is the fundamental theorem of calculus? Write a short history of this important
theorem.
2. Use the fundamental theorem of calculus to nd derivative
df
dx
if f (x) =
x
0
sin t dt.
3. The binomial theorem, as s
Homework for Lecture 35
July 23, 2014
1. What is the Cauchy criterion for the convergence of an innite sequence of real numbers
? Use the Cauchy criterion to tell whether or not the following series is convergent:
cos x cos 2x
cos nx
+
+ . +
+ .
2
2
2
2n
Homework for Lecture 33
July 23, 2014
1. Read some of Gausss letters on non-Euclidean geometry in the appendices.
2. A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the
base. Let ABCD and A B C D be two Saccheri quadrilat
The Project Gutenberg EBook of General Investigations of Curved Surfaces
of 1827 and 1825, by Karl Friedrich Gauss
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use
Comment. acad. sc. Petrop. 7. 1740 (p. 124 onwards) :
L. EULER. De summis serierum reciprocarum. (E041). Translated by Ian Bruce.
1
Concerning the sums of series of reciprocals.
Leonhard Euler.
. 1.
Now the series of the reciprocals of the powers of the n
Gauss
And
Non-Euclidean Geometry
Crash Course Notes
by Stan Burris
Waterloo, September 2003
1
BRIEF TIMELINE
1796 1798
Gauss and Bolyai (senior) are both studying in Goettingen; they were the best
of friends. Gauss left Goettingen in September of 1798, Bo
Appendix: Weierstrass, Father of Modern
Analysis
Weierstrass and M
unster
A teacher in secondary school German mathematician Karl Weierstrass (1815-1897)
is considered the father of modern analysis. Not like many great mathematicians succeed in
earlier ag
Hippasus and Evolution
All things being equal, one might imagine the study of mathematics to being pretty straight forward,
especially compared to other subjects. After all, 2 + 2 is simply 4 and even ei 1 can be shown to be
equal to zero. There is no Hei