Download History of Math - Texas Woman`s University

HISTORY OF MATHEMATICS, Fall 2007
Day and Time: T, 6PM-8:50 PM, Room: MCL 410
Instructor: Dr. Ellina Grigorieva, Office: MCL 414, Phone: (940) 898-2452,
e-mail: [email protected]
Office Hours: 2:30 p.m.-4:00 p.m.
5:00 p.m.-6:00 p.m.
Tuesday
W-TH

Class Materials: * Text: An introduction to history of mathematics, 6th
Edition, by Howard Eves.

Course Objective and Description: The purpose of this course is not
only study who created mathematics but mostly what mathematics was
created and how it affects other branches of mathematics and our life.
We will work out problems offered by Babylonian and Egyptian
mathematicians thousands years BC or problems solved by Newton in
17th century in order to see the cultural connections. This course will be
very helpful for those whose profession is mathematics teaching,
because they will learn numerous of methods for problem solving, which
they can always apply in their every day teaching.
B
c
R
a
O
R
R
A
b
C
1
a  ha
2
1
A  a  b  sin C
2
A  pr
A
abc
4R
A  p( p  a )( p  b)( p  c )
A
Tests: There will be 1 take home test and 1 major test –the comprehensive final exam, 100
points each. I will give you homework every day, and I will grade your class work every day. There
will be one presentation over a selected topic, 100 points maximum for a research paper. The final
exam will be given on December 11, at 6PM.
Grades: At the end of the semester three grades (class work, a project, and the final test) will
be added and their average will become your course performance grade. Your everyday class
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work will be evaluated using scale of 5, such that “5” is excellent, “4” is good, “3” is
satisfactory, and “2” is bad; you will receive “2” if you refused to go to the board or did not
prepare your homework while I was checking it. At the end of the semester everyone will have
many class work grades, I will find their average and then multiply that number by 20. For
example, if your average is 3.78, and then your class work grade will be 76. The grading scale is:
270-300=A
240-269=B
210-239=C
180-209=D
0-179=F
Class procedure: We will work out some problems from HW, and than I will introduce new material.
Attendance: Students who regularly attend class generally have a better understanding of
course material, which is reflected by higher exam scores. Because I will grade your class work
on regular basis your attendance is crucial. In addition, your attendance record could be the
determining factor in your final course grade if your grade is on borderline.
Note: Since communications between students and their instructors is crucial in any learning
process, you are always welcome to discuss our coursework with me. My main duty here is to
serve you. In order to best serve you, I welcome and encourage your comments and
suggestions. It is my belief that through our joint effort we can make this course both beneficial
and enjoyable. State law requires that a student be awarded credit of three hours for successful
completion of 40 contact hours. Hence attendance is an integral part of education.
Texas Woman’s University seeks to provide reasonable accommodations for all
qualified individuals with disabilities. This university will adhere to all applicable
federal, state, and local laws, regulations, and guidelines with respect to providing
reasonable accommodations as required affording equal educational opportunity. It is
the student’s responsibility to register with Disability Support Services and to contact
the course instructor during the first two weeks of the course to arrange for
appropriate accommodations.
The last day to drop class without academic penalty is October 5.
The last day of Fall 2007 classes is December 8, 2007.
Possible Research Topics:
1.
2.
3.
4.
5.
6.
Inductive mathematics or deductive mathematics
Normal numbers
The importance of unsolved problems
Three women of mathematics
History of decimal fractions
Five great French mathematicians of the 17th Century
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7. James Stirling
8. George Green
9. Famous Russian mathematicians of 20th century.
Day
1
Date
08.28
Plan
HW
CH1: Egyptian Mathematics
2
09.04
CH2: Babylonians
3
09.11
4
09.18
5
09.25
6
10.02
7
8
9
10.09
10.16
10.23
10
10.30
11
11.06
12
11.13
CH3: Greeks
Take home test 1 is given
CH4:Thales and Pythagoras
No class self study
CH5: Elements of Euclid
Test 1 (Take home test) due
CH6: Archimedes and
Inscribed Figures
CH 6
Review CH 1-6
CH7: Roman Era (Heron,
Diophantus )
CH8: quadratic formula.
CH9: Trigonometry
Ch10 and 11: Cardano’s
formula for cubic equation.
Francois Viete (11.5)
CH12: Fermat and Cavalieri
13
14
15
17
11.20
11.27
Projects Presentations
12.04 (Last class)
Review for Final
12.11 (Final Exam) FINAL EXAM
CH.1: Numeral Systems
CH 2: Babylonian and Egyptian Mathematics
CH 3: Pythagorean mathematics
CH 4: Duplication, Trisection and Quadrature
CH 5: Euclid and his Elements
CH 6: Greek mathematics after Euclid
CH 7: Chinese, Hindi, Arabian Mathematician
CH 8 European Mathematics
3
Problems:
P 27-31 # # 1.1, 1.2, 1.5, 1.6, 1.10
P 57-67 ## 2.1-2.17
p. 93-103 ## 3.2, 3.4, 3.6, 3.7, (3.8), 3.15
p. 124-133 ## 4.1, 4.2, 4.3, 4.5, 4.13, 4.15
p.155-163 ## 5.1, 5.2, 5.5, 5.7, 5.9, 5.12
p. 187-202 ## 6.1.6.2, 6.7, 6.10, 6.16, 6.21
p. 237-249 ## 7.1, 7.3, 7.6, 7.9, 7.11, 7.15, 7.16
p. 283-296 ## 8.1,.8.2, 8.5, 8.14, 8.15
Possible Plan:
1. review of trigonometry on the unit circle. Some problems. Derivation of the formula of
truncated pyramid volume.
2. Solid geometry review, cross-section construction for cubes and pyramids. Formula of the
area of projected figure (Area of the actual section x cosB= Area of projection).
Some problems of quadratic equations, such as
3. Continuation of cross-sections.
Solving HW problem about the pyramid’s cross-section. L is the midpoint of SC, BK: KC=1:3,
O is the point of gravity. Construct a cross-section that passes through K,O, and L and
evaluate its area, if pyramid id a tetrahedron with side of 1.
4
Using Thales theorem and Heron theorem . Answer is
63
72
.
Techniques of solving equations with modulus and different substitutions. 1).
x1
x 1 3


x 1
x1 2
2) x  3  4 x  1  x  8  6 x  1  1
HW: Solve:
1
1
1. 2( x 2  2 )  7( x  )  0
x
x
2
2
2. x  x  1 / x  1 / x  4
5
3.
x 2 48
x
 2  5(  4 / x )
3
3
x
4. 4 x 2  (3a  2) x  a 2  1  0 . For what values of a parameters a the sum of the roots is
greater than 1.25?
5. For what values of parameters p and q the difference of the roots of x 2  px  q  0 equals
5, but the difference of their cubes 35?
4.
Review of HW. Thales theorems (chapter 3) we proved that each angle supported by diameter
is 90 degree. Pythagorean Theorem, different proofs. Proofs (general form of a
number) of divisibility of a number by 2, 4, 3, 4, 8, 9 , 5, and 11.
5. Test 1 for two hours.
6. review of the test and handouts with solutions of the most difficult problems. Relationship
between arithmetic and geometric means. Regarding to one of the problems in the
Test. Continue chapter 3. Geometric proof for some algebraic relationships.
(a  b) 2  a 2  2ab  b 2 or h 2  ab . Some problems on Greeks (1-6). Started
problem 6.
Proof using similar triangles if you drop a perpendicular from the vertex of the right triangle.
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Discussion how to find such a point H on AB that x 2  a(a  x ) , a is a side of a square.
7. Problem 3.2c prove that for any perfect number (equals the sum of its proper divisors) the
sum of reciprocals of all divisors including the number itself equals 2.
Two numbers m and n are amicable if the sum of all proper divisors of n equals m and vice
versa. Finding amicable number if one is given, check the formula:
Let m is the number (1210), find its prime factorization :2, 5, 11, 11.
p n1
Then the product
, where n is the multiplicity of the factor is evaluated, then subtract
p1
original number and get its amicable number.
211  1 511  1 1121  1


 3  6  133  2394
21
51
11  1
2394-1210=1184!
Prove algebraic identities using geometry: a 2  b 2  (a  b)(a  b)
7
.
8.
We showed how to construct a point H on segment AB, such that AH^2=AB*HB, AH the
largest segment. (Problem from Greeks) Paul showed one way and said that also it can be
constructed on a circle and an equilateral triangle subscribed into it. Take a midpoint on its side
draw a line parallel to the basis a point of intersection of the line and the circle is H.
We also inscribed pentagon and 15th gon into a circle.
9. We have presentations
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