Download Review 7

MathEd / Math 300
Course Review - History
1.
Know the following counting systems we discussed in class:
a.
Tallies and tokens
b.
Knots systems and counting boards
2.
Know what is meant by:
a.
2-counting
b.
4-counting
c.
5-counting
d.
10-counting
e.
5-10-counting
f.
5-20-counting
g.
5-10-20-counting
3.
Know where 2-counting systems, 5-20 systems and 5-10 systems are generally found.
4.
Know what is meant by, and how to recognize, 2-cycles, 5-cycles, 10-cycles, and 20cycles.
5.
Explain the Babylonian number system
a.
What symbols were used? What bases?
b.
How were whole numbers and fractions represented?
c.
How were addition and subtractions done?
i.
What kinds of helps (e.g. tables) were used or needed for addition and
subtraction?
d.
How were multiplication and division done?
i.
What kinds of helps (e.g. tables) were used or needed for multiplication
and division?
e.
What other kinds of numerical calculations were done?
i.
Powers, roots, reciprocals...
f.
What were the strengths and weaknesses of the Babylonian number system?
g.
Be able to do some simple calculations in the Babylonian system (NOT ON
FINAL)
6.
How do we know about Babylonian mathematics? What are the major sources?
7.
What kinds of problems could the Babylonians solve?
a.
Classify in terms of our linear, quadratic, simultaneous etc. equations
b.
Anything special or unique about their methods?
8.
What kinds of geometry problems did the Babylonians solve?
9.
Explain the Egyptian number system
a.
What symbols were used? What bases?
b.
How were whole numbers and fractions represented?
c.
How were addition and subtractions done?
i.
What kinds of helps were used or needed for addition and subtraction?
d.
How were multiplication and division done?
i.
What kinds of helps were used or needed for multiplication and division?
e.
What other kinds of numerical calculations were done?
i.
Powers, roots, reciprocals .. .. .. .
f.
What were the strengths and weaknesses of the Egyptian number system?
g.
Be able to do some simple calculations in the Egyptian system (NOT ON
FINAL).
10.
How do we know about Egyptian mathematics? What are the major sources?
11.
What kinds of problems could the Egyptians solve?
a.
Classify in terms of our linear, quadratic, simultaneous etc. equations
b.
Anything special or unique about their methods?
12.
What kinds of geometry problems did the Egyptians solve?
13.
Miscellaneous:
a.
Explain what a Red Auxiliary is and how it was used.
b.
Talk about Plimpton 322. Why is it important?
c.
Why did the Babylonians use base 60?
d.
What are some vestiges of Babylonian mathematics in culture?
14.
Know the differences between ancient Greek mathematics and that of the Babylonians
and Egyptians. What effect did Greek mathematics have on our mathematical culture?
15.
How do we know about ancient Greek mathematics? What are the major sources?
16.
Be able to describe the major accomplishments and contributions of the following
mathematicians:
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
Apollonius
Archimedes
Diophantus
Euclid
Eudoxus
Hippocrates
Hypatia
Pappus
Ptolemy
Pythagoras
Thales of Miletus
17.
Be able to discuss the content and importance of Euclid’s Elements.
18.
What characterized ancient Chinese mathematics? What are our sources, and what are
the important books from that period?
19.
Explain the ancient Chinese number system
a.
What symbols were used? What bases?
b.
How were whole numbers and fractions represented?
c.
What kinds of numerical calculations were done?
i.
Powers, roots, reciprocals...
20.
What kinds of problems could the ancient Chinese solve?
a.
Classify in terms of our linear, quadratic, simultaneous etc. equations
b.
Anything special or unique about their methods?
21.
What kinds of geometry problems did the ancient Chinese solve?
22.
Is there anything unique about ancient Chinese mathematics?
23.
What characterized ancient Indian mathematics? What are some of its accomplishments
and peculiarities?
24.
What were the mathematical accomplishments of the Islamic Empire?
25.
Discuss the accomplishments of these Islamic mathematicians:
a.
Al-Khwarizmi
b.
Khayyam
c.
Al-Tusi
26.
What were the mathematical accomplishments of the Medieval period? Who was the
major mathematical figure from this period, and what did he accomplish? What was his
major work?
27.
Name some mathematical accomplishments and figures from the Renaissance period.
a.
Who predicted the cubic equation could not be solved by radicals?
b.
Who solved the cubic and quartic equation by radicals? Be able to tell this story.
c.
Be able to discuss the accomplishments of:
i.
Johannes Müller, Regiomontanus
ii.
Luca Pacioli
iii.
Scipioine del Ferro
iv.
Copernicus
v.
Tartaglia
vi.
Cardano
vii.
Ferrari
viii. Bombelli
ix.
x.
xi.
xii.
xiii.
Viète
Brahe
Napier
Stevin
Kepler
28.
What were the major mathematical accomplishments of the 1600's?
29.
Be able to discuss the accomplishments of:
a.
René Descartes
b.
Pierre de Fermat
c.
Isaac Barrow
d.
Isaac Newton
e.
Leibniz
f.
Johann and Jacob Bernoulli
g.
L’Hospital
30.
What were the major mathematical accomplishments of the 1700's
31.
Be able to discuss the accomplishments of:
a.
Saccheri
b.
George Berkeley
c.
Johann Lambert
d.
Euler
e.
Ruffini
f.
Gauss
32.
What were the major mathematical accomplishments of the 1800's?
33.
Be able to discuss the accomplishments of:
a.
Cauchy & Weierstrass
b.
Lobachevsky
c.
Bolyai
d.
Able
e.
Galois
f.
Beltrami
g.
Cantor
34.
Be able to discuss the accomplishments of:
a.
Hilbert
b.
Russell
c.
Gödel
d.
Cohen
35.
Describe the accomplishments or “state of the art” of algebra in each of these periods:
a.
b.
c.
d.
e.
f.
g.
h.
Ancient Egypt
Ancient Mesopotamia
Ancient Greece (don’t forget Diophantus)
Ancient and Medieval China
The Islamic Empire
The European Middle Ages and Renaissance
The 1600's
The 1800's
You should consider at least: a) the kinds of problems or equations that could be solved;
b) the methods of solution available; 3) algebraic notation; and 4) algebra’s relation to
geometry. Don’t forget systems of linear equations or indeterminant equations)
36.
Describe the “state of the art” of geometry in each of these periods:
a.
b.
c.
d.
Ancient Egypt, Mesopotamia, China, India (as a group)
Ancient Greece
The 1600's (what happened?)
The 1800's (what happened?)
37.
Speaking of geometry, discuss the significance of Euclid’s fifth postulate from its
inception through the development of non-Euclidean geometries. Why was that postulate
an issue in the first place? How early, historically, did it become an issue for
mathematicians? Who are the major figures who worked on the problem? How was it
resolved, by whom, and what were the consequences for mathematics as a whole?
38.
Name 10 important mathematical works (e.g. books) by title and/or author and
or/description, and give a sentence telling why each is important or what it dealt with.
Look up the names, titles, etc if you need to.
39.
Today, a mathematician tries to publish new discoveries as quickly as possible so as to
have priority – that is, so that everyone knows she was first. Give some examples of how
that was not necessarily true in the past.
40.
Some questions about number systems:
a.
b.
41.
Compare and contrast the number systems of the Ancient Babylonians, Egyptians,
and Chinese cultures.
What do the number systems of the Greeks and Hebrews have in common, and
what kind of nonsense did it lead to?
OK, here’s a mean one: Describe what Brian Butterworth calls the number module and
discuss some of the evidence for its existence.
42.
Compare and contrast the lives and mathematical contributions of Euler and Gauss.
43.
Below are listed some important mathematical texts. Be able to give their author(s) and
provide a few sentences about them; include their importance to the history of
mathematics.
The Almagest
Liber Abaci
Principia Mathematica
The Elements
The Ahmes Papyrus
Plimpton 322
The Nine Chapters
The Sulbasutras
The Compendious Book on Calculation by Completion and Balancing
Liber Algorizmi
The Bakhshali Manuscript
On Triangles of Every Kind
Ars Magna
La Géométrie (an appendix to the Discourse on Methods)
Proofs and Refutations
Philosophiae Naturalis Principia Mathematica
44.
How did the mathematics of the Greek culture differ from that of the Egyptian,
Babylonian, Chinese, and Indian cultures? Give specific examples to illustrate the
difference.
45.
What roles were played by the Library at Alexandria and the House of Wisdom in
Baghdad in the history of mathematics?
46.
Below are listed some pairs of mathematicians. Describe in a couple of sentences what
they have in common.
Fermat and Descartes
Newton and Gauss
Lobachevsky and Bolyai
Cauchy and Euler
Galois and Abel
Newton and Leibniz
47.
Give an example of how the Greek’s strong preference for geometric thinking affected
later developments in algebra. (Hint: Think about Cardano’s work on cubics, or the
notations that were used in early algebra.)
48.
How does what we know about “Pascal’s Triangle” support a Platonic view of
mathematics?
49.
How would a mathematical humanist like Reuben Hersh respond to the above argument
about Pascal’s Triangle?