Download Mat320 Exercises - My Illinois State

Mat 320 Exercises
The exercises are due with each Project submission (the first class of the following
week). The solutions will be discussed the class following the day they were assigned.
Week 1
1) Explain what is meant by Pythagorean triples. Parameterize them as a family indexed
by two natural numbers (Research the web / library for such a formula).
2) Systems of one linear and one quadratic equation appear in ancient “land problems”
(“To trade or not to trade the land?”, this was their question ).
Show that for rectangles, maximizing the area under fixed perimeter yields the same
shape as minimizing the perimeter under fixed area constraint.
3) What are square and triangular numbers in Greek mathematics?
4) What is “Squaring a circle” and why it is not possible using straight edge and
compass?
Week 2
5) Explain the circular uniform motion. What was Plato’s challenge?
6) Describe the Greek and Apollonius’ models of the motion of the sun, assuming earth
stationary: concentric model, eccentric, epicyclic, eccentric with epicycles.
7) What is a “Diophantine equation”?
Week 3
8) What is the Pell Equation and where does the name come from? (Short story)
9) What is the importance (“use”) of the Pell equation? What method is used to solve the
Pell equation?
10) What is the Chinese Remainder Theorem (State the problem in general and then the
Theorem for two congruence equations).
11) Solve the system of congruencies: n=2 (mod 3), n=3 (mod 5), n=2 (mod 7).
12) What are “permutations without repetitions” and “combinations without repetitions”?
(Explain and give the corresponding formula).
13) Write down and explain al-Khwarizmi’s geometric justification for the solution of
x2+10x=39.
(14) Write down and explain al-Khwarizmi’s calculation of the area of a triangle with
sides 13, 14, 15.
(15) Explain were does the RHS of the following identity comes from, and then prove it.
A B 
A  A2  B 2

2
A  A2  B 2
2
(16) Describe Heytesbury mathematical model of uniformly accelerated motion (1335).
(17) Explain the rudiments of group theory used in Malekula, South Pacific, to analyze
kin relationship (Hint: explain the two relationships with the corresponding constraints;
then explain the correspondence with the dihedral group presented by generators and
relations).
(18) Show how to reduce the general cubic equation to its reduced form: x3+cx=d.
(19) Prove that a solution (u,v) of the system:
u-v=d,
uv=(c/3)3
yields a root of the cubic equation x3+cx=d (Cardano’s formula):
x=u1/3-v1/3.
(Hint: check directly that x satisfies the reduced cubic equation).
(20) Use Viete’s substitution x=y+b/y to find a solution of the reduced cubic equation x33bx=2d (No need to transform it into Cardano’s formula).
(21) (a) Explain how to add ax2+b to complete the square in both sides of the reduced 4th
degree equation x4+cx2+e=dx.
(b) What system does (a,b) must satisfy, for ax2+b to complete the square in both sides?
(c) Show (explain) why the system stated at (b) has a solution.
(d) Using the solution from (c), explain how to solve the 4th degree equation (Ferrari’s
solution, 1522-1565).
(21) What are the Viete’s relations between the roots of a (a) quadratic equation, and (b)
of a cubic equation.
(22) What are the symmetric polynomials? Give a couple of examples.
(23) What are decimal fractions? Who introduced the notation for decimal fractions in
Europe? When? (Roughly).
Week 4 (Homework due: Thursday, the day of the Final)
(24) Perspective was used by painters of the Renaissance as a rendering technique.
Determine the central projection with center C(0,0,1) and projection plane OXY, as a
map f:R3->R2.
(25) Descriptive geometry uses parallel projection to specify solid objects by their plane
drawings. Determine the parallel projection with direction v=(1,1,1) and projection plane
OXY, as a map f:R3->R2.
(26) The idea of logarithm was suggested by the trigonometric identities relating addition
and multiplication:
sin(a+b)=sin(a)cos(b)+sin(b)cos(a), cos(a+b)=coas(a)cos(b)-sin(a)sin(b).
Prove the identities by using Euler’s formula: eix=cos(x)+i sin(x).
Questions and Problems
(Preparation for the Final Exam)
Tell the story of Archimedes and the king’s crown.
Tell the (“navy”) story of Archimedes and the “power of … levers”.
Tell the story of Hypathia.
Describe the difference between Euclid’s method (Elements’ style) and Papus’s On the
Domain of Analysis.
What is the 5th Postulate of Euclidean geometry? What other geometries are there from
the point of view of the various forms of the fifth postulate?
What is the ecliptic coordinate system for the celestial sphere? What is the equatorial
coordinate system of the celestial sphere? (Name the coordinates and sketch a picture).
What is the place-value decimal number system and where was it introduced and then
perfected?
What is combinatorics, and which sciences needs it and why?
Tell the story of the solution of the cubic equation. (Hint: Tartaglia, Cardano, Scipione
del Ferro).