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Math 4820: History of Mathematical Ideas,
Spring 2012
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Homework
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Assignment
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Assigned
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Due
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Problems
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| HW1 |
1/25/12
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2/1/12
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Read Sections 1.1-1.7.
1.
Use the methods of Section 1.3 to
find a parametrization by rational functions for the hyperbola defined by
x2-2y2=1.
2. True or False? Every integer N>2 occurs in some Pythagorean Triple.
(Justify your answer.)
3. The following fact was used during the lecture on 1/25/12:
If N is an integer and √N is rational, then √N is an integer.
Prove this using ideas from Section 1.5.
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| HW2 |
2/3/12
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2/8/12
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Read Sections 2.1-2.6.
1. What is the height of a regular tetrahedron of side length 1?
2.
Consider a regular polyhedron whose faces are congruent
regular p-gons and which has q faces meeting at each vertex.
Let v be the total number of vertices, e be the
total number of edges and
r be the total number of faces.
(i) Explain why 2e=pr and 2e=qv.
(ii) Show that v = 4p/(2p+2q-pq), e = 2pq/(2p+2q-pq), and
r = 4q/(2p+2q-pq).
3. Exercise 2.2.2 from the text.
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| HW3 |
2/8/12
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1. Exercise 2.3.3 from the text.
2. Exercise 2.3.4 from the text.
3. It is not possible to construct an angle
of Pi/13 radians. Show that it is nevertheless possible
to trisect an angle of Pi/13. (That is, if you are given an
angle of Pi/13, then from it you can construct an angle
of Pi/39.)
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| HW4 |
2/16/12
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2/22/12
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1. Show that if a polygon is constructible, then its
area is a constructible number.
2. Show that if a regular polygon of circumradius 1
has constructible area,
then it is possible to construct a copy of the polygon.
(The circumradius is the radius of the circumscribing circle.)
3. Give an example of a nonconstructible polygon whose area
is a constructible number.
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| HW5 |
2/23/12
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2/29/12
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Read Sections 3.1-3.4.
1. Find integer solutions to 81x+64y=1
using each of the following methods:
(i) the extended Euclidean algorithm.
(ii) the continued fraction algorithm.
(Find one solution per method.)
2. Find the regular continued fraction expansion of
√p for the values p=7,11,13.
3.
Find an integer solution to x2-py2=1
for each p=7,11,13.
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| HW6 |
3/8/12
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Read Sections 4.1-5.2.
1. Find the volume formula for a tetrahedron by completing:
(i)
Exercise
4.3.5
from the text.
(ii) Exercise 4.3.6 from the text.
2. Evaluate the sum
(1/n)2+
(1/n2)2+
(1/n3)2+…
using the method of exhaustion. Here n is an integer greater than 1.
3. Find the smallest positive integer x satisfying the system
x≡1 (mod 3)
x≡2 (mod 4)
x≡3 (mod 5)
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| HW7 |
3/16/12
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3/21/12
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Read Sections 5.3-5.7.
1.
Find the smallest positive integer x satisfying the system
2x≡1 (mod 3)
3x≡1 (mod 5)
5x≡1 (mod 7)
2.
A band of 17 pirates stole a number of gold coins.
When the pirates divided the coins into equal piles,
3 coins were left over. When they fought over who should
get the extra coins, one of the pirates was slain.
When the remaining pirates divided the coins into equal piles,
10 coins were left over. When the pirates fought again over
who should get the extra coins, another pirate was slain.
When they divided the coins in equal piles again, no coins
were left over. What is the least number of coins
that could have been stolen?
3. Find the general solution to 8x+16y≡14 (mod 10).
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| HW8 |
3/22/12
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4/4/12
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Correct your midterm for extra points! (Due April 4, 2012.)
Read Sections 6.1-6.5.
1. Exercise
6.5.2
from the text.
2. Exercise
6.5.3
from the text.
3. Find all of the roots of
x6 - 15x2 - 4 = 0 using
an adapted form of the Cardano formula. (That is, let
y=x2, solve y3 - 15y - 4 = 0
with the Cardano formula, then find x.)
Which of your three roots is equal to the root x=2?
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| HW9 |
4/6/12
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4/11/12
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Read Sections 6.6-6.8.
1. Sketch the parabola
y=x2-x and the hyperbola xy=1 together,
and locate all (real) points of intersection of these curves.
2. Find the 1rst, 2nd and 3rd power sums for the roots of
x3-2x2+x-3=0. (This means: find
the sum of the roots, the sum of the
squares of the roots, and the sum of the cubes of the roots.
You don't have to know the roots to solve this problem!)
3. Solve the system:
x+y+z=4
x2+y2+z2=4
x3+y3+z3=4
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| HW10 |
4/13/12
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Read Sections 24.1, 24.2, 24.4, 24.5.
1. Show that ω and
ω2 are equipotent. (ω2 is
the set {0,1,2,…,ω,ω+1,ω+2,…,
ω+ω=ω2,ω2+1,…,
ω3,…}. In the ∈-order, this set looks like
ω copies of ω.)
2.
Show that the open unit interval (0,1) is equipotent with the real line
by constructing a 1-1, onto function from one set to the other.
(Hint: Some function you know from calculus could work here.)
3. Show that the real line is equipotent with the real plane.
(Hint: the Cantor-Schroeder-Bernstein Theorem simplifies this problem.)
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| No HW |
4/18/12
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4/25/12
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No HW
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| HW11 |
4/26/12
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1.
Show that the Dehn invariant of a regular octahedron of side length 1
is not zero, hence a regular octahedron is not scissors congruent
to any cube.
2.
Compute the Dehn invariant of the following solid:
From a 3x3x3 Rubik's cube delete the 1x1x1
cubes from the center of the cube and from the center of each face.
(A total of seven 1x1x1 cubes have been deleted from the
total twenty-seven 1x1x1 cubes.)
3. Show that a regular tetrahedron is not scissors congruent
to the union of two regular tetrahedra, each half the volume of the original.
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