MATH 401   Spring 1997

Grader: Petra Henneberger  MS 607

Office Hours: Wednesdays 10 - 12  or after my classes at the classroom
              Tuesday and Thursdays   10:30 - 11:45   REC 301 
              Tuesday and Thursdays   3:00 - 4:15     UNIV 303
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Books: Text: Victor J. Katz, A History of Mathematics.
       Reserve: G. Polya, How to Solve it.

GRADING POLICY:

The grading system is designed to foster habits which aid in problem solving.
Suppose a problem is worth  p  points. 
If the student cannot solve the problem and states "I can't solve the problem
because of .... ", he or she will be rewarded with at least p/2 points.
False statements and incorrect answers
can lower the grade to  0  .  A completely correct solution is worth  p  
points. If the student cannot solve the problem, but can explain why he 
cannot solve the problem, the score will vary from  p/2  to  p  depending
upon the reason given by the student. Students who solve the problem will
be eligible for extra credit by reflecting profitably upon the problem.

Students are encouraged to look up solutions or to discuss the problem with
their friends if they wish. I only require that the solution be written in 
the student's own words, and that any substantial help be cited. These 
citations will not harm the student's grade. Giving due credit is part of the
complete honesty necessary to carry out mathematics.
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EXAMS:

There will be a FINAL EXAM. It will cover vector algebra for Euclidean
Geometry as well as spherical geometry.

Students who miss less than three classes in the semester are exempt from
the exam.

401 Final  10:20 - 12:20  Univ 303  On May 7, 1997.
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COMPUTERS:


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ANNOUNCEMENTS:

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Assignment 1: Write a page essay detailing the mathematics you have studied
and your strengths and weaknesses and enthusiasm for mathematics.


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Assignment 2: Counting and Arithmetic to the ancients.

Chapter 1, problems 2, 3, 4, 10, 11, 12 .

Each problem is worth 10 points. In problem 11, just do as many of the 
reciprocals as is necessary to answer the question in the second sentence
of the problem.
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Assignment 3: 
  a)  Let a set of numbers contain exactly one each of the ten digits
      0,1,2,3,4,5,6,7,8,9. Can the set add up to 100 ? (Polya, p. 163.)

  b)  Propose a problem about the digits of a number. Hint:
                Use Theorem 401

Theorem 401: Let N be a number which is expressed in terms of digits between
zero and nine. Let  d  be a number. Divide  d  into  1, 10, 100, 1000, 10000,
and so on and take the remainders. These form a Sequence of integers beginning
with  1  and continuing. Call this Sequence the "Decimal Remainder Sequence
for  d " Now multiply the unit digit of  N  by  1, the first number of the
Decimal Remainder Sequence
for  d , and then multiply the tens digit of  N  by the second number of the
Decimal Remainder Sequence
for  d , and multiply the 100's digit by the third number in the Sequence and
so on until there are no more digits. Then add all these products up to get
a number  R  . Then  d  divides  N  if and only if  d  divides  R.

Note that the conclusion is still true if any number in the 
Decimal Remainder Sequence
for  d  is replaced by itself plus or minus a multiple of  d  .
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Assignment 4:
  Katz, Chapter I, # 18, 19, 20 34, 36, 37, 38.
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Assignment 5:

1. Suppose you sum the digits of a positive integer 
and then sum the digits of that 
number and continue this process until you reach a number with a single
digit. Show that the original number was divisable by 9 if and only if 
the single digit is 9.

2. Give necessary and sufficient conditions on the digits of an integer
so that it is divisable by 
a) 10
b) 5
c) 2
d) 3
e) 4 
f) 6
g) 7
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Assignment 6:

  Katz, Chapter 2, Page 90: # 1, 2, 3, 4, 5, 6, 7, 9.
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Assignment 7: Give examples of the following types of statements.

1. An axiom
2. A proposition
3. A hypothesis
4. A conclusion
5. A converse
6. A contrapositive
7. An equivalence
8. A definition
9. An example of some statement
10. A counter example to a statement
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Assignment 8:

1. Give an example illustrating each of the concepts defined in Side Bar 2.1 on
Page 57.

2. Give examples illustrating each of the Axioms and Postulates on Pages 57-58.

3. For each of the propositions displayed in 2.4.2 of Katz, state:

   1. The hypotheses
   2. The conclusions
   3. The contrapositive
   4. The converse
   5. An example illustrating the true converse or a counterexample destroying
      the false converse.
* For all these problems, if you cannot understand what to do, identify the
  words you don't understand and write a description of your attempts to
  find out their meanings.
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Assignment 9:

1. What does it mean to divide a line in extreme and mean ratio? Compare
   Side Bar 2.4 and problem 24 on page 91 and show their agreement.

2. Solve by any means problem 24.
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Assignment 10:

Katz, P. 90; # 12,13, 14, 15, 17.
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Assignment 11: 

Find the distance from the South Pole for which it is possible
to walk due South 10 miles, due East 10 miles and then due North 10 miles to
get to the original starting point. 
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Assignment 12:

Katz: P. 121; # 1, 2, 5, 12.
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Assignment 13:

Katz: P. 154, #22. 

Extra credit: 4 ships A, B, C, D, are sailing in a fog with constant and
different speeds along different straight line courses. Five of the six
possible simple near collisions are known to have occurred. 
Explain why the sixth near collision must occur.

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Assignment 14:

Katz, Chapter 5: # 8, 9, 15, 20, 21, 22, 23.

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Assignment 15:

P. 219, # 32.
P. 262, # 22a.
P. 297, # 2.
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Assignment 16:

Katz, Chapter 9, P. 348; # 1, 2, 4, 7, 18, 19, 20, 21, 22*, 26, 35.
   * Use the book's answer to write the problem in modern English.
   ** c > 0  and  d > 0  .
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Assignment 17:

Davis and Snider, Page 6, # 1, 2, 3, 4, 5, 6, 7.
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Assignment 18:

Vector Problems, # 1, 2, 3, 4, 5, 6.
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Assignment 19:

;Nothing.
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Assignment 20:

Use a geometric theorem from euclidean geometry as a space-time diagram to
invent a problem about time distance and motion.
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Assignment 21:

Use the cross ratio and the figure for cross ratio = -1  to construct a space
time motion problem.
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Assignment 22:

Problems 7 and 8 in the vector problem set.
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Assignment 23:

Do problem 10 in the vector problem set.
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Assignment 24:

Do problems 9, 11, and 12 in the vector problem set.
Where problem 12 is to find the the locus of  (x - a) dot (x + a) = 0.
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Assignment 25:

Show that if the velocity has constant speed along a curve, the acceleration
is orthogonal to the velocity.
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            Problems about Functions: Assignment 26


Suppose you have 4 pennies, 2 nickels, 3 dimes and 4 quarters in your pocket.

1. How many cents do you have?

2. How many coins do you have?

3. Let  X  be the set of all the different subsets of those coins, including 
the empty set of coins and the complete set of coins.
How many elements (i.e. subsets of coins) does  X  have?

4. Let  Y   be the set of ``combinations'' of coins in your pocket.
By a combination I mean something like  { 2 pennies and a nickel} as opposed 
to a set where  { the 1942 penny and 1997 penny and the shiny nickel }
is different from  { the 1990 penny, the 1992 penny and the dirty nickel }.
How many ``combinations'' are in  Y  ?

5. Let  $   denote the rule which takes a bunch of coins and gives their value 
in cents. For example, $\cents$ assigns the value of 7 to the combination 
2 pennies and a nickel.

6. Is  $: X -->  {1,2,3,...}  a function?

7. Is  $: Y --> {0,1,2,3,...,144}  a function?

8.
   a) How many combinations yield 4 cents?
   b) How many elements are in the fiber of  4  contained in  Y  ?
   c) How many sets yield 4 cents?
   d) How many elements are in the fiber of  4  contained in  X ?

9.
   a) Is  $: Y --> {0,1,2,...,144}  an onto function?
   b) Is every amount between 0 and 144 given by a ``combination'' of coins 
from your pocket?

10.
   a) Is  $: Y --> {0,1,2,...,144}  one to one?
   b) Is it impossible to find two ``combinations'' of coins from your pocket 
with the same value?

11.
   a) Which numbers have fibers with only one element for 
      $: Y --> {0,1,2,...,144}  .
   b) For which amounts is there only one ``combination'' which yields the 
amount?

12. Which numbers have fibers with only one element for the function 
    $: X --> {0,1,2,...,144}  ?

13. Can you find a pocket containing 144 cents worth of coins for which the 
function   $: Y --> {0,1,2,...,144}   is not onto?

14. Let  p: X --> Y  which take a set of coins to its underlying 
``combination'' of coins.
   a) Which combinations have the fiber with the most sets in it.
   b) How many sets are in this largest fiber?
   c)What is the value of  $   evaluated on this fiber?
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Assignment 27.  More problems about functions.

How many elements are in the fibre of 7 of the real valued functions whose rules
are written below?

a) The identity function.

b) Multiplication by -1.

c) The square of a number.

d) The square root of a number.

e) The area of a rectangle.

f) The perimeter of a triangle.

g) The radius of a circle.

h) The number of pages of a book.

i) The value of a set of coins.

j) The grade point average of a student.

k) The sum of the squares of two real numbers.

l) The sum of the squares of two integers.

m) The derivative of a function of one variable evaluated at zero.

n) The composition of the function in b with the function in c.

o) The composition of the function in c with the function in b.

In each case write out the meaning of the question without mentioning any
functional concept, and then solve the question.  Answers such as zero, one,
finite, infinite are acceptable.
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Bibliography:

Lecture 8: Journey Through Genius by William Dunham. Pages 29-32.
           
           Euclid, The Thirteen Books of the Elements. Translated by 
           Sir Thomas L. Heath. 2nd Edition, Page 251.

Lecture 12: Journey Through Genius by William Dunham. Pages 84-89.

Lecture 15: Journey Through Genius by William Dunham. Pages 134 - 142.

Lecture 17: Introduction to Vector Analysis by Davis and Snider.