Date: Friday, April
1
Time: Usual class time (12 noon - 12:50 p.m.)
Place: WEL 1.308 (Same room as first exam; see map on information for first exam.)
What you may and may not bring to the exam:
Exam procedures:
Bonus points: You will receive 1 bonus point for every one of your Platonic Solids that you have brought to the exam, provided:
Exam coverage: The quiz will cover Sections 2.6, 2.7, 3.1, 3.2, 3.3, 4.1, 4.5, 7.1, and 7.2 through p. 530 only.
Possible Exam Questions
1. What is a rational number? Show why each of the following
numbers is rational:
a. 7 3/4 b. 0. 0043 c. 6.4123
3. Is the product of two rational numbers always rational, never rational, or sometimes (but not always) rational? Justify your answer by giving reasons if you say "always" or "never" and examples if you say "sometimes."
4. Is the product of two irrational numbers sometimes
irrational, always irrational, or never irrational? Justify your answer
by
giving reasons if you say "always" or "never" and examples
if you say "sometimes."
5. Explain why 100 x 0.1 =
10 without using the rule
about moving the decimal point over
so many places.
6. Give an example of an irrational number between 0.11 and
0.12. Explain how you know your number is irrational.
7. Suppose you know the hypotenuse of a right triangle has
length 4 inches and one leg has length 3 inches. Find the area of the
triangle.
b. Express 1.262626 ... as a quotient of two whole numbers.
c. What is 1.262626 ... - 1.262626? Express it as a decimal and also express it as a quotient of two integers.
d. Which number is larger, 1.262626 or 1.262626 ...?
e. Find a rational number between 1.262626 and 1.262626 ....
f. Find an irrational number between 1.262626 and 1.262626 ... .
10. True or false: If a number has an infinitely long
decimal expansion, then it must be an irrational number. Explain.
11. What characterizes the decimal expansion of a rational
number?
12. a. Define what is meant by "set A has the same
cardinality as set B."
b. Let S be the set of all positive square roots of positive whole numbers. Exhibit (using both arrows and either words or formulas) a one-to-one correspondence between S and the set of all positive whole numbers.
c. What does your one-to-one correspondence tell you about the cardinality of S?
13. List the positive rational numbers in one list so that
the pattern of the order is clear and so that all the positive rational
numbers
would eventually appear on the list. Explain the pattern and why every
positive
rational number will eventually be on the list.
14. I have an infinite list of real numbers between zero and
one. The first six numbers on my list are given below. Describe how you
can
construct a number between 0 and 1 whose decimal expansion consists of
3's and
0's and which is not on my list. Explain why the number you construct
cannot be
on my list. Give the first six digits in the decimal expansion of this
number.
Explain what this procedure tells us about the cardinality of the set
of real
numbers between 0 and 1.
First number: 0.436727384 ...
Second number: 0.728458988 ...
Third number: 0.433378349 ...
Fourth number: 0.444444444 ...
Fifth number: 0.222222222 ...
Sixth number: 0.033003303 ...
15. For each of the following numbers, determine if the number is rational or irrational. Give brief reasons justifying your answers. (In particular, if the number is rational, express it as the quotient of two whole numbers.)
a. three times the square root of 2 b. 3.14159 1.856
16. True or False: If the number M is rational, then 1/M must also be rational. If you say "True," explain why the statement is true. If you say "False," give an example to show that the statement is false -- that is, give an example of a number M which is rational but whose reciprocal 1/M is irrational.
17. True or False: If the number M is irrational, then 1/M must also be irrational. If you say "True," explain why the statement is true. If you say "False," give an example to show that the statement is false -- that is, give an example of a number M which is irrational but whose reciprocal 1/M is rational.
18. Is there a rational number between 0.999
... and 1? Explain.
19. Is there a one-to-one correspondence between the set {1, 2, 3, 4, 5} and the set {x, ?, &, m, @, A}? If there is, show one. If there isn't, explain why there can't be.
20. 1. What is the probability of getting a
sum greater than
4 when you roll two dice? (Hint: there is a hard way of doing this and
an
easier way.)
21. I have rolled two dice. I tell you that
the sum of the
two dice is 4 or less. What is the probability that the numbers on the
dice sum
to 3? (Drawing a chart of possibilities might help.)
22. a. What do we mean when we say that two
solids are dual?
b. Fill in the following chart for
the regular solids and indicate all the places on the chart where the
concept
of duality is shown.
|
|
Shape of faces |
# of faces |
# of edges |
# of vertices |
# of edges at each vertex |
|
Tetrahedron |
|
|
|
|
|
|
Cube |
|
|
|
|
|
|
Octahedron |
|
|
|
|
|
|
Dodecahedron |
|
|
|
|
|
|
Icosahedron |
|
|
|
|
|
c.
What is the relationship between the number of faces, number of edges,
and
number of vertices that is true for all of these solids?
d.
Why do the tetrahedron, octahedron, dodecahedron, and icosahedron have
those
names?