Math 113 Final Exam Review Fall 2017

Pigeonhole principle

Prime numbers: Why are there infinitely many? How does the proof work?

Know about the Fibonacci numbers and how to play Fibonacci nim.

Understand the golden rectangle and its properties and how to sketch the golden spiral

Be able to estimate numbers - like how many people attend Neyland Stadium in a year.

Understand the various sets of numbers and their properties.

Natural numbers, real numbers, irrational numbers, rational numbers

Decimal expansions: how can you tell if a number is rational from the decimal expansion?

Be able to prove that a number like 2 is irrational.

Understand 1-1 correspondence and cardinality and how to show infinite sets have the same cardinality.

Know who Georg Cantor is and how his diagonalization proof works and what it shows.

Know how to compute modular arithmetic and for example, to verify UPC codes.

Know Fermat's little theorem and how it relates to raising numbers to high powers in modular arithmetic.

Know public key encryption

Pythagorean theorem

Symmetries - rigid and symmetries of scale. Understand the pinwheel diagram and what it shows.

The art gallery theorem

Rubber sheet geometry. What does it mean for two objects to be equivalent by distortion? Take the letters of the alphabet and divide them into categories so that you group together letters which are equivalent by distortion.

How can you show that a sphere and a torus are not equivalent by distortion?

Moebius Strip - properties

Platonic solids - how many and their properties

Euler characteristic

Planar graphs and map coloring

Know who Leonhard Euler is and how to pronounce his name.

Everything else that I forgot to include

Some Review Questions

1. Know all about stories 1, 2, 3, 5, and 7 from chapter 1. Notice that the complete

2. Thompson-Boling Arena holds 24,535 people. It was a sell-out crowd, and the

Vols won! Everyone was so happy that they decided to have a party every day

for the next year. They decided that each person would attend the party which

occurred on his or her own birthday. Show that at least one of these parties

will have at least 50 people.

3. Your mom's office is having one of those jelly bean jar contests for Easter.

Someone got a really big jar and filled it with jelly beans, and you enter the

contest by guessing how many jelly beans are in the jar. Explain how you would

4. You have a deck of cards. How many cards do you have to draw in order to be

sure that you have two cards of the same number? Two of the same suit? Two

of the same color?

5. The Fibonacci numbers, where we see them in nature, and how we compute

them.

6. How do we write a number as the sum of Fibonacci numbers? Try for 200 and 232. (Hint: 144 is a Fibonacci number.)

7. How do we win at Fibonacci Nim? Suppose that it is your turn and there are 20 sticks left and that the previous player took 2 sticks. How many do you take?

8. What is the Golden Ratio? What other topics in this course is it related to?

What is its exact value (not a decimal)? What is its approximate (decimal)

value? Is it rational or irrational?

9. What is a prime number? How can we tell that a number is (or isn't) prime?

Is 51 a prime? Is 71 a prime?

10. The prime factorization of the natural numbers. What is it? What does it

mean? Justify that it is true. What is the prime factorization of 270?

11. How many primes are there (answer: infinitely many)? How do we prove that? (This means: understand the proof.) How can you convince me that there must be a prime number larger than 600? (Note: you can't write out - or multiply out - the quantity you'd need and show all its digits. It's just too big. But with judicious use of 3 dots . . . you can get your point across.)

12. List some values of n for which 2n - 1 is a prime number. List some values of

n for which it is not a prime number.

13. What is a natural number? A rational number? An irrational number? A

real number? Can you give examples of each type? What about an example of a rational

number which is not a natural number? A real number which is not a rational

number?

14. Be able to explain the following: If we have a natural number n and we know

that 2 divides evenly into n2, then it must be the case that 2 divides evenly into

n.

15. Show that is not a rational number. Show that is not a rational number.

16. Show that the product of two rational numbers is another rational number.

17. Explain the idea of "no holes, no next door neighbors" for the real numbers. There is no

'next' real number.

18. Decimal expansions. How can you tell if a decimal expansion of a number

describes a rational or an irrational number?

19. What are periodic decimals? What steps do you take to rewrite them as fractions?

What is 0.22999999999999. . . when written as a fraction?

20. When you consider all the real numbers, are there more rationals or irrationals?

21. Know all about the dodgeball game. Explain the strategy for any size board.

22. What are the natural numbers? The rational numbers? The real numbers?

23. Who is Georg Cantor?

24. State a definition of one-to-one correspondence.

25. State a definition of what it means for two sets to have the same cardinality.

26. For each pair of sets, determine whether they have the same cardinality. Either give a one-to-one correspondence .or explain why there isn't oue.

(a) {1, 2, 3, 4, 5, . . .} and {2, 4, 6, 8, 10, . . .}

(b) {1 , e, 2} and {7, 13, 42}

(c) {1, 2, 3, 4, 5, . . .} and {5, 6, 7, 8, 9, 10, . . .}

(d) {1, 10, 100, 1000} and {7, 77, 777}

(e) {1, 2, 3, 4, 5, . . .} and {-1, -2, -3, -4, -5 . . . .}

(f) {1, 2, 3, 4, 5, . . .} and {1, 2, 3, 5, 8, 13, 21, 34, . . .}

(g) {1, 2, 3, 4, 5, . . .} and {2, 3, 5, 7, 11, 13, 17, 23, 29, . . .}

(h) {1, 2, 3, 4, 5, . . .} and {2, 3, 4, 5, . . . }

(i) The natural numbers and the integers.

(j) The natural numbers and the rational numbers.

(k) The integers and the rational numbers.

(l) The natural numbers and the real numbers.

(m) The natural numbers and the real numbers whose decimal expansion consists only of twos and threes.

(n) States in the United States and members of the US Senate.

(o) States in the United States and current governors.

27. Neyland Stadium seats 104,079 people. Is there a one-to-one correspondence between the people in a sell-out crowd and their birthdays? Is there a one-to-one correspondence between the people in a sell-out crowd and the day that they die?

28. Is this number prime? Factor it into primes. Why is 28 perfect?

29. Consider the set of people living in the United States and the set of phone

numbers in service. Is the pairing which matches people to their phone numbers

a one-to-one correspondence?

30. Given a list of decimal numbers (real numbers), find a number which is certainly not on the list.

31. The Hotel Cardinality (sometimes called the Hilbert Hotel).

32. From the textbook: section 3.2: 16-18, 21.

33. From the textbook: section 3.3: 12.

34. Be able to state the Pythagorean Theorem in words.

35. Given the lengths of two sides of a right triangle, use the Pythagorean Theorem to find the length of the other side.

36. Explain why the "puzzle proof" shows that the Pythagorean Theorem is true.

37. State the Art Gallery Theorem completely accurately and in your own words.

38. Know how to apply the Art Gallery Theorem.

39. Be able to choose the best-suited points to put the cameras in the Art Gallery.

40. (a) Sketch a gallery with 18 vertices which needs 6 cameras.

(b) Sketch a gallery with 12 vertices which needs 4 cameras.

(c) Sketch a gallery with 12 vertices which needs only 3 cameras.

41. (a) How many cameras will you need for a gallery with 8 vertices?

(b) Can you draw a gallery with 12 vertices which needs 5 cameras? Why or

why not?

(c) Draw a gallery with 6 vertices which needs 1 camera (or explain why it's

impossible).

(d) Draw a gallery with 6 vertices which needs 2 cameras (or explain why it's

impossible).

(e) Draw a gallery with 6 vertices which needs 3 cameras (or explain why it's

impossible).

(f) Same question as above, but for a gallery with 10 vertices for 1, 2, 3, or 4

cameras.

42. Be able to divide the museum into triangles and label/color the vertices with

their respective labels.

43. Sketch a Golden Rectangle.

44. Given any rectangle, determine whether or not it is a Golden Rectangle.

45. What happens if you start with a Golden Rectangle and remove the largest

square from it?

46. Given the base (long side) of a Golden Rectangle, find its height (short side).

47. Given the height (short side) of a Golden Rectangle, find its base (long side).

48. Be able to find the area of a Golden Rectangle given either its base or its height.

49. Be able to draw the Golden Spiral and find its center.

50. Define symmetry.

51. What is rigid symmetry?

52. Know what symmetry of scale means. What are supertiles?

53. What shape is the basic building block of the Pinwheel Pattern?

54. Produce supertiles of the Pinwheel Pattern from five smaller tiles.

55. Given an illustration of the Pinwheel Pattern, outline a five-tile supertile.

56. Given an illustration of the Pinwheel Pattern, show how five five-tile supertiles

can be combined to form a 25-tile supersupertile.

57. What's the difference between a polygon and a polyhedron.

58. The definition of a regular polygon.

59. The definition of a regular polyhedron.

60. The names of the five Platonic Solids.

61. Given a drawing of a Platonic Solid, identify it by name.

62. How many vertices, faces, and edges dies each Platonic Solid have.

63. Know the concept of duality. What is the dual of each Platonic Solid?

64. Know the Euler characteristic equation.

65. Know how the Euler characteristic applies to planar graphs.

66. Given two of the following: number of vertices, number of faces, and number of edges. Compute the third quantity.

67. What does it mean for two objects to be equivalent by distortion?

68. How do you take off a rubber vest without taking off your jacket?

70. Take the letters of the alphabet (from any language)and divide them into categories so that you

group together letters which are equivalent by distortion.

71. Can a crawling bug tell the difference between a sphere and a torus?

72. How can you show that a sphere and a torus are not equivalent by distortion?

73. How many sides does a cylinder have? How many edges? What happens if you cut it down the middle?

74. What is a Mobius strip? How do you make one? How many sides does it have? How many edges? What happens if you cut it down the middle? What if you cut it a third of the way over from the edge?

75. Find the missing digit of the UPC code: 2 5 5 5 3 2 5 3 – 0 0 2

76. What day of the week was it 4,229 days ago?

77. What happens if you compute all of the powers of 3 modulo 17? That is, look at the sequence {3, 32 (mod 17), 33 (mod 17), 34 (mod 17), . . . } What if you change 3 to another number like 2?

78. Try the public key encryption on the website. Choose 2 secret primes and determine the public encryption key and the secret decryption key. Decode the message posted.

79. How can you tell whether a planar graph has an Eulerian circuit?

80. Explain why a political map of the United States requires more than 3 colors. How many colors are needed?