M302 Final Exam

probability

P(E) = # of outcomes in event/ total # of outcomes

Which of our famous knots are alternating?

Trefoil, Figure-eight, Granny

icosahedron

V - 12 E - 30 F - 20

dodecahedron

V - 20 E - 30 F - 12

octahedron

V - 6 E - 12 F - 8

cube

V - 8 E - 12 F - 6

Euler characteristic

V - E + F = 2

how many V, E, & F does a Mobius Band have? What is its Euler Characteristic?

V: 0 E: 1 F: 1 V - E + F = 2 0 - 1 + 1 = 0 It's Euler Characteristic is 0.

is it true the product of any two irrational numbers is irrational?

While the product of 2 irrational numbers can sometimes be irrational, there are cases where 2 irrational numbers produce a rational number. example: √8 * √2 = √16 = 4

Does it matter where you cut it?

Yes, it must be cut down its core because when you cut too close to either side it results in 2 bands. One band will be same length as original and contain a half twist (so another Mobius band) whereas the other will be the doubled length and two half twists.

How many 3-dimensional faces does a hyper-cube have?

a hyper-cube has 8 3D faces because its unfolded diagram consists of 8 cubes.

what does it mean for a knot to be alternating?

a knot is an alternating knot if when you follow along the string you see an over-crossing, under-crossing, over-crossing, under-crossing, (...) until you return to your starting point

What is a mathematical knot?

a mathematical knot has a closed loop with no loose ends

Euler circuit

a path that traverses each edge once, reaches every vertex, and returns to the starting point

Unknot

a simple loop of rope. Many forms of the un-knot exist. If a knot can be undone to a simple loop, the starting knot is equivalent to the un-knot

regular polygon

all the sides have the same length and all the angles are the same

integer (Z)

any natural number (1,2,3,...) or any negative natural number (-1,-2,-3,...), or 0.

composite number

any natural number that is not prime.

Natural Numbers (N)

any positive counting number like 1,2,3,4,....

rational number (Q)

any real number that we can write as (p/q), where p and q are integers.

irrational number

any real number that we cannot write as a fraction no matter what

Which sets have the same size as Natural Numbers?

evens, integers, rationals

T or F: every integer is a natural number

false

T or F: if a real number has an infinitely long decimal expansion, then it must be an irrational number

false

T or F: there are no even primes

false - 2

real number (R)

-any number that corresponds to a location on the number line -All real numbers are either rational or irrational.

regular solid

-faces are all identical regular polygons -number of edges coming out of any vertex is the same for all vertices

platonic solid

-faces of the solid are identical regular polygons -number of edges coming out of each vertex is the same for all

The Golden Ratio

-the number φ = (1+√5)/2 = 1.618 -It is the number you get when you make fractions out of the Fibonacci numbers (each Fib. number divided by the previous one)

What are the elements of effective thinking?

1) Understand Deeply 2) Make Mistakes 3) Ask Questions 4) Follow the Flow of Ideas 5) CHANGE (CUMAF)

first 10 numbers of the Fibonacci sequence

1, 1, 2 , 3, 5, 8, 13, 21, 34, 55

How many regular solids are there?

1. tetrahedron 2. cube 3. octahedron 4. dodecahedron 5. icosahedron

Equivalence by Distortion

2 objects are equivalent by distortion if we can stretch, shrink, bend, or twist one to form into the other (without cutting or gluing)

rigid symmetry

motion that preserves original shape or pattern without shrinking, stretching, or distorting

alphabet letters (equivalent by distortion)

(D, O) (E, F, T, Y) (K, X) (K, H) (P, Q) (A, R) (B)

prime number

-a natural number whose only divisors are 1 and itself -It can't be divided evenly by any other number.

How many edges and faces does a Klein bottle have?

E: 0 F: 1

How many F & E does Mobius band have?

F: 1 E: 1

What order does Natural, Integer, Real, and Rational?

N<Z<Q<R

prove there are infinitely many primes

given any natural number we can always find a prime number that is larger than that natural number proven by Euclid

event

group or set of outcomes (Ex: roll a "double" with 2 dice)

golden rectangle

has the base (b) and height (h) that satisfy the golden ratio (1+√5)/2 = 1.618

pythagorean theorem

if a triangle is a right triangle then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other 2 sides

pigeonhole principle

if there are more X objects than X spaces then at least one space will have to hold more than one object

Which sets are bigger than Natural Numbers?

irrationals, reals

What happens when you cut the Mobius Band in half?

its length doubles and has two half twists

no repeats & order does not matter

nCr = N! / (n-r)! * r!

graph

object drawn on a connected plane using a finite set of edges and vertices

fractal

object that displays self-similarity on all scales

outcome

one specific result (Ex: HHT)

Euler Circuit Theorem

only possible if there are an even number of edges for every vertex

random

randomly generated patterns usually exhibit more clustering and often have large empty spaces

examples of reflection

reflection in water

cardinality

size of a set

example of translation

square tiles/hexagon (bee comb)

example of rotation

starfish

self-similarity

the characteristic of looking the same as or similar to itself under increasing magnification

duality

the number of faces of a platonic solid correspond with the number of vertices of another platonic solid and vice versa

examples of rigid symmetry

translation, reflection, rotation

T or F: between every two rational numbers there is a rational number

true

T or F: every integer is a rational number

true

T or F: every integer is a real number

true

T or F: every irrational number is a real number

true

T or F: every natural number is either a Fibonacci number, or it can be written as a sum of non-consecutive Fibonacci numbers

true

one-to-one correspondence

two collections of objects that are equally numerous

tetrahedron

v- 4 e- 6 f- 4

are there infinitely many nonprimes?

yes there are infinitely many non-primes proven by the sequence (4, 6, 8, 10, 12, ....) because all of these numbers can be divided by 2 and the sequence goes on forever