A Fraction Ahead Review

Question 1

In geometry, what does congruent mean?

Answer 1

Two shapes that are identical in every way

Question 2

What is a regular polygon?

Answer 2

A polygon whose side lengths are all equal, and whose angles measurements are all equal

Question 3

What is a Platonic solid?

Answer 3

A 3 dimensional shape whose faces are congruent regular polygons, with the same number of faces meeting at each vertex

Question 4

How many Platonic solids exist, and what are their names?

Answer 4

5: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron

Question 5

What shapes are the faces of a tetrahedron, and how many faces does a tetrahedron have?

Answer 5

Triangles with 4 faces

Question 6

What shapes are the faces of a cube, and how many faces does a cube have?

Answer 6

Squares with 6 faces

Question 7

What shapes are the faces of an octahedron, and how many faces does an octahedron have?

Answer 7

Triangles with 8 faces

Question 8

What shapes are the faces of a dodecahedron, and how many faces does a dodecahedron have?

Answer 8

Hexagons with 12 faces

Question 9

What shapes are the faces of an icosahedron, and how many faces does an icosahedron have?

Answer 9

Triangles with 20 faces

Question 10

The Platonic solids were named after which philosopher?

Answer 10

Plato

Question 11

Where did Plato live and at what time period?

Answer 11

Ancient Greece, approximately 400 BCE

Question 12

What are some of Plato’s most popular philosophical ideas?

Answer 12

Theory of Forms, Plato’s Republic, Plato’s Cave

Question 13

What is the formula for the addtion of numbers from 1 to a natural number n?

Answer 13

Question 14

How do you derive the formula for the addtion of numbers from 1 to a number n?

Answer 14

This video is a visual explanation of the derevation of the formula

Question 15

What symbol is used to describe a factorial?

Answer 15

!

Question 16

How do you calculate a factorial?

Answer 16

Multiply each number in descending order up until 1. Example: 4! = 4x3x2x1 = 24

Question 17

What does a factorial tell us?

Answer 17

How many ways you can order a certain amount of objects

Question 18

How many ways can you order a deck of cards?

Answer 18

52! ways, or approximately 8.0658175e+67 ways

Question 19

Is 52! a large number?

Answer 19

Yup

Question 20

Just how big is 52!?

Answer 20

This video explains

If the video starts at the beginning, the explanation starts about 14 minutes in.

Question 21

What is this symbol called, and what does it represent?

Answer 21

It is a “blackboard bold” N, and it represents the set of natural numbers

Question 22

Which numbers are included in the set of natural numbers?

Answer 22

{1, 2, 3, 4, …}. The “…” means this goes on forever. Some mathemeticians include 0 in the natural numbers.

Question 23

What does this symbol represent?

Answer 23

The set of integers

Question 24

Which numbers are contained within the set of integers?

Answer 24

{…, -2, -1, 0, 1, 2, …}

Question 25

What does this symbol represent?

Answer 25

The set of rational numbers

Question 26

Which numbers are included in the set of rational numbers?

Answer 26

Any number (positive or negative) that can be represented as a fraction / ratio

Question 27

What does this symbol represent?

Answer 27

The set of real numbers

Question 28

Which numbers are included in the set of real numbers?

Answer 28

All of the rational numbers and all of the irrational numbers

Question 29

What is an irrational number?

Answer 29

A number that cannot be represented by a ratio of integers. Their decimal expansions never end and never repeat

Question 30

What are some examples of irrational numbers?

Answer 30

π, e, the square root of any number that cannot be represented as a ratio of perfect squares, and many more

Question 31

What is a perfect square?

Answer 31

A number that can be found by multiplying 2 of the same integers. Examples: 4 (2*2), 9 (3*3), 100 (10*10), etc

Question 32

What is this symbol called and what does it represent?

Answer 32

Aleph Null, it is the cardinality of a countably infinite set (such as the sets of natural numbers, integers, and rational numbers)

Question 33

What is cardinality?

Answer 33

The number of elements in a set. For example, the cardinailtiy of the set {apple, pear, orange} is 3, because it contains 3 elements

Question 34

In regards to set theory, what is an element?

Question 35

What is a set?

Answer 35

A collection of elements contained within curly brackets. Example: Set A = {3, 4, red, 9, banana}

Question 36

What is a union of 2 sets?

Answer 36

The combination of 2 sets into 1 set, without any repetitions. Example: Set A = {1,5,7} and Set B = {2,5,7,8,10}

The union of set A and B is: A∪B = {1,2,5,7,8,10}

Question 37

What is the intersection of 2 sets?

Answer 37

Creating a new set with shared elements of 2 sets. Example: Set A = {1,2,3} and set B = {2,3,4}.

The intersection of sets A and B is: A∩B = {2,3}

Question 38

What is the empty set?

Answer 38

A set containing no elements, denoted by the symbols ∅ or {}

Question 39

What is the complement of a set?

Answer 39

A set containing all elements that are not in the specified set but are in the universe of numbers that are being worked with.

Example: There exists a set A = {-1,0,1} in ℤ. The complement of set A would be: A^c = {…, -3, -2, 2, 3, …}

Question 40

What is the name of this mathematical tool?

Answer 40

The Cartesian plane / coordinate plane

Question 41

What was the name of the mathematician / philosopher who invented the Cartesian plane?

Answer 41

René Descartes

Question 42

Which two branches of math does the Cartesian plane unify?

Answer 42

Algebra and Geometry

Question 43

What is the name of this method that approximates the area under a curve?

Answer 43

Riemann Approximation Method

Question 44

What calculus operation is used to find the area under a curve?

Answer 44

The integral