mathematics in modern world midterm

Patterns of visual

Patterns of flow

Patterns of movement

Patterns of rhythm

Patterns of texture

Geometric patterns

Different kinds of pattern

Patterns of visual

Are often unpredictable, never quite repeatable and often contains fractals.

Patterns of flow

The flow of liquids provides an and exhaustible supplies of natures patterns.

Patterns of movement

In the human walk, the feet strike the ground in a regular rhythm: the left-right-left-right-left rhythm. When a horse, a four-legged creature walks, there is more of a complex but equally rhythmic pattern.

Patterns of rhythm

is conceivably the most basic pattern in nature. Our hearts and lungs follow a regular repeated pattern of sounds or movements whose timing is adapted to our body's needs.

Patterns of texture

is a quality of a certain object that we sense through touch. It exists as a literal surface that we can feel, see, and imagine. Textures are of many kinds. It can be bristly, and rough, but it can also be smooth, cold, and hard.

Geometric patterns

is a kind of pattern which consists of a series of shapes that are typically repeated. These are regularities in the natural world that are repeated in a predictable manner.

Reflectional symmetry

sometimes called line symmetry or mirror symmetry captures symmetries when the left half of a pattern is the same as the right half.

Rotational symmetry

Captures symmetries when it still looks the same after some rotation (of less than one full turn). The degree of rotational symmetry of an object is recognized by the number of distinct orientations in which it looks the same for each rotation.

Sequence

is a set of things (usually numbers) that are in order.

Each number in the _____ is called a term (or sometimes "element" or "member").

Arithmetic sequence

check for common difference; the difference between one term and the next is a constant.

An=A1+(n-1)d

Arithmetic sequence formula

Sn=n/2(2A1+(n-1)d)

Arithmetic series formula

Translational symmetry

Exist in patterns that we see in nature and in man-made objects.

Translations acquire symmetries when units are repeated and turn out to have identical figures, like the bees' honeycomb with hexagonal tiles.

Geometric Sequence

check for common ratio; each term is found by multiplying the previous term by a constant.

Reflectional symmetry

Rotational symmetry

Translational symmetry

Types of symmetries

an = a1 * r ^ (n - 1)

Geometric sequence formula

Sn= a1(r ^ n - 1)/r - 1

Geometric series formula

Harmonic Sequence

is a sequence of numbers such that the difference between the reciprocals of any two consecutive terms is constant.

Fibonacci Sequence

The next number is found by adding up the two numbers before it.

xn=phi ^ n - (1-phi) ^ n/√5

Fibonacci Sequence formula

Standard Fibonacci Sequence

1,1,2,3,5,8,13,21,34,55,89,144,233,…

1.618034

Whats the value of phi or golden ratio

Language

is a system of communication consisting of sounds, words, and grammar, or the system of communication used by people in a particular country or type of work.

Precise

Powerful

Concise

Characteristics of Mathematical Language

Precise

able to make a very fine distinction

Concise

able to say things briefly

Powerful

- able to express complex thoughts with relative cases

set

A _____ is a collection of well-defined objects.

element

Each member of the set is called an ___ and the E notation means that an item belongs to a set.

E notation

means that an item belongs to a set.

{}

Set Notation

Operations

+ , -, ×, ÷

Relations

<, >, =, ≠

Logic

E (element of), c (subset), u (union), n [intersection)

Sets

are denoted by capital letters (A, B, C).

Elements

are lowercase letters (a, b, c).

Roster/Tabular Method

Rule/Set-builder Method

Ways to Describe a Set

Roster/Tabular Method

Listing elements (e.g., A = (1, 2, 3))

Rule/Set-builder Method

-Describing elements (e.g.. A = (x x is an even number < 10}}

Unit Set

: One element

Empty or Null or void sets

: No element

Finite Set

: Countable elements

Infinite Set

Uncountable elements

Equal Sets

: Contain same elements

Equivalent Sets

: Contain same number of elements.

Universal Set (U)

Contains all elements under discussion

Joint Sets

: Have common elements

Disjoint Sets

: Have no common elements

Cardinal number (n)

the number of elements found in a set

Subset

: Every element of A is also in B

Formula for number of ___: (2^n J, where n = number of elements

Union (U)

Elements in A or B or both (AUB={x|XEA\text{or) x ∈ B))

Intersection (n)

Elements common to both [Αn Β = {x | x ∈ A \text{and) X E B)

Difference (-)

→ Elements in A but not in B

(A-B = {x | x∈ A \text{ and } x E B})

Complement (A')

→ Elements not in A but in universal set U (A'= {x | x EU \text{ and } x E A)

Cartesian Product (A × B)

-All ordered pairs (x, y) (AxB = {(x, y) | ΧΕΑ, γε Β))

Relations

A rule that relates elements from one set (domain) to another (range).

Functions

A ____ is a relation where each element of the domain corresponds to only one value in the range.

Table of Values

: shows x and y relationships

Ordered Pairs

: (x, y)

Mapping Diagrams

: arrows showing connection between x and y

Graphs

: use the Vertical Line Test (a valid function touches a vertical line once only)

function of functions

If a function is substituted to all variables in another function, you are performing a composition of functions to create another function. Some authors call this operation as "______".

Inductive Reasoning

is the process of getting a general conclusion by observing the specific examples or set.

CONJECTURE

is the conclusion formed by using inductive reasoning.

Inductive Reasoning

Application of ____ is very essential to solve some practical problems that you may encounter. With the use of ______ we can easily predict a solution or an answer to a certain problem.

deductive Reasoning

is the process of reaching a conclusion by general assumption, procedures, or principle.

Mathematical intuition

is coming across a problem, glancing at it, and using your logical instinct to pull out an answer without asking further questions.

proof

The old, colloquial meaning of "prove" is: test, try out, determine the true state of affairs.

certainty

is something that is accurate and absolute.

George Polya

was a Hungarian who immigrated to the United States in 1940.

His major contribution is his work in problem-solving.

Aristotle

According to ______ , logic is defined as "the science of correct reasoning".

statement

A ____ is a declarative sentence that is either true or false, but not both true and false.

simple statement

compound statement

TWO KINDS OF STATEMENT

simple statement

A ______ is a statement that conveys a single idea.

compound statement

is a statement that conveys two or more ideas

^

Conjunction

(And)

v

Disconunction

(Either…. Or)

~

Negation

(not/ It is not the case)

→

Conditional

(If... then)

←→

Biconditional

(if and only if)

parentheses

1.In symbolic form, the ____ are used to indicate the simple statements that are being grouped.

Comma

2.In sentence form, a ____ is used to indicate which simple statements are grouped. That is, statements of the same side and a _____ are grouped.