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Mathematics

The study of quantity, structure, space, and change.

Number Patterns

Lists of numbers that follow a specific rule or pattern.

Leonardo Pisano Bigollo

Created the Fibonacci Sequence

Language of Mathematics

system used by mathematicians to communicate mathematical ideas

symbols

used to express formula or constant

syntax

to make the expression well formed to make the characters and symbols clear and valid that do not violate the rule of mathematics

Precise, Concise, Powerful

the language of mathematics should be: 1. 2. 3.

Precise

able to make fine distinction

concise

able to say things briefly

poweful

able to express complex thoughts with relative ease

Arithmetic Sequence

These sequences have a constant difference between consecutive terms.

Geometric Sequence

These sequences have a constant ratio between consecutive terms.

Geometric Sequence

Is a sequence where each term is multiplied by a constant value (the common ratio) to get the next term.

Number Series

collection of numbers in which all the terms are formed according to some particular rule or pattern.

Prime Series

the terms are prime numbers in order

Alternate Primes

the terms are prime numbers alternating patterns.

Logic Patterns

Recurring arrangements or sequences that follow a specific rule or relationship

Logic Patterns

These patterns can be found in numbers, shapes, objects, or even in sequences of events or ideas.

Counting

___ happens when there is pattern

Pattern

____ and counting are correlative

Regularity

when a same thing happens in the same circumstances

Symmetry

An agreement in dimensions , due proportion and arrangement.

Symmetry

refers to a sense of harmonious and beautiful proportion and balance.

Symmetry

In mathematics, ___ means than an object is invariant to any various transformations including reflection,rotation or scaling

reflection, rotation, scaling

In mathematics, symmetry means than an object is invariant to any various transformations including ____, ____ or ____

Bilateral Symmetry

the property of being divisible into symmetrical halves on either side of a unique plane.

Bilateral Symmetry

Also called “mirror symetry”

Radial Symmetry

a type of symmetry where parts are arranged around a central axis.

Translational Symmetry

the property of a shape or pattern remaining unchanged after being shifted or translated (moved) by a certain distance in a specific direction, without any rotation or reflection

Fractals

A never-ending pattern.

Fractals

are infinitely complex patterns that are self-similar across different scales

Fractals

They are created by repeating a simple process over and over in an ongoing feedback loop.

Tesselation

Is the tiling of a plane using one or more geometric shapes called tiles, with no overlaps and no gaps

Spirals

A curve that emanates from a point and winds around it, either continuously moving closer to or farther away from the center

0,1,1,2,3,5,8,13,21,34,55

first eleven numbers in the Fibonacci Sequence

(1+ square root of √5)/2 1.618

The Golden Ratio

Mathematical Expression

is a finite combination of symbols that is well-formed according to rules that depend on the context.

Mathematical Expression

a correct arrangement of mathematical symbols used to represent a mathematical object of interest.

number, sets, and function

Most common mathematical expressions

Mathematical Sentence

The analogue of an English sentence.

Mathematical Sentence

It is a correct arrangement of mathematical symbols that states a complete thought.

always true, always false, sometimes true: sometimes false

A sentence can be___ ___, ___ ___ , or ____ ___, ___ ____

Mathematical convention

is a fact, name, notation, or usage, which is generally agreed upon by mathematicians.

Set

A collection of well-defined objects that contains no duplicates.

Elements

Objects in a set

Infinite set

a set that contains an unlimited number of elements

Finite set

contains sets that can be counted and terminates at a certain natural number

List Notation/Roster Method

list names of elements of a set, separate them by commas and enclose them in brackets

Predicate notation/ Rule Method/Set Builder Notation

By stating property of its elements. It has a property that the members of the set share a condition or a predicate which holds for members of this set.

Recursive rule

by defining a set of rules which generates or defines its members.

Equal Set

sets that contain exactly the same elements.

Equivalent Set

sets that contain the same number of elements.

Universal set, U

A set that contains all the elements considered in a particular situation and is denoted by ___

Venn Diagram

visual tool that uses overlapping circles to represent the relationships between sets of items, showing their similarities and differences.

Subset

__ of a set of another set if every element of that set is contained in another set.

Proper Subset

is a subset that is not equal to the original set

Improper Subset

is a subset that is equal to the original set

Cardinality of the Set

It is the number of a distinct elements belonging to a finite set.

Power Set, Power A

It is the family of all the subsets of A denoted by ___

Union

Is an operation for sets A and B in which a set is formed that consists of all the elements included in A or B or both denoted by U as AUB.

Intersection

It is the set containing all common elements to both involved sets, denoted by ⋂.

Complementation

Is an operation on a set that must be performed in reference to a universal set denoted by A’.

Relation

A rule that pairs element in one set, called the domain, with one or more elements from a second set called the range.

Function

A rule that assigns to each input exactly one output.

Binary Operations

can be understood as a function f (x, y) that applies to two elements of the same set S, such that the result will also be an element of the set S.

Binary Operation

is a rule that is applied on two elements of a set and the resultant element also belongs to the same set

Closure Property

when an operation is performed on elements within a set, the result is also within that same set

Commutative Property

The order in which you perform addition or multiplication does not change the result.

Associative Property

The way numbers are grouped when adding or multiplying does not change the result. This means you can rearrange the parentheses in an expression without altering the final answer

Commutative Property

What property is shown: for addition, a + b = b + a, and for multiplication, a b = b a.

Associative Property

What property is shown: for addition, (a+b) + c = a + (b+c), and for multiplication, (a*b)*c = a * (b*c)

Euclid

who mentions as the “extreme and mean ratio” in the Elements.

Pattern

is a discernible regularity in the world or in a man-made design.

Predictable

the elements of a pattern repeat in a ____ manner