DVM 1 : MMW (Chap 1-3)

Mathematics

is the study of the relationships among numbers, quantities, and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics and calculus.

Mathematics

nurtures human characteristics like power of creativity, reasoning, critical thinking, and others.

mathematics

helps organize patterns and regularities in the world.

Patterns

are visible regularities found in the natural world. These persist in different contexts and can be modelled mathematically.

natural patterns of productivity and species richness

may consists of sprials, symmetries, mosaics, stripes, spots etc.

Joseph Plateau

examined soap films, leading him to formulate the concept of minimal surface.

Ernst Haeckel

a german biologist and artist that painted hundreds of marine organisms to emphasize their symmetry.

D'arcy Thompson

pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth.

Alan turing

a british mathematician that predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes.

Aristid Lindenmayer and Benoit Mandelbrot

a hungarian biologist and french american mathematician that showed how the mathematics of frantals could create plant growth patterns.

Gary Smith

adopts eight patterns in his landscape work.

scattered, fractured, mosaic, naturalistic drift, serpentine, sprial, radial, and dendritic

these patterns occur in plants, animals, rocks, formations, river flow, and stars or in human creations.

snowflakes, honeycomb, tiger's stripes and sunflower

patterns in nature

spiral, radial, scattered, dendritic

some patterns adapted in landscape design.

Leonardo Pisano Bogollo

lived between 1170 and 1250 in italy. His nickname "fibonacci" rougly means "son of bonacci"

Leonardo pisano bogollo

aside from being famous for the fibonacci sequence, he also helped spread hindu arabic numerals through europe in place of roman numerals.

November 23

date of fibonacci day

1, 1, 2, 3

it has the digits that is part of the sequence, which he developed.

Golden ratio

is a mathematical ratio. It is commonly found in nature, and when used in a design, it fosters organic and natural-looking compositions that are aesthetically pleasing to the eye.

golden spiral

is a logarithmic sprial whose growth factor is this (empty set symbol) golden ratio.

three petal iris, five petal columbine, eight petal delphiniums, thirteen petal-ragwort, 21-petal aster, 34-petal pythethrum

examples of fibonacci flowers

concept of symmetry

fascinates philosophers, astronomers, mathematicians, artists, architects, and physicists.

motion of a pendelum

the reflection in a plane mirror, the motion of a falling object and the action-reaction paper of forces are all guided and organized by mathematics.

mathematics of a pendelum

is quite complicated but harmonic. its period or time it takes to swing back to its original position is related to its length, but the relationship is not linear.

free falling object

is an object that is falling under the sole influence of gravity. any object that is moving and being acted upon only be the force of gravity is said to be in a state of free fall.

mathematics

predict the behavior of nature and phenomena in the world.

bird's two wings, clover's three leaflets, deer's four hooves, buttercup's five petals, insect's six legs, rainbow's seven colors, octopus' eight arms

mathematician noticed that numbers appear in many different patterns in nature

fibonacci

son of bonacci

fractal patterns

patterns that build into a simple repetitive shapes that are reduced in size every time they are repeated.

law of reflection

an image formed by an object in a plane mirror can be explained mathematically.

forces

always comes in pairs - equal and opposite action-reaction pairs.

mathematics

is a universal language in different places, in different times, in different settings and different circumstances.

Language

It is the system of words, signs and
symbols which people use to express ideas,
thought and feelings.

Mathematical language

It is the system used to communicate
mathematical ideas.

Mathematic language

It is more precise than any other
language one may think of.

Language

It consist of the words, their
pronunciation and the methods of combining
them to be understood by a community.

Numbers, sets, relations, functions, perform operations

Fundamental Elements of the Language of
Mathematics

Mathematics

deals with ideas, relationships, quantities, processes, ways of figuring out certain kinds of things, reasoning,
generalizing and many more. It uses words, but it
is not about words.

equilateral, quotient, probability

Technical terms specific to mathematics

line, factor, frequency

Specialist use of more general terms

function, expression, difference, area

Mathematical terms that we use every day
for conveying ideas

set

is a well-defined collection of distinct
objects.

set

It usually represented by capital letters.

set

It is said to be well-defined if the elements in a
set are specifically listed or if its elements are
described to determine whether an object in
question is an element or not an element of the
set.

Tabular or Roster Form

A set that can be represented by listing its
elements between braces.

Set Builder Notation or Rule Form

Let S denote and let P(x) be a property that
elements of S may or may not satisfy. We define
a new set to be the set of all elements x in S
such that P(x) is true. We denote this set as
follows:

finite

if the number of elements in the
set is a whole. It contains only a countable
number of elements.

universal set

The set of all elements that are being
considered is called the

infinite

if the counting of elements has
not countable. The set of integers, natural
numbers or whole numbers are infinite sets.

empty set or null set

is the set that
contains no elements. The symbol ⊘ or { } is
used to represent the empty set.

unit set

The set with only one element is a _______

equal set

are set with exactly the same elements and cardinality.

equivalent sets

are set with the same number of elements and cardinality.

joint sets

are set with common elements.
(intersection)

disjoint sets

are set with no common
elements.

subset

Set A is a _____ of Set B denoted by A ⊆ B, If
every element of A belongs to B.

proper subset

If there is at least one element found in B but
not in A, then A is a ______ of B
denoted by A ⊂ B.

improper subset

A subset which contains all the elements of the
original set is called an ___

power set

P of A, denoted P(A) is defined
as the set of all subsets of A.

complement

The ______ of set A, denoted by A', is
the set of all elements of the universal set U that
are not elements of A

universal set

contains all elements under consideration

empty set

the complement of the universal set is the

john venn

an english logician who developed venn diagram to illustrate sets and relationship between sets.

commutative law

the order in which the sets are taken does not affect the result.

associative law

the grouping in which the
sets are taken does not affect the result.

identity laws

A set operated to another set
called the identity element gives the set itself.

A, for union of sets, the identity is
the empty set

A ∪ ⊘

A, for intersection of sets, the
identity element is the universal set.

A ∩ U

Inverse or Compliment Laws

This involves
inside and outside of a set.

Distributive Laws

These laws involve three
sets with two different operations, distributing the
first operation over the second one.

relation

is a correspondence between two
things or quantities.

relations

It is a set of ordered pairs such that the set of
all first coordinates of the ordered pairs is
called _____

range

domain and the set of all the second coordinates of the ordered pairs is called _____

relation

can be expressed as a statement, by
arrow diagram, through table, by an equation or
graphically.

Types of Relations

1. One - to - One Relation
2. One- to - Many Relation
3. Many- to - One Relation

Equivalence Relation

1. Reflexive
2. Symmetric
3. Transitive

function

is a relation such that each element
of the domain is paired with exactly one
element of the range.

functional notation

allows us to
denote specific values of a function. To evaluate a
function is to substitute the specified values of the
independent variable in the formula and simplify.