MMW prelims

Mathematics

is used to organize and systematize

our ideas about patterns, we have discovered a

great secret: nature’s patterns are not just there to

be admired, they are vital clues to the rules that

governs natural processes.

Patterns

are regular, repeated, or recurring forms

or designs. We see patterns everyday – from the

layout of tiles, design of skyscrapers, to the way we

tie out shoelaces.

▪ The snowflake

▪ The honeycomb

▪ The sunflower

▪ The snail’s shell

▪ Flower’s petals

▪ Weather

ex of patterns:

honeycomb conjecture

states that a

regular hexagonal grid or honeycomb has the

least total perimeter of any subdivision of the

plane into regions of equal area.

Flowers

are easily considered as thins of

beauty. Their vibrant colors and fragrant odors

make them very appealing as gifts or

decorations.

protoconch

Snails are born with their shells called

equiangular spiral.

As the

snail grow, their shells expand proportionally so

that they can continue to live inside their

shells. This process results in a refined spiral

structure that is even visible when the shell is

sliced. This figure is called an ___________________

cycle of seasons

The occurrence of seasons one after the other

in a year leads to the _______________.

terms

A sequence is an ordered list of numbers, called_______

definite term.

The arrangement of these term is set by a ____________

finite or infinite

A sequence may be ___________.

finite sequence

_________________has a definite number of terms.

Number patterns may be described by examining

how terms are being generated.

pattern

A __________ may have a list of numbers in which a

constant number is added to get the succeeding

terms.

FIBONACCI SEQUENCE

It is named after the Italian mathematician

Leonardo of Pisa,

Leonardo of Pisa

he was better known for his nickname “FIbonacci”

Leonardo of Pisa,

He said to have discovered this sequence as he

looked how a hypothesized group of rabbits

breed and reproduced.

34; 55

The clockwise spirals of a sunflower is ___ and the counter

clockwise spirals is number of number of ___.

8;13

The nubs on many

pineapples form __ spirals that rotate

diagonally upward to the left and __ spirals

that rotate diagonally upward to the right.

BINET’S FORMULA

you can determine

the nth Fibonacci number without finding the two

preceding Fibonacci numbers.

Jacques Philippe Marie Binet

Binet’s formula is derived by

Golden Ratio

It is also interesting to note that ratios of successive

Fibonacci numbers approach the number Φ (Phi),

also known as __________

1.618

Golden ratio is equal to:

Golden Ratio

The _______________ can also be expressed as the

ratio between two numbers, if the latter is also the

ratio between the sum and the larger of the two

numbers.

fractals

_________ is a mathematical formula of a pattern that repeats

over a wide range of size and time scales. These

patterns are hidden within more complex systems.

Benoit Mandelbrot

__________________ is the father of fractals, who

described how he has been using fractals to find

order within the complex systems in nature, such

as the shape of coastlines.

Fractal Geometry

___________________has been applied in different

fields of knowledge such as in engineering,

computer graphics, medicine, etc.

Sequence

___________ refers to an ordered list of numbers

called terms, that may have repeated values. The

arrangement of these terms is set by a definite

rule.

Arithmetic sequence

It is a sequence of

numbers that follows a definite pattern. The

terms have a common difference.

Geometric sequence

If in the arithmetic

sequence we need to check for the common

difference, in geometric sequence we need

to look for the common ratio.

Harmonic Sequence

In the sequence, the

reciprocal of the terms behaved in a manner

like arithmetic sequence.

Symmetry

means that one shape becomes

exactly like another when you move it in some

way: turn, flip, or slide.

Reflection symmetry

__________________ sometimes called line

symmetry, bilateral symmetry or mirror

symmetry, captures symmetries when the left

half of a pattern is the same as the right half.

Rotations

___________also known as radial/ rotational

symmetry, captures symmetries when it still

looks the same after some rotation (of less

than one full turn).

Translations

This is another type of symmetry.

Translational symmetry

________________e xists in patterns that

we see in nature and in man-made objects.

VITRUVIAN MAN

The drawing depicts a nude male figure in two

superimposed positions with his arms and legs

apart, inscribed in a circle and square.

Vitruvius

Roman architect that outlined the

ideal proportions of the human body.

sets

_____ are collections of distinct elements. They can

be finite or infinite and are denoted by braces,

domain

_______ of a function is the set of all possible

input values (typically denoted as xxx) for which

the function is defined. In simpler terms, it includes

all the x-values that you can plug into the function

without causing it to be undefined.

range

The ______ of a function is the set of all possible

output values (typically denoted as y or f (x)) that

the function can produce.

finite set

A_________ is a set that contains a specific

number of elements.

infinite set

An _________ is a set with an unlimited

number of elements.

empty set

is a set with no elements.

singleton set

A __________ is a set that contains

exactly one element.

subset

A _______ is a set whose every element is

also an element of another set.

power set

The _________ of a set SSS is the set of all

subsets of SSS, including the empty set

and SSS itself.

universal set

The ___________ is the set that contains all

elements under consideration in a

particular discussion or problem.

complement set

The complement of a set AAA with

respect to a universal set UUU contains all

elements of UUU that are not in AAA.

Intersection

Shows common elements between

sets. For example, if Set A and Set B represent two

groups, the intersection (A ∩ B) is shown where the

circles overlap.

Union

Represents all elements that are in either of

the sets or in both. This is depicted as the total

area covered by both circles.

Difference

Shows elements that are in one set but

not in another. For example, A - B is represented

by the part of Circle A that does not overlap with

Circle B.

Complement

Represents elements not in a

particular set. For example, the complement of Set

A (A') is the area outside Circle A.