Mathematics in the Modern World

Term Flashcards for MMW

Patterns of Visuals.

Often unpredictable, never quite repeatable, and often contain fractals.

Patterns of Visuals.

These patterns can be seen from the seeds and pinecones to the branches and leaves. They are also visible in self-similar replication of trees, ferns, and plants throughout nature.

Patterns of Flow.

Liquids provides an inexhaustible supply of nature’s patterns.

Patterns of Flow.

This pattern is usually found in the water, stone, and even in the growth of trees.

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 Movement.

This prevalence of pattern in locomotion extends to the scuttling of insects, the flights of birds, the pulsations of jellyfish, and also the wave-like movements of fish, worms, and snakes.

Patterns of Rhythm.

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

Patterns of Rhythm.

Many of nature’s similar movement are most likely similar to a heartbeat, while others are like breathing. The beating of the heart, as well as breathing, have a default pattern.

Patterns of Texture.

It 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.

Patterns of Texture.

They are of many kinds. It can be bristly, and rough, but it can also be smooth, cold, and hard.

Geometric Patterns.

A __________________ 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.

Geometric Patterns.

They are usually visible on cacti and succulents.

Waves and Dunes

A __________ is any form of disturbance that carries energy as it moves. Likewise, ripple patterns and _______ are formed by sand wind as they pass over the sand.

Mechanical waves

which propagate through a medium ---- air or water, making it oscillate as waves pass by.

Wind waves

are surface waves that create the chaotic patterns of the sea.

Water waves

are created by energy passing through water causing it to move in a circular motion.

Spots and Stripes

Patterns like this are commonly present in different organisms are results of a reaction-diffusion system (Turing, 1952). The size and the shape of the pattern depend on how fast the chemicals diffuse and how strongly they interact.

Spots

We can see patterns like ______ on the skin of a giraffe.

Stripes

________ are visible on the skin of a zebra.

Spirals

These patterns exist on the scale of the cosmos to the minuscule forms of microscopic animals on earth. This are also common and noticeable among plants and some animals.

Spirals

It appears in many plants such as pinecones, pineapples, and sunflowers. On the other hand, animals like ram and kudu also have these patterns on their horns.

Spirals

Milky Way

Symmetries

In mathematics, if a figure can be folded or divided into two with two halves which are the same, such figure is called a _____________. It has a vital role in pattern formation. It is used to classify and organize information about patterns by classifying the motion or deformation of both pattern structures and processes.

Reflection symmetry

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

Rotations

also known as rotational symmetry, captures symmetries when it still looks the same after some rotation (of less than one full turn).

Rotations

The degree of the symmetry of an object is recognized by the number of distinct orientations in which it looks the same for each turns.

Translations

This is another type of symmetry. _________ symmetry exists in patterns that we see in nature and in man-made objects.

Translations

Acquire symmetries when units are repeated and turn out having identical figures, like the bees’ honeycomb with hexagonal tiles.

Symmetries in Nature

From the structure of subatomic particles to that of the entire universe, symmetry is present.

Human Body

The _______________ is one of the pieces of evidence that there is symmetry in nature. Our _______ exhibits bilateral symmetry. It can be divided into two identical halves.

Animal Movement

The symmetry of motion is present in their movements. When animals move, we can see that their movements also exhibit symmetry.

Sunflower

One of the most interesting things about this flower is that it contains both radial and bilateral symmetry. What appears to be "petals" in the outer ring are actually small flowers also known as ray florets.

Snowflakes

have six-fold radial symmetry.

Starfish

have a radial fivefold symmetry. Each arm portion of this is identical to each of the other regions.

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. To determine if the series of numbers follow an _______________, check the difference between two consecutive terms. If common difference is observed, then definitely this sequence governed the pattern.

Geometric sequence

If in the arithmetic sequence we need to check for the common difference, in this sequence we need to look for the common ratio.

What type of Sequence is this? 1, 4, 7, 10, , ?

What type of Sequence is this? 80, 40, 20, , ?

What type of Sequence is this? 1, 1, 2, 3, 5, 8, , ?

What type of Sequence is this? 56, 46, 36, 26, , ?

What type of Sequence is this? 2, 20, 200, 2000, , ?

What if Fib (13) ?

What is Fib (20) ?

What is Fib (8) + Fib (9) ?

What is Fib (1) * Fib (7) + Fib (12) – Fib (6) ?

What is the sum of Fib (1) up to Fib (10) ?

What is the sum of Fib (10) + Fib(5) ?

60

What is Fib (12) ?

233

What are the next two terms of the sequence, 8, 17, 26, 35?

44, 53

What type of sequence is 5, 8, 13, 21, 34, 55, ... ?

Fibonacci Sequence

∪

Union

∩

Intersection

∈

Element

∉

Not an element of

⊂

Subset

A number increased by five

Twice the square of a number

The square of the sum of two numbers

The sum of the squares of two numbers

A number less by three

Twice of a number added by four

The cube of a number less than five

A set of counting numbers from 1 to 5

A = { 1, 2, 3, 4, 5 }

A set of all even positive integers.

C = { 2, 4, 6, 8, ... }

is a set that contains only one element.

Unit Set

Unit Set

A = { 1 }; B = { c }; C = { banana }

Empty set or Null set

is a set that has no element.

Empty set or Null set

{ }

A finite set is a set that the elements in a given set is countable.

Finite set

Finite set

A = { 1, 2, 3, 4, 5, 6 }

Infinite set

An infinite set is a set that elements in a given set has no end or not countable.

Infinite set

A = { ...-2, -1, 0, 1, 2, 3, 4, ... }

Equal set

A = { 1, 2, 3, 4, 5} B = { 3, 5, 2, 4, 1}

Equivalent set

A = { 1, 2, 3, 4, 5 } B = { a, b, c, d, e }

Universal set

is the set of all elements under discussion.

Universal set

A set of an English alphabet

Joint Sets

sets if and only if they have common element/s.

Disjoint Sets

disjoint if and only if they are mutually exclusive or if they don’t have common element/s.

Roster or Tabular Method

It is done by listing or tabulating the elements of the set.

Rule or Set-builder Method

It is done by stating or describing the common characteristics of the elements of the set. We use the notation A = { x / x ... }

Subsets

A subset, A  B, means that every element of A is also an element of B.

Ordered Pair

pairs (a,b) and (c,d) are equal if a = c and b = d.

Union of Sets

A = {1, 2, 3} and B = {4, 5}, then A U B = {1, 2, 3, 4, 5}

Intersection of Sets

1: If A = {1, 2, 3} and B = {1, 2, 4, 5}, then A ∩ B = {1, 2}

Difference of Sets

If A = {1, 2, 3} and B = {1, 2, 4, 5}, then A - B = {3}

Compliment of Set

Let U = { a, e, i, o, u } and A = { a, e } then Ac = { i, o u }

Cartesian Product

Given sets A and B, the Cartesian product of A and B, denoted by A x B and read as “A cross B”, is the set of all ordered pair (a, b) where a is in A and b is in B.

If f(x) = 2x + 1 and g(x) = 3x + 2, what is (f+g)(x)?

5x + 3

What is (f • g)(x) if f(x) = 2x + 1 and g(x) = 3x + 2?

6x2 + 7x + 2

What is (f/g) (x) if f(x) = 2a + 6b and g(x) = a + 3b?

2

If f(x) = 2x + 1 and g(x) = 3x + 2, what is (g o f)(x)?

6x + 5