Mathematics in the Modern World

Nothing can beat the beauty of nature. Finding mathematics in nature at a first glance may seem less obvious. However, if one does take time to examine nature, then one can describe its beauty mathematically.

PATTERNS IN NATURE AND THE REGULARITIES OF THE WORLD

• Patterns and counting are correlative. Counting happens when there are patterns and logic happens when there is counting.

• Consequently, a pattern in nature goes with logic or logical set up.

PATTERN

• Is a discernible regularity in the world or in a man-made design

• Can be sequential, spatial, temporal, and even linguistic.

• All these phenomena create a repetition of names or events called regularity.

• Arrangement that helps observers anticipate what they might see or what happens next. It also shows what may have come before.

REGULARITY IN THE WORLD

• States the fact that the same thing always happens in the same circumstances.

PATTERNS IN NATURE

• Visible regularities of form found in the natural world.

• These patterns recur in different contexts and can sometimes be modeled mathematically.

• Includes symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks, and stripes.

SYMMETRY

• An agreement in dimensions, due proportion, and arrangement.

• Refers to a sense of harmonious and beautiful proportion and balance.

No symmetry, Radial symmetry, Bilateral symmetry

SPIRAL

• A curve that emanates from a point, moving farther away as it revolves around the point.

MEANDER

• One of a series of regular sinuous curves, bends, loops turns, or windings in the channel of a river, stream, or other water courses.

• Produced by a stream or river as it erodes the sediments compromising an outer, concave bank, and deposits this and other sediments downstream on an inner, convex bank which is typically a point bar.

WAVE

• Oscillations or vibration of a physical medium or a field, around relatively fixed locations.

FRACTURE OR CRACK

• Separation of an object or material into two or more pieces under the action of stress.

• Fracture or a solid - usually occurs due to the development of certain displacement discontinuity surfaces within the solid.

• Normal tensile crack/simply a crack - A displacement develops perpendicular to the surface of displacement.

• Shear crack - Displacement develops tangentially to the surface of displacement.

STRIPES

• Made by a series of bands or strips, often of the same width and color along the length.

FRACTAL

• A never-ending patterns

• Infinitely complex patterns that are self-similar across different scales.

• Created by repeating a simple process over and over in an ongoing feedback loop.

• Math is a subject that makes students either jump for joy or rip their hair out. However, math is inescapable as you become an adult in the real world. Consider this list of reasons why;

RESTAURANT TIPPING

• It is a common courtesy to pay your waiter a generous tip.

• You need to have the most basic math skills to calculate how much a 15% or 20% tip would be.

NETFLIX FILM VIEWING

• Let’s say you have approximately 1 hour until you have to go somewhere very important.

• For example, an episode of Friends on Netflix is about 20 minutes and you would be able to fit 3 episodes in that hour.

CALCULATING BILLS

• Math is required to calculate these payments and subtract them from your savings.

DOING EXERCISE

• You need to know how many more reps to curl, how many seconds to cut off your mile time, or how many more pounds to lose to achieve that goal.

SURFING INTERNET

• We have to thank mathematics for establishing technology and the social media that consumes our lives.

• Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its  intrinsic interest.

PATTERN AND RELATIONSHIPS

• Mathematics is the science of patterns and relationships.

• As a theoretical discipline, mathematics explores the possible relationships among abstractions without concern for whether those abstractions (can be anything)  have counterparts in the real world.

ABSTRACTION AND SYMBOLIC REPRESENTATION

• Mathematical thinking often begins with the process of abstraction (noticing a similarity between two or more objects or events.)

• Aspects that they have in common.

• For example, whole numbers are abstractions that represent the size of sets of things and events or the order of the things within a set.

MANIPULATING MATHEMATICAL STATEMENTS

• The strings of symbols are combined into statements that express the ideas or propositions.

• For example, the symbol A for the area of any square can be used for the length of the square’s side to form a proposition A = s^2. This equation specifies how the area is related to the side.

MATHEMATICS, SCIENCE, AND TECHNOLOGY

• Mathematics is abstract. Its function goes along well with Science and Technology.

• Because of its abstractness, mathematics is universal in a sense that other fields of human thought are not.

MATHEMATICAL INQUIRY

• Using mathematics to express ideas or to solve problems involving at least three phrases:

1. Representing some aspects of things abstractly;

2. Manipulating the abstractions by rules of logic to find new relationships between them; and

3. Seeing whether the new relationships say something useful about the original things.

APPLICATION

• Mathematical processes can lead to a kind of model of a thing, from which insights can be gained about the thing itself.

• Any mathematical relationship arrived at by manipulating abstract statements may or may not convey something truthful about the thing being modeled.

• Mathematics is appreciated as a human endeavor because all professionals and ordinary people apply its theories and concepts in the office, laboratory, and marketplace. According to Mark Karadimos (2018), the following professions use Mathematics in their field of work

ACCOUNTANT

• Assists businesses by working on their taxes and planning for upcoming years. They work with tax and codes and forms, use formulas for calculating interest, and spend a considerable amount of energy organizing paperwork.

ARCHITECTS

• Design buildings for structural integrity and beauty. They must know how to calculate loads for finding acceptable materials in design which involves calculus.

BIOLOGISTS

• Study nature to act in concert with it since we are very closely tied to nature. They use proportions to count animals as well as use statistics or portability.

CHEMISTS

• Find ways to use chemicals to assist people in purifying water, dealing with waste management, researching superconductors, analyzing crime scenes, making food products, and in working with biologists to study the human body.

ENGINEERS

• Build products/structures/systems like automobiles, buildings, computers, machines, and planes. They cannot escape the frequent use of a variety of calculus.

LAWYERS

• Argue cases using complicated lines of reason. That skill is nurtured by high-level math courses. They also spend a lot of time researching cases, which means learning relevant codes, laws, and ordinances. Building cases demands a strong sense of language with specific emphasis on hypotheses and conclusions.

MANAGERS

• Maintain schedules, regulate worker performance, and analyze productivity.

MEDICAL DOCTORS

• Understand the dynamic systems of the human body. They research illnesses, carefully administer the proper amounts of medicine, read charts/tables, organize their workload and manage the duties of nurses and technicians.

NURSES

• Carry out the detailed instructions doctors give them. They adjust intravenous drip rates, take vials, dispense medicine, and even assist in operations.

POLITICIANS

• Help solve the social problems of our time by making complicated decisions within the confines of the law, public opinion, and budgetary restraints.

G. H. HARDY

• A British mathematician who characterised mathematics as the study of patterns.

LOGIC PATTERNS

• Deals with the characteristics of various objects, orders, or sequences while others possess similar attributes.

• Seen on aptitude tests in which takers are shown a sequence of pictures and asked to select which figure comes next among several choices.

GEOMETRIC PATTERNS

• Deals with a motif or design that depicts abstract shapes, like lines, polygons, and circles, and typically repeats like a wallpaper.

WORD PATTERNS

• Deals with the metrical patterns of poems and the syntactic patterns of how we make nouns plural or verbs past tense are both word patterns, and each supports mathematical as well as natural language understanding.

NUMBER PATTERNS

• Deals with the prediction of the next term in a sequence Working with number patterns leads directly to the concept of functions in mathematics: A formal description of the relationships among different quantities.

FIBONACCI SEQUENCE

• LEONARDO PISA

• European mathematician (1175-1250)

• Discovered the Fibonacci sequence by investigating how fast rabbits could breed under ideal circumstances.

• The fibonacci sequence is the series of numbers: 0,1,1,2,3,5,13,21,34…

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

• The 2 is found by adding the two numbers before it (1+1)

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WHY IS FIBONACCI SIGNIFICANT?

• Fibonacci numbers show up unexpectedly in architecture, science, and nature (sunflowers & pineapples).

• Fibonacci numbers have useful applications with computer programming, sorting of data, generation of random numbers, etc.

FIBONACCI SPIRAL

• Fibonacci numbers can be represented as spirals also known as Fibonacci spirals.

GOLDEN RATIO

• Two quantities are in the Golden ratio if their ratio is the same as their sum to the larger of the two quantities. “De divina Proportione” by Luca Pacioli

• It is known as the famous irrational number 1.618803398 called phi. Denoted by 𝝋. It is used extensively by Ancient Greeks in architecture.

GOLDEN SECTION IN ARCHITECTURE

• Appears in many of the proportions of the Parthenon in Greece.

• Front elevation is built on the golden section (o.618 times as wide as it is tall).

• Can be found in the Great pyramid in Egypt.

MATHEMATICAL LANGUAGE

• The system used to communicate mathematical ideas.

• Consists of a substrate of some natural language. For example, English, using technical terms and grammatical conventions that are peculiar to mathematical discourse, supplemented by a highly symbolic notation for mathematical formulas.

• Mathematics as language has symbols to express a formula or to represent a constant.

• It has syntax to make the expression well-formed to make the characters and symbols clear and valid that do not violate the rules.

• Can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations, and other aspects of logical syntax.

• A mathematical concept is independent of the symbol chosen to represent it. In short, convention dictates the meaning.

PRECISION

• Means able to make very fine distinctions.

Example: The use of mathematical symbols is only based on its meaning and purpose. Like “+” means add, “-” means subtract, x means multiply and ÷ means divide.

CONCISE

• Able to say things briefly.

Example: The long English sentence can be shortened using mathematical symbols. Eight plus two equals ten which means 8+2 = 10.

POWERFUL

• Able to express complex thoughts with relative ease.

Example: The application of critical thinking and problem solving skills requires the comprehension, analysis, and reasoning to obtain the correct solution.

• An expression or mathematical expression is a finite combination of symbols that is well-defined according to rules that depend on the context.

• A correct arrangement of mathematical symbols used to represent the object of interest, it does not contain a complete thought and it cannot determine if it is true or false.

• The most common type involving an expression is simplify.

• To simplify an expression means to get a different name for the expression, that in some way is simpler.

The notion of simple can have different meanings:

• Simpler means using fewer symbols.

• Simpler means using fewer operations.

• Simpler means better suited for the current use.

• Simpler means in a preferred style or format.

In simplifying mathematical expressions, the following order of operations is one critical point to observe.

Order of operations is the hierarchy of mathematical operations. It is the set of rules that determines which operations would be done before or after others.

The order of operations is merely a set of rules that prioritize the sequence of operations starting from the most important to the least important. Follow these steps:

• Do as much as you can to simplify everything inside the parentheses first.

• Simplify every exponential number in the numerical expression/

• Multiply or divide whichever comes first, from left to right.

• Add or subtract whichever comes first, from left to right.

MATHEMATICAL SENTENCE

• Makes a statement about two expressions, either using numbers, variables, or a combination of both. A mathematical sentence can also use symbols or words like equals, greater than, or less than.

• A correct arrangement of mathematical symbols that states a complete thought and can be determined whether it's true, false, sometimes true/sometimes false.

CLOSED SENTENCE

• A sentence with a truth value of true (or false).

OPEN SENTENCE

• A sentence when it is not known if it is true or false.

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

• For instance, the fact that one evaluates multiplication before addition in the expression (2+3) x 4 is merely conventional.

SET THEORY

• Branch of Mathematics that studies sets or the mathematical science of the infinite.

• It became a fundamental theory in mathematics in the 1870s which was introduced by Georg Cantor, a German Mathematician.

SET

• A well-defined collection of objects (Elements/members).

• To describe a set, we use braces {}, and use capital letters to represent it.

• The symbol ∈ is used to denote that an object is an element of a set.

Roster Method or Tabulation Method

• The method where the sets are enumerated or listed and each element is separated by comma.

• A = {a,e,i,o,u}

Rule Method or Set Builder Notation

• A method used to describe the elements or members of the set.

• A = {x|x is a collection of vowel letters}

Finite Set

• A set whose elements are limited or countable, and the last element can be identified.

• A = {x|x is a positive integer less than 10} or A = {1,2,3,4,5,6,7,8,9}.

Infinite Set

• A set whose elements are unlimited or uncountable, and the last element cannot be specified

• A = {x|x is a set of whole numbers} or A = {1,2,3,4,5,6,7,8,9…} The three dots are called ellipsis or a continuing pattern.

• Unit Set

• A set with only one element.

• A = {1}

Empty Set/Null Set

• A unique set with no element.

• A = { } or ∅

The cardinality Number of a Set

• Number of elements or members in the set.

• Denoted by n (A).

• For finite sets A, n(A) is the number of elements of A.

• For infinite sets A, write n(A) = ∞

Equal Sets

• Two sets are equal if they have the same element

• Let A = {1,2,3,4,5} and B = {5,4,3,2,1}

Universal Set

• A set that contains everything.

• Denoted by U.

• A set U that includes all of the elements under consideration in a particular discussion.

Subset

• When we define a set and we take pieces of that set, we can form what is called a subset.

• For each set A, A ⊆ A

• For each set B, ∅ ⊆ B

• A is proper subset of B if A is a subset of B, A ⊆ B and A is not equal to B, A ≠ B.

UNION OF SETS

• The union of A and B is the set of all elements in x in U such that x is in A or x is in B.

• Denoted by A ∪ B

• The union of two sets A and B in symbol is Defined by A ∪ B = {x|x ∈ A or x ∈ B}.

• The word “or '' is inclusive.

INTERSECTION OF SETS

• The intersection of A and B is the set of all elements X in U such that x is in A and x is in B.

• Denoted by A ∩ B.

• The intersection of A and B in symbol if defined by A ∩ B = {x|x ∈ A or x ∈ B}.

• The word “and” is inclusive.

COMPLEMENT OF A SET

• The complement of A is the set of all elements x in U such that x is not in A.

• It is denoted as A’.

• THe complement of A in symbol is defined as A’ = {x  ∈ U | x ∉ A}.

DIFFERENCE OF SETS

• Difference between A and B is the set of elements x in U such that x is in A and x is not in B.

• It is denoted as A - B.

• A diagram with any circles, or overlapping circles to express logical relationships between sets.

• The set U represents the universal set.

• It is the set of all objects or elements of all other sets in consideration.

• The overlapping circles indicate the intersection of A and B.

• Illustration of Venn Diagram set A.

• Union of A and B

• Intersection of A and B

• Difference between Set A and B.

• Complement of Set A

• Union of A, B, and C.

DATA MANAGEMENT

• Development, execution, and supervision of plans, policies, programs, and practices that control, protect, deliver, and enhance the value of data and information assets.

• Administrative process by which the data is acquired, validated, stored, protected, and processed, and by which its accessibility, reliability and timeliness is ensured to satisfy the needs of the data users.

STATISTICS

• The word statistics originated from the word “status” meaning “state”.

• Science that deals with the collection, classification, analysis, and interpretation of numerical facts or data, in such a way that valid conclusions and meaningful predictions can be drawn from them.

GENERAL PURPOSES OF STATISTICS

• Statistics are used to organize and summarize the information so that the researchers can see what happened in the research study and can communicate the results to others.

• Statistics help the researcher to answer the questions that initiated the research by determining exactly what general conclusions are justified based on the specific results that were obtained.

METHODS OF DATA GATHERING

1. Direct or Interview

• Person-to-person encounter between the source of information, the interviewee, and the one who gathers information, the interviewer.

2. Indirect or Questionnaire

• Technique in which a questionnaire is used to elicit the information or data needed.

3. Registration

• Obtains data from the record of government agencies authorized by law to keep such data or information and makes these available to researchers.

4. Observation

• Technique in which data particularly pertains to the behaviors of individuals or groups of individuals during the given situation.

• To notice using the full range of appropriate senses. To see, hear, smell, and taste.

5. Experimental

• A system used to gather data from the results of a series of experiments on some controlled and experimental variables. This is commonly used in scientific inquiries.

• Independent Variable - The variable that is systematically manipulated by the investigator.

• Dependent Variable - Variab;e that the investigator measures to determine the effect of the independent variable.

6. Scientific Method

• The data from the experiment force a conclusion consonant with reality. Thus, scientific methodology has a built-in safeguard for ensuring that truth assertions of any sort about reality must conform to what is demonstrated to be objectively true about the phenomena before the assertions are given the status of scientific truth.

DESCRIPTIVE STATISTICS

• Involves a collection and classification of data.

INFERENTIAL STATISTICS

• Involves an analysis and interpretation of data.

POPULATION

• Set of measurements corresponding to the entire collection of units about which the information is sought. It is the group of objects/subjects about which conclusions are to be drawn.

SAMPLE

• Set of individuals selected from a population, usually intended to represent the population in a research study.

SAMPLE SIZE

Sample Determination Formula

DATA

• Measurements or observations. A data set is a collection of measurements or observations.

• A Datum is a single measurement of observation and is commonly called a score or raw score.

• The measurements that are made on the subjects of an experiment are also called data.

• Usually data consists of measurements of the dependent variable or of other subject characteristics, such as age, gender, number of subjects, and so on. The data as originally measured are often referred to as raw or original scores.

QUALITATIVE DATA

• Data that deal with categories or attributes.

• Color of skin, courses in Computer Engineering.

QUANTITATIVE DATA

• Data that deal with numerical values.

• Number of units in one semester, Grade point average.

DISCRETE DATA

• Data that is obtained by counting.

• Number of students in the classroom, number of cars in the parking lot.

CONTINUOUS DATA

• Data that is obtained by measuring.

• Area of a mango farm in Pampanga, volume of water in a pool in Pansol, Laguna.

PARAMETER

• Usually a numerical value that describes a population. A parameter is usually derived from measurements of the individuals in the population.

STATISTIC

• Usually a numerical value that describes a sample.

• Statistics are usually derived from measurements of the individuals in the sample.

SAMPLING ERROR

• Naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter.

VARIABLE

• Any property or characteristic of some event, object, or person that may have different values at different times depending on the conditions.

LEVELS OF MEASUREMENTS

• Nominal, ordinal, interval, or ratio (interval and ratio are sometimes called continuous or scale).

LEVELS OF MEASUREMENT

Nominal

• Labels qualitative data into mutually exclusive categories.

Example:

1. What is your civil status?

• Single

• Married

• Separated

• Annulled

2. Where do you live?

• Queen

• Manila

• Makati

• Cavite

Ordinal

• Ranks qualitative data according to its degree.

Example:

1. How satisfied are you with our food service?

• Very satisfied

• Satisfied

• Dissatisfied

• Very dissatisfied

2. What is your level of anxiety?

• Low

• Average

• High

Interval

• Numerical data that has order and its differences can be determined; they do not have a “true” zero.

Example:

1. Temperature

Ratio

• Numerical data that has order, differences can be determined and has a “true” zero.

Example:

1. Speed

2. Height

3. Weight