mathematics in the modern world, Essays (high school) of Mathematics

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Table of Contents the ld Chg Snmetly a) Blated Smmetry b) Rati Symmetry Fractals Parallel Lines Fibonacci Spiral Fibonacci Spiral Galen Ratio Galoen Rectangles Mtherretics for Q-garization Matherratics for Prediction Mtherratics for Contral JeonguageOynbals Cowention Letters Metherratical Srbols Expression Versus Sentences Differentiation of Matherratical Expression and Euatiar/Sentence Basic Cancepts an Sts Set Herrents ‘Methods of Defining Set Typesaf Set Theary Syrrtal (THS IS OPTIONAL) Vem Dagrars Set Qoeration Properties of the tion Qneration Properties of the Intersect Operation Srrtls to Var Expressions (THS/S OPTIONAL) Losic Introduction to Lagic Saterent Truth value and truth table Negation Conpaund Saterrentsand Grouping Symhals Altermative Method for Truth Table Conditional Saterrent Basic Cancepts m Sets Set Herrents ‘Methods of Defining Set Typesof St Theay Sb Vem Dagrars Set Qreration Properties of the tion Qneration Properties of the Intersect Qperation ‘Syrrbolsto Word Expressions Sole Reasoning Wantageneat Qualitative Data Qantitative Data ‘Types of Quantitative Variable Typesof Satistics Descriptive Statistics Inferertial Statistics LEMLS OF MASLREMAT Narn Dita Otiral Data Interval Data Retio Data Measure of Gntral Tendency Mok Measure of Variance ar Dispersion Range Variance Standard Deviation Definition of symbols and Variables Formula of Measures for Lagroyped an Grauped Data Qareatin Analysis Interpretation When the value of “1” is Pearson's Product Marert Regression Mthad Least Square Regression Equation Regression Method Formula: Carrdation Between Ordinal Varialle ‘Spearman Rank Order Correlation Coefficient Kinds OF Data DSTRIEUTION Symmetrical a Narral Dstribution Positively Skened Distribution Negptively Heved Distribution Pararretric tests t-test for Dependent Sarples (paired) t-test for Independent Sarrples (unpaired) Z-tet Fest No-peraretric tests Hpothesis Nil Hypothesis (+4) Alternative Hypothesis (+) eownettic Design Translation Retatios Gide Feflections RON Rosette patterns (finite cen) 1 Odic Symmetry 2 Dhecral Syrrretry Friezepatierns Typesof Frieze Patterns 1. HopPattem Sep Sde Spin hop Spin ste Spin jurp MRO RON Mathematics op Graphs Termindlgies OF Concepts Of Graphs Qnplete Gah Euivdent Gas Ellerian Graph Thearem Aller Path Theorem Drac's Theoren The Geedy Agarithm The Ellge-Ficking Agarithm Applications of Véighted Gaphs Four-Calar Theorem 2-Cdaralle Gaph Thearem Mathematical Systems Arithmetic Operations Modulo n Examples: Arithmetic Operations Modulo n Examples: Arithmetic Operations Modulo n Examples: Arithmetic Operations Modulo n Solving Congruence Equations Problems: Solving Congruence Equation Additive and Multiplicative Inverses — Calder Rectangles, o Ratio of the length is longer and the widthis shorter o A rectangle that can be cut up into a square and a rectangle similar to the original one. The longe e shorter side ° —— Mathematica, for 6 ° expresses itself everywhere, in almost every facet of life it is considered as the language of science and engineering used as a tool to help us make sound analysis and better decision you can probably think of different situations with mathematical tools being used it also develop strategies of problem-solving ° prediction through the analysis and interpretation of existing data probability and patterns usually predictions is used in weather forecasting, and also as a basis for predicting patterns based on your observation —— Mathematica, pow Control —— ° Influence the behavior of a system and has the control in order to achieve a desired goal Money - mathematics of manipulation Man is able to exert control over himself and effects of nature through math Human Behavioral pattern can change the society and the natural world LANGUANGE ° ° OF MATHEMATICS facilitates communication and clarifies meaning for many things system of communication which consists of sets of sounds and written symbols which are used. by the people. allows people to express themselves and maintain their identity it consists of words, letters, and symbols that are known as natural language. Non-verbal signs or images that can be used to communicate can be called as mathematical langauge Math is a universal language o the principles and foundations of math are the same everywhere around the world o improves our mental ability as it teaches us logical ways of thinking o makeit easier to express their thoughts because it is: » Precise - able to make very fine distinctions » Concise — able to say things briefly « Powerful - able to express complex thoughts with relative ease not rules, but often used that way some letters have special uses constants (fixed values) positive integers (for counting) variables (unknown) greater than or equal to TRANSLATION FROM ENGLISH TO MATHEMATICAL STATEMENTS —Nounw, Verb, Senter 0) o Nouns could be your integers, numbers, or expressions Ex. 5, 2(5-1/2) o Verb may be your equals sign “=”, or an inequality (>,<) o Pronouns are your variables x and y Ex. 5x-2, xy o Sentence would be formed. Ex. 3x + 7 = 22 —Expreasion vo. SOM —$<— Noun (person, place, thing) Ex. Carol, Expression Ex: 5, 2+3, 1/2 Sentence Ex: The capital ! ofIdahois : Sentence Yeni of Mitral Epa, tgstan— FUE FURES OH MATHEW ATICN! —Basio Concept av Seta, Expressionis a ! i ser | mathematical : An equationisa o awell-defined collection of objects (elements) P combines : mathematical : similar/related or dissimilar/non-related in : statement wherein : some ways. numbers, i : i iables and ; ‘Wo expressions are i ; var i setequaltoeach : o Set theory - created by Georg Ferdinand erates f° i other. i Ludwig Philip cantor of something. : o elements are enclosed in braces “{ }” Asentence fragment, that : A sentence that standsfora : shows equality eal : eeprescione o objects/ random symbols/numbers/names that value. belong to a set, denoted by “U” Simplification Solution o elements in a set do not repeat. repeat, they seeneseeeeeneaseassennsvndeeseessenesseesseeseneseneaseaes DO NOT repeat. No relation Yes, equal sign (=) symbol equal sign o order of elements in a set is not important, but Two sided. left and used for organizing purposes PEt Method, op Dopining Set expression sossessssnesenesensseeecsegerseseeecseseeesseeseesereseeees o A={I,2, 3, 4, 5} Tx - 2(3x + 14 One sided o defined by enumerating the elements of the Five increased by ten set. Twenty subtracted from twelve a: |_RULE METHOD | Two more than the sum of six and four “2 o A= {x|xis a counting number less than 6} Thirty decreased by eight o Isby using descriptive phrases in the form of The quotient of twenty- four by three, plus x”such that x” —Typew, of Set Theary Synhol— Note: Some examples tend to change the elements in Set A or/and B to illustrate how the symbol functions in a statement properly. a collection of A= {3,7,9, 14}, elements B (9.14.28) A={x| xc. : 7 ch that that : Five years ago, John’s age was half of the ou s° R, x<0} age he will be in 8 years. How oldis he intersection objects that belong AnB= now? to set A and set B {9,14} union objects thatbelong AUB= toset A or set B {8,7,9,14,28} by disjunctions from left to right, — Tegatian followed by conditionals from left to right, and finally by biconditionals from Say Pis a statement. left to right. o Thenegation of Pmeans not Pand is denoted by ~P. Tautology — Statement that is always true o Ifthe statement is false, its negation is true. Self - Contradiction - Statement that is always false o Ifthe statement is true, its negation is false. , Cc Qiti b St t o Thenegation of the negation of a statement is the original Conditional Converse Inverse Contrapositive statement A> .B BOA A> 2B BO? A . TI|T |T T/T |T FO\F UT FOLF \T —Compound Statementa, & Grouping Symbol — Tete) tte) fede tere FIT [T FiT |F TIF IF TIFT +> Given a compound statement in symbolic form, — parentheses are used to indicate which simple T]t |T statements are grouped together. TIF IE Ex: FIT [F F[F [T P, (Q f~R) @r-QfR = P-QE)>e FQ) +> Given a compound statement written in English, comma is used to indicate which simple statements are grouped together. Difference when using commas P,andQorR P,(QfR) PandQ, orR (P,Q)fR Say P and Q be statements. The conjuction denoted by P~ O is TRUE if and only if BOTH P and Q are true. The disjuction denoted by P v Qis TRUE if and only if Pis TRUE, Q is TRUE, or BOTH P and Q are true. —Aomative Metta fon Tale —§ 2 = # of rows in a truth table; n = # of Statement ¢ Use the truth values for each simple statement and their negations to enter the truth values under each connective within a pair of grouping symbols— parentheses (), brackets [ ], braces { }. ¢ Ifsome grouping symbols are nested inside other grouping symbols, then work from the inside out. ¢ Inany situation in which grouping symbols have not been used, then we use the following order of precedence agreement. First assign truth values to negations from left to right, followed by conjunctions from left to right, followed o atoolin the field of general mathematics, Sul 0 (ist Reasoning probability, and statistics that helps calculate the number of possible outcomes of an event or problem, and to cite those potential : outcomes in an organized way. Problem Sabsing 3 » Problem -an inguiry starting from given | POLYACS PROBLEM-SOLVING srRAGETEGY — conditions to investigate or demonstrate a fact, result or law. o George Polya was a mathematician in the 1940s o He devised a systematic process for solving problems that is now referred to by his name: the Polya 4-Step Problem-Solving Process o Problem Solving is a fundamental means of developing mathematical knowledge at any level. — Inductive Reasoning a ‘Look back o Itis an approach to logical thinking that eesearc) involves making generalizations based on specific details. — Conyechine a mathematical statement that has not yet been rigorously proved. o One counterexample means that the statement is false. ws Centncber en 0 pl, o Anexample that opposes or contradicts an idea or theory. Devise a plan (PLANNING) o Itisa type of logic where general statements, or premises, are used to form a specific conclusion. o It moves from generalities to specific conclusions. Wantagtiveat STATISTICS o Branch of science that deals with the collection, presentation, organization, analysis, and interpretation of data » Population - collection of all elements under consideration in a statistical inquiry @ Sample - subset of a population Variable - characteristics & attributes of the elements in a collection that can assume different values for the different elements CLASSIFICATION OF DATA =» Descriptive Statistics o Collecting, organizing, presenting, and analyzing numerical data o Focuses on quantitatively describing the collection of data o Summary of the samples with corresponding measures is stated o Thisis the organizing and summarizing data using numbers and graphs = Inferential Statistics o Analyzing the organized data o Leading to prediction o Assume from the sample data of the population might probably be o Using sample Data to make an interference or conclusion of the population © Measurement -Process of determining the value or label of the variable based on what has been observed — Qualitative Doto»— | BEVEES OF MEASUREMENT | | o Represents differences in quantity, character or kind but not amount o Measure of “types” and may be represented by names or bols o Describes individuals or objects by their categories or groups o Descriptive data based on observations and usually involves 5 senses (see, feel, taste, hear, smell) — Quantitative Data, Numerical in nature and can be ordered/ ranked o Measure of “values” or “counts” and expressed in numbers = Discrete Quantitative Variable (Countable) Ex: 8 cats (base in whole numbers and can be counted) = Continuous Variable Quantitative Values (integers, rational, irrational values) Ex: 5.5m (anything can be measured such as distance, speed, weight) =» Nominal Data o Labeling variables, without quantitative value o Nominal scales -"labels" o Cannot be arranged in an ordering scheme = Ordinal Data o Measures of non-numeric concepts o Difference between the values of the data cannot be determined o Interval is meaningless o The order of the values is important and significant = Interval Data o Quantitative measurements used to identify and rank o Hasnegative values o Can't compare two values o Hasnotrue zero point = Ratio Data o Similar to interval scale but has a true zero point o Nonegative values o Multiples are significant MEASURES OF CENTRAL TENDENCY AND VARIANCE/DISPERSION CORRELATION AND REGRESSION ANALYSIS —Canelotion Anobypia-—— ° a statistical method used to determine whether a relationship between two variables exists SCATTER PLOT EXAMPLES O] Positive * O] Negative Correlation Correlation of No * Correlation no correlation perfect correlation very low moderately low high very high ° Measure of the linear association between 2 variables that are measured on interval or ratio scales. ° It was developed by Kar! Pearson that is why the correlation coefficient is sometimes called "Pearson's r“. The formula is defined by: N=IXY—NXZY [NEx2 - xy] [vey - cry] EXAMPLE. ‘An education researcher wishes to determine the extent of relationship of the between the reading comprehension and vocabulary tests among 12 students. Determine r,, and interpret the result. ‘STUDENTS | READINGOX) [ VOCABULARY(Y) |__X? y xY 1 3 1 9 121 33 2 7 1 49 1 Z 3 2 19 4 367 38 4 9 5 ar 25 45; 5 8 17 64 289 136) 6 4 3 16 9 12 7 1 15 2 225 15 @ 10 9 100 87 90. 9 16 15 256 225 240 10 5 8 25 64 40 1 3 12 9 144 36 12 8 4 64 16 32. Ex=76 By=119 Ex? = | By? =1567 | Exy=724 678 12(724) — 76(119) rxy= [12678) - (76)'| [12561 - 119} Note:madeacorrectiononone value; "Dx?" from 670 to 678 r= -011 Therefore, there is a very low correlation between the results of the scores obtained by the reading comprehension and vocabulary test ° A linear regression is used to make predictions about a single value. a+ 0X WHERE: Y= DEPENDENT X= INDEPENDENT a= Y-int (x=0) b= SLOPE _ 222) - vey) n&x2 — (Ix) nixy— (2x) (Zy) ~ ndx2— (=x)? From the previous example: 119678) - 76(724) qa = = 1087 12678) - (76) 12(724) - 76(119) ba) 5-015 12678) - (76) Y = 10.87 —0.15X wn Comnelation Betuscorn Ordinal Variable, ° Used to calculate the correlation of ordinal data w/c are classified according to order or rank. ‘WHERE: d = difference between ranks n= number of paired observations Example In a contest for miss university, two judges gave their ratings to 8 candidates. Transform the ratings to ranks and compute the coefficient of rank correlation. Interpret the results Candidate | Judge1 | Judge2| Rt R2 id] & 1 1 98 94 1 4 3 9 2 97 97 2 2 oO 0 3 95 98 3 1 2 8 4 90 95 4 3 1 1 5 89 92 5 5 oO 0 6 88 90 6 6 oO oO 7 85 89 75 7 0.5 0.25 8 85 85 75 8 0.5 0.25 14.5 6(=d2 6(14.5) pap - He r= loa = 0.88 5 _ n (n2 -1 ) ( ) CONCLUSION: « Ther, =0.83 indicates that there is a very high positive correlation between the two judges. KINDS OF DATA DISTRIBUTION me Symmetrical Normal Dish ihiti me the mean, median, and mode all fall at the same point or equal symmerical distribution mean = median = mode mean median mode —Panitivaly Showed Distribution <<< the extreme scores are larger, thus the mean is larger than the median frequency positively skewed curve symmetrical curve 2 Long tai £ it —» —= Negatively Sbowed Distrib uta The order of the measures of central tendency would. be the opposite of the positively skewed distribution, with the mean being smaller than the median, which is smaller than the mode Negatively skewed curve symmetrical dist. ~~, tong tat —Panamatic Teat, ° — Now Panametnic Toat HYPOTHESIS TESTING Method of using simple data to decide between two competing claims(hypothesis) about a population characteristic. Concerns itself with the decision-making rules for choosing alternatives while controlling and minimizing the risks of wrong decisions. TYPES OF HYPOTHESIS TESTING The parametric tests are tests applied to data that are normally distributed. A statistical test, in which specific assumptions are made about the population parameter is known as parametric test. It is assumed that the measurement of variables of interest is done on interval or ratio level. The measure of central tendency in the parametric test is mean. There is complete information about the population. tests that do not require a normal distribution a statistical test used in the case of non-metric independent variables, is called non- parametric test the variable of interest is measured on nominal or ordinal scale the measure of central tendency in the parametric test is median there is no information about the population t-test for Dependent Samples (paired) A parametric test applied to one group of samples It can be used in evaluation of a certain program or treatment It is applied when the mean before and the mean after are being compared t-test for Independent Samples (unpaired) Used when we compare the means of two independent groups Used when the sample is less than 30 Z-test It is used to compare two means: the sample means and the perceived population mean. It is also used to compare the two sample means taken from the same population. When samples are equal to or greater than 30. It can be applied in two ways: the One-sample mean test and the two sample mean test. F-test It is another parametric test used to compare the means of two or more independent groups. It is also known as the analysis of variance (ANOVA) Kinds of ANOVA: One-way, two-way, three-way We used ANOVA to find out if there is a significant difference between and among the means of two or more independent groups. Ifa figure is rotated all : the way around back : to where it started, it is : a full rotation with an angle of 360°. : I a figure is rotated : only half of the full \ i rotation, it has an angle of 180°. Real-life examples of rotation: [Center oF Rotation Ferris wheel Bicycle wheels, — Adjidereflectiisa cortiretin of a translation and a reflection — Theaisof reflection must be parallel to the direction of the translation — Youcan reflect and then translate or vice versa. > a “=v “Sw Examples of glide rotation in patterns: Real-life ecarples of glide retatione d &. Man’s footsteps Leaves in branches Aplane pattern has a symretry if there isan isometry present in the plane. Atransformetion of a pattem isa synrretry of the pattern if the pattem o there is a rotation symmetry around a center point but no mirror lines; it only goes continuously in circles. Examples of cyclic symmetry in patterns: Real-life examples of cyclic symmetry: Dart board © rdation syntretry around a center point with mirrar lines through the center point. You can distinguish the different equal pats. Banpled dihedral erretry inpatiens London eye (Ferris wheel) Real-life examples of dihedral symmetry: © Frieze patterns are patterns that have trandiational symmetry in ‘one direction © — they goonto infinity directions, both left and right [eyerele TLL. Example of cyclic symmetry in patterns: — Types of Friege Pattorrtom _Horpartern | ~ just a translation dein a repetitive manner vB; oth oth ot; eee SSS ze | - invdveshavizontal reflections followed by translations 99771 sm_| all are glide reflections Ea - invdvestranslations and vertical reflection lines with a 180° rdtation Ez - involves translations and a 180° rotation. Smnspie__| — tranclation then either vertical reflection or 180° ration. Senor _| — horizontal and vertical reflections and translations i@~ 7B: {ew ve ao! tq sof te +++ WALLPAPER PATTERN Wallpaper pattern is a pattern with translation in two directions, done in arepetitive manner. Itisan arrangerrent of frieze patterns stacked upon ane another to fill the entire plane. Awallpaper pattern can be made up of a combination of rotation, reflection, and glide reflection. Qovering any flat surface with a pattem of multiple shapesand styles ‘such that no part remains uncovered or overlaps The Edge-Picking Algorithm ¢ Another method of finding a Hamiltonian circuit in : : GC. a complete weighted graph is given by the a= Ng I ni Onan 70. pl Vay following edge-picking algorithm. ¢ Hamiltonian circuit is a circuit that visits * Mark the edge of smallest weight in the graph. vertex once with no repeats. Being a circuit, it must start and end at the same vertex. ¢ Hamiltonian path also visits every vertex once with no repeats, but does not have to start and end at the same vertex. ¢ Continue the process until you can no longer mark any edges. Then mark the final edge that * Mark the edge of the next smallest weight in the graph, as long as it does not complete a circuit and does not add a third marked edge to a single vertex. 4__§ _¢ completes the Hamiltonian circuit. b z 4 Planarity and Graph Coloring ¢ Aplanar graph is a graph that can be drawn so G —Weighted Graph that no edges intersect each other (except at vertices). « Aweighted graph is a graph in which each edge ) is associated with a value, called a weight. . . ky pee eS Planar Non Planar The Greedy Algorithm * greedy algorithm is basically choosing the smallest value option at every chance we get. * it focuses on picking the vertex with the minimum amount all throughout until it travels along all the vertices. ¢ finding efficient Hamiltonian circuits in complete weighted graphs * choose a vertex as a starting point, and travel along the edge that has the smallest weight. — Copy CL —$>=$>=$=>=>=-——= ¢ Ifthe map is divided into regions in some manner, what is the minimum number of colors required if the neighboring regions are to be colored differently? ¢ There is a connection between map coloring and graph theory. Maps can be modeled by graphs using the countries as the vertices and two vertices (countries) are adjacent if they share a common boundary. ¢ In graph coloring, each vertex of a graph will be assigned one color in such away that no two adjacent vertices have the same color. The interesting idea here is to determine the minimum number of (distinct) colors to be used so that we can color each vertex of a graph with no two adjacent vertices have the same color. Four-Color Theorem Every planar graph is 4-colorable. The Chromatic Number of a Graph The minimum number of colors needed to color a graph so that no edge connects vertices of the same color is called the chromatic number. 2-Colorable Graph Theorem « Agraph is 2-colourable if and only if it has no circuits that consist of an odd number of vertices. ahencatical Dystens MODULO N To say that two integers are considered as congruent modulon, wherein n is anatural nurber, ir is equivalent to an integer. vewiteit as a= b moduon Moreover, the integer value of modulo nis equal to the rerrainder left of the nurber when it is divided by the n. The statarent a = b mod niscalleda congruence. Modulon exanple 29= 87 60= Onad15 17 2mod5 Auithmdio Operations, Module w performthe indicated operations first, afterwards divide your answer by modulus The nurber you will get isthe remainder and should be awhde number and less than the modulus Note: The result of an arithrretic operation mod nis alwaysa whole nunber less than n. Performthe modular arithmetic. 1249 = 1mod5 123 = Ormod5 EXAMPLES: ARITHMETIC OPERATIONS MODULO N 1. Dsregarding AMor PM if it isnow5 dclodk, what tire will be 45 hours fromnon? The time can be deterrrined by (5 +45) mod 12 Observe that (5 +45) mod 12=50mod 12. Bu 50 mod 12=2. Therefore, if it’s 5 ddlock now, 45 hours framnowis 2 dclock. EXAMPLES: ARITHMETIC OPERATIONS MODULO N 2 Osregarding AMor PM if it isnow5 dclock, what tire wasit 71 hours ag? The time can be deterrined by (5 - 71) mod 12. Observe that (5 - 71) mod 12=(-66) mod 12 Find a whole number x less than 12 such that - 66 =x mod 12. —66—x —66=xmod12@ So that, - 66 mod 12=6. Therefore, if it's 5 dclock now, 71 hours ago is 6 dclodk. EXAMPLES: ARITHMETIC OPERATIONS MODULO N 3. In 2005, April 15 fell on a Friday. On what day of the week will April 15 fall in 20137 There are 8 years between two dates Each year has 365 days (except for 2008 and 2012). So the total number of daysin between two dates are 8(365) +2 =2,922 Thus we want 2,922 mod 7 But 2,922 +7 = 417 remainder 3. We went a day that is sare as the day 3 days after April 15, 2006. Therefore, April 15, 2013 is Monday. It isdefined asfinding of all the values (whichis a whole nurrber less than the modulus) of the variables for which the congruent is true. It isnot your ordinary matherratical equation that needs certain solution to be solved It is only satisfied once the value of the variable is found. It does Not require any specific sd.ution rather ance a single solution was found additional solutions can also be found by just repeatedly adding the modulus to the original equation For instance, just like for commm problems, we lodk for solutions that would be best to solve our problems without the need to base it toany matherratical formulas. Renarks Accongruence equation can have mare than one solution amang the whole numbers less than the modulus. Accongruence equation can have no solution PROBLEMS: SOLVING CONGRUENCE EQUATION Find all whole number solutions of the congruence equation. x=7mod4 2x=5mod9 2x +1E 6mod 5 5x +1 3mod 5 PWNS