Mathematics in the Modern World Modules

Introduction to Mathematics in the Modern World

• Mathematics in the Modern World is a general education course focusing on the nature of mathematics, its practical, intellectual, and aesthetic applications in daily life.
• The course aims to equip students with intelligence to become leaders in nation-building and skills to combat challenges in daily living.
• The learning material is guided by CHED CMO No. 20, series of 2013, and is divided into modules covering the nature and utility of mathematics.
• Modules include patterns in the world, mathematical language, problem-solving, mathematical systems (modular arithmetic/group theory), data management, logic, and graph theory.
• The content was developed by Jose Alejandro R. Belen, Neil M. Mame, and Israel P. Piñero in 2020 at Batangas State University.

Module 1: The Nature of Mathematics

The Philosophy of Nature’s Numbers

• Based on Ian Stewart’s book, Nature’s Numbers, mathematics is a formal system of thought developed in the human mind and evolved within human culture.
• Human ancestors initially realized the existence of "form" in their surroundings, leading to a system of thought for understanding measures (counting, gauging, assessing, and quantifying).
• Recognizing shapes led to classifying contours and using designs to build human culture and flourish as a civilization.
• The natural world is embedded in a mathematical realm of patterns; natural order utilizes mathematical patterns efficiently to its advantage.

Mathematics as a Tool and Sense

• Math is not merely about "crunching numbers," formulas, or symbols; it is about forming new ways to see problems using insight and imagination.
• It acts as a "sense" (like sight or touch) used to decipher patterns, relationships, and logical connections.
• It deals with the logic of shape, quantity, and arrangement.
• Functions of mathematics include serving as problem-solving tools, providing explanations for occurrences, and acting as an art form involving inductive/deductive reasoning.

Patterns in the Natural World

• Patterns of Visuals: Often unpredictable and contain fractals. Seen in seeds, pinecones, branches, leaves, and self-similar replications of trees or ferns.
• Patterns of Flow: Found in the flow of liquids, water, stone, and the growth of trees. Includes undulating lines in meandering rivers.
• Patterns of Movement (Locomotion): Regular rhythms such as the left-right-left human walk, complex rhythmic horse gaits, insect scuttling, bird flight, jellyfish pulsations, and wave-like movements of snakes/fish.
• Patterns of Rhythm: The most basic pattern in nature. Hearts and lungs follow regular repeated patterns adapted to body needs.
• Patterns of Texture: Qualities sensed through touch (bristly, rough, smooth, cold, hard).
• Geometric Patterns: Repeated series of shapes found in cacti and succulents.
• Waves and Dunes: Disturbances carrying energy through media (air/water). Surface waves in the sea or ripple patterns/dunes formed by wind over sand.
• Spots and Stripes: Visible in giraffes and zebras. Alan Turing (1952) proposed a reaction-diffusion system where the size/shape depends on chemical interaction and diffusion speed.
• Spirals: Scale from the barred spiral Milky Way galaxy to microscopic animals. Present in pinecones, pineapples, sunflowers, and the horns of rams and kudu.

Symmetry in Nature

• A figure is symmetric if it can be divided into identical halves.
• Reflection Symmetry: Also called line or mirror symmetry; the left half is the same as the right.
• Rotational Symmetry: Remains the same after a rotation of less than one full turn. Degree depends on distinct orientations where it looks identical.
• Translational Symmetry: Units are repeated to create identical figures (e.g., hexagonal tiles in a honeycomb).
• Specific Examples:
  • Human Body: Exhibits bilateral symmetry.
  • Sunflower: Contains radial symmetry in the dark inner ring (disk florets) and bilateral symmetry in the outer ring (ray florets).
  • Snowflakes: Possess six-fold radial symmetry; single arms are nearly identical to others.
  • Honeycombs: Examples of wallpaper symmetry where patterns cover a plane.
  • Starfish: Radial fivefold symmetry.

The Fibonacci Sequence

• Named after Leonardo Pisano Bigollo (1170–1250), an Italian mathematician who discovered it while studying rabbit populations.
• Defined by the rule that a number is obtained by adding the two previous numbers: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$
• Fibonacci in Nature:
  • Flower Petals: Buttercups (5 petals), Lilies (3), Delphiniums (8), Marigolds (13), Asters (21).
  • Spirals: Nautilus shells (logarithmic growth), pineapples, and red cabbages.
• Formula for the nth term ($X_n$):
  • $X_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$
  • Here, $\phi$ represents the Golden Ratio, valued at approximately $1.618$.
• Golden Rectangle: Composed of squares with sizes matching the Fibonacci sequence. It generates a spiral line found from infinite to infinitesimal scales.

Module 2: Mathematical Language and Symbols

Language Characteristics

• Precise: Able to make very fine distinctions.
• Concise: Able to say things briefly.
• Powerful: Able to express complex thoughts with relative ease.
• Like any language, it has vocabulary (nouns/objects of interest) and rule-based sentences (completed thoughts).

Expressions and Sentences

• Expression: The analogue of a noun. A correct arrangement of symbols representing an object (e.g., $x+2$, $5x$). It does not state a complete thought and is neither true nor false.
• Sentence: The analogue of an English sentence. States a complete thought (e.g., $3+4=7$). It contains a verb (most commonly the equal sign "$=$").
• Connectives: Used to connect objects to create compound objects (e.g., the "$+$" sign in $1+2$).

Set Theory Basics

• Set: A collection of well-defined objects, introduced by Georg Cantor in 1879. Notated with braces $\{ \}$.
• Element ($\in$): An object belonging to a set.
• Unit Set: Contains only one element.
• Empty/Null Set ($\emptyset$ or $\{ \}$): Contains no elements.
• Cardinal Number ($n$): Measures the number of elements in a set.
• Equal Sets: Sets with the same cardinality and identical elements.
• Equivalent Sets: Sets with the same cardinality (1-to-1 correspondence).
• Universe Set ($U$): The set of all elements under discussion.
• Subsets ($A \subseteq B$): Every element of $A$ is in $B$. A proper subset ($A \subset B$) has at least one element in $B$ not in $A$.
• Total Subsections: For a set with $n$ elements, there are $2^n$ possible subsets.

Operations on Sets

• Union ($A \cup B$): All elements in $A$ or $B$.
• Intersection ($A \cap B$): Elements present in both $A$ and $B$.
• Difference ($A - B$): Elements in $A$ but not in $B$.
• Complement ($A^c$ or $A'$): Elements in the universal set $U$ but not in $A$.
• Cartesian Product ($A \times B$): Set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

Relations and Functions

• Relation: A subset of $A \times B$. A set of ordered pairs where $x$ is related to $y$ ($xRy$).
• Function: A relation where every input is paired with exactly one output. No two distinct ordered pairs have the same first component.
• Properties of Relations:
  • Reflexive: Every element is related to itself ($aRa$).
  • Symmetric: If $aRb$, then $bRa$.
  • Transitive: If $aRb$ and $bRc$, then $aRc$.
  • Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
• Function Operations: Sum ($f+g$), Product ($f \cdot g$), Quotient ($f/g$), and Composition ($f \circ g(x) = f(g(x))$).

Module 3: Problem Solving and Reasoning

Types of Reasoning

• Inductive Reasoning: Reaches a general conclusion (conjecture) by examining specific examples. It does not guarantee a true result but provides a means for prediction.
• Deductive Reasoning: Reaches a specific conclusion by applying general ideas, assumptions, procedures, or principles (Syllogisms).

Intuition, Proof, and Certainty

• Intuition: Immediate understanding or knowing something without reasoning. Often relies on abstract information in memory to make decisions from limited data.
• Ponzo Illusion (1911): An example where intuition can be misleading (two identical lines on converging railway tracks appear to be different lengths).
• Mathematical Proof: An inferential argument demonstrating the truth of a statement using axioms, rules of inference, and previously derived theorems.

Methods of Proof

• Direct Proof: Assumes the premise ($P$) is true and shows the conclusion ($Q$) must follow.
• Indirect/Contrapositive Proof: To prove $P \rightarrow Q$, assume $\sim Q$ and prove $\sim P$.
• Proof by Counterexample: Disproving a universal statement by finding one instance where it is false.
• Proof by Contradiction: Assume the implication is not true (assume $P$ is true and $Q$ is false) and derive a logical contradiction (e.g., $2=1$).

Polya’s Four Steps in Problem Solving

• George Polya: Known as the "Father of Problem Solving."
• Step 1: Understand the Problem: Identify what is asked, what is given, and if there is missing data.
• Step 2: Devise a Plan: Use heuristics like drawing diagrams, identifying patterns, working backward, formulating equations, or guessing.
• Step 3: Carry out the Plan: Work carefully/accurately. If the plan fails, try another.
• Step 4: Look Back: Check if the answer is correct and interpret the solution in context.

Module 4: Mathematical Systems

Modular Arithmetic

• A type of arithmetic for integers focusing on remainders. Often called "clock arithmetic."
• Definition: $a \equiv b \pmod{n}$ if $n$ divides $(a - b)$.
• Additive Inverse: The number $x$ such that $(a + x) \equiv 0 \pmod{n}$. In modulo $11$, the inverse of $5$ is $6$ because $5+6=11$.
• Multiplicative Inverse: The number $x$ such that $ax \equiv 1 \pmod{n}$. In modulo $7$, the multiplicative inverse of $2$ is $4$ because $2 \times 4 = 8 \equiv 1 \pmod{7}$.

Applications of Modular Arithmetic

• ISBN-13: Uses 13 digits. The check digit ($d_{13}$) formula is:
  • $d_{13} = 10 - [d_1 + 3d_2 + d_3 + 3d_4 + d_5 + 3d_6 + d_7 + 3d_8 + d_9 + 3d_{10} + d_{11} + 3d_{12}] \pmod{10}$
  • If the result is $10$, the check digit is $0$.
• UPC (Universal Product Code): 12 digits. The check digit formula is:
  • $d_{12} = 10 - [3d_1 + d_2 + 3d_3 + d_4 + 3d_5 + d_6 + 3d_7 + d_8 + 3d_9 + d_{10} + 3d_{11}] \pmod{10}$
• Luhn Algorithm: Used to validate credit cards (MasterCard, Visa, Amex, Discover). Steps:
  1. From the next-to-last digit, read right to left.
  2. Double every other digit. If the result is two digits, add them (e.g., $18 \rightarrow 1+8=9$).
  3. Sum all resulting digits. Final sum must be congruent to $0 \pmod{10}$.
• Cryptography:
  • Plaintext: Original message.
  • Ciphertext: Encrypted code.
  • Cycle Shift: $c \equiv (p + m) \pmod{26}$.
  • Affine Cipher: $c \equiv (ap + m) \pmod{26}$ where $gcd(a, 26) = 1$.

Group Theory

• Definition: A set $G$ with a binary operation $\ast$ is a group if it satisfies:
  1. Closure: $a \ast b \in G$.
  2. Associativity: $(a \ast b) \ast c = a \ast (b \ast c)$.
  3. Identity: There exists $e$ such that $a \ast e = a$.
  4. Inverse: There exists $a^{-1}$ such that $a \ast a^{-1} = e$.
• Abelian Group: A group that also satisfies the commutative property ($a \ast b = b \ast a$).
• Finite Groups: Represented using Cayley Tables.

Module 5: Data Management

General Fields

• Descriptive Statistics: Describing and summarizing data in symbolic or abbreviated forms.
• Inferential Statistics: Predicting values or generalizing from a sample to a population.

Scales of Measurement

• Nominal: Categorical labels (e.g., gender, marital status).
• Ordinal: Ranked data (e.g., 1st, 2nd, 3rd); tells us "greater than" but not by how much.
• Interval: Measurement data with meaningful distances between points but no absolute zero (e.g., intelligence scores, grades).
• Ratio: Measurement data with an absolute zero value.

Measures of Central Tendency

• Mean ($\bar{x}$): The arithmetic average ($\sum X / N$). Heavily affected by outliers (skewness).
• Median: Midpoint that separates the upper 50% from the lower 50%.
• Mode: The most frequently occurring score.

Measures of Dispersion (Variability)

• Range: Difference between the highest and lowest score.
• Standard Deviation ($SD$): Measures how much scores normally differ from the mean using every score in the distribution.
  • $SD = \sqrt{\frac{\sum X^2}{N} - \bar{x}^2}$
• Variance ($V$): The square of the standard deviation ($SD^2$).

Relative Position

• Z-Score: Converts raw scores from different distributions to a common scale ($\mu=0$, $\sigma=1$).
  • $z = (X - \mu) / \sigma$
• Percentiles: The point below which a given percentage of scores fall (e.g., 97th percentile means 97% did worse).
• Quartiles: Divide distribution into four quarters ($Q_1=25th$, $Q_2=50th/Median$, $Q_3=75th$).
• Box-and-Whisker Plot: Five-point summary visualizing Min, HS, $Q_1$, $Q_2$, and $Q_3$.

Normal Distribution and Correlation

• Normal Curve: Symmetrical bell-shaped curve where Mean = Median = Mode. Empirical Rule: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
• Pearson r: Measures linear association between two variables.
  • $r = \frac{\sum XY / N - (\bar{x})(\bar{y})}{SD_x SD_y}$
  • Range is $[-1, +1]$. Interpretations by Guilford: $.40-.70$ is substantial, $.70-.90$ marked, $.90-1.00$ very dependable.
• Linear Regression: Finding the least-squares regression line ($y = mx + b$) to predict values.
  • $m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$
  • $b = \frac{\sum y - m(\sum x)}{n}$

Module 6: Logic

Logic Statements

• Definition: A declarative sentence that is true or false, but not both. Examples: "Batangas is a province" (true), "$x+1=5$" (true or false depending on $x$).
• Simple Statement: Conveys a single idea.
• Compound Statement: Conveys two or more ideas connected by "and," "or," "if…then," "if and only if."
• Negation ($\sim p$): False if $p$ is true; true if $p$ is false.

Logic Connectives

• Conjunction ($p \wedge q$): True only if both $p$ and $q$ are true.
• Disjunction ($p \vee q$): True if at least one of $p$ or $q$ is true.
• Conditional ($p \rightarrow q$): False only if antecedent $p$ is true and consequent $q$ is false.
• Biconditional ($p \leftrightarrow q$): True if both statements have the same truth value.

Equivalent Statements and Tautologies

• Logically Equivalent ($p \equiv q$): Same truth values in all cases.
• De Morgan’s Laws:
  • $\sim(p \wedge q) \equiv \sim p \vee \sim q$
  • $\sim(p \vee q) \equiv \sim p \wedge \sim q$
• Tautology: Statement that is always true.
• Contradiction: Statement that is always false.

Logic Gates and Networks

• Switching Network: Closed if current flows; open if not. Series ($\wedge$), Parallel ($\vee$).
• Logic Gates:
  • NOT-gate: Inverse signal ($1 \rightarrow 0; 0 \rightarrow 1$).
  • AND-gate: Output is 1 only if both inputs are 1.
  • OR-gate: Output is 0 only if both inputs are 0.
• Boolean Variables: Variables taking only two values (bits represented as 0 or 1).

Module 7: Mathematics of Graphs

Graph Fundamentals

• A graph $G$ consists of a non-empty set of vertices ($V$) and edges ($E$).
• Order: Number of vertices. Size: Number of edges.
• Adjacency: Two vertices joined by an edge. Incident: Vertices that are endpoints of an edge.
• Simple Graph: No loops (edge starting/ending at same vertex) or parallel edges (two edges between same vertices).
• Isomorphic (Equivalent) Graphs: One graph can be redrawn to form the other.

Degree and Walks

• Degree ($deg(x)$.): Number of edges incident with vertex $x$.
• Walk: Sequence of adjacent vertices. Length: Number of edges in the walk.
• Trail: A walk where no edge is repeated.
• Path: A walk where no vertex is repeated.
• Circuit: A closed trail (starts/ends at same vertex).
• Cycle: A closed path.

Eulerian and Hamiltonian Systems

• Eulerian Circuit: Travels every edge once. Eulerian Graph: Connected graph with every vertex degree even.
• Eulerian Trail Theorem: Exists if exactly two vertices have odd degree; must start at one odd vertex and end at the other.
• Hamiltonian Cycle: Travels through every vertex exactly once. Hamiltonian Graph: Contains a Hamiltonian cycle.
• Dirac’s Theorem: A connected graph with $n \geq 3$ vertices is Hamiltonian if every vertex degree is at least $n/2$.

Routing and Optimization

• Weighted Graph: Edges are assigned values/weights.
• Greedy Algorithm: At each vertex, choose the edge with the smallest weight connecting to an unvisited vertex. Does not guarantee the best route.
• Edge-Picking Algorithm: Successively select the smallest weights in the graph as long as they don't form early cycles or add a third edge to a vertex.
• Graph Coloring: Assigning colors to vertices so adjacent vertices have different colors.
• Chromatic Number ($\chi(G)$.): Minimum colors required.
• Four-Color Theorem: Every planar graph is 4-colorable. Used for scheduling problems where common participants cannot meet simultaneously.