Mathematics in Our World and Mathematical Language

Institutional Overview and Policies

• Vision: A premier institution in Region IV focused on developing globally-competitive professionals for community development and nation-building.

• Mission: Committed to equipping individuals with knowledge, skills, and values to achieve service for national development.

• Grading System:

  • Written/Skills-based Quizzes: $20\%$

  • Class Participation (Recitation): $10\%$

  • Other Requirements (Research Projects/Case Studies): $20\%$

  • Major Examinations: $50\%$

• Attendance Policy: Per CHED policy, missing $20\%$ of contact hours results in failure. Absence limits are based on meetings per week:

  • $3$ meetings/week: Absences should not exceed $17$

  • $2$ meetings/week: Absences should not exceed $10$

  • $1$ meeting/week: Absences should not exceed $7$

Patterns in Nature

• Definition: Patterns are visible regularities of form found in the natural world. Counting and patterns are correlative.

• Types of Patterns:

  • Symmetry: Agreement in dimensions, proportion, and arrangement.

  • Spiral: A curve emanating from a point, moving farther away as it revolves.

  • Meander: Sinuous curves, bends, and loops in river channels.

  • Wave: A disturbance transferring energy through matter or space.

  • Foam: Gas pockets trapped in a liquid or solid.

  • Fracture: The separation of an object into pieces.

  • Stripes: Bands of uniform width and color.

  • Fractals: Never-ending, self-similar patterns.

The Fibonacci Sequence and the Golden Ratio

• Natural Occurrences: Evident in pinecone spirals, petal arrangements in flowers, sea shells (Nautilus), and leaf arrangements on stems.

• Mathematical Characteristics:

  • Seed Migration: Seeds fill space by migrating outward in Fibonacci spirals.

  • Branching: Visible in the structure of stems and new plant growth.

  • Golden Ratio: Expressed through the Golden Rectangle and logarithmic spirals.

The Nature and Importance of Mathematics

• Essence: Mathematics is a blend of logic and creativity, exploring relationships between abstract concepts (numbers, figures, equations).

• Perspective: Albert Einstein stated, "Pure mathematics is the poetry of logical ideas."

• Everyday Applications:

  • Managing finances (bank accounts, taxes, mortgage).

  • Daily tasks (tipping $15\%$ or $20\%$, tracking health reps, cooking measurements).

  • Professional Utility: Used by engineers, computer programmers, medical doctors, meteorologists, and lawyers.

• Interdisciplinary Influence:

  • Psychology: Uses statistical analysis for social phenomena.

  • Actuarial Science: Uses math to match insurance premiums to future liabilities.

  • Music: Leibnitz described it as a hidden arithmetic exercise; Pythagoras linked harmony to numbers.

  • Arts and Philosophy: Goethe based architecture on geometry; math acts as the language of the universe.

Mathematical Language and Symbols

• Characteristics:

  • Precise: Expressed clearly without ambiguity.

  • Concise: Communicates complex ideas efficiently.

  • Powerful: Simplifies high-level abstractions.

• Parts of Speech:

  • Nouns: Represented by numbers (quantities).

  • Connectives: Operation symbols ($+$, $-$, $\bullet$, $\div$) link expressions.

  • Verbs: Relation symbols ($=$, <, >, $\cong$) compare quantities to complete a sentence.

• Classification:

  • Expression: A name given to an object (e.g., $2$, $3x$, $ax + by + c$).

  • Sentence: States a complete thought that can be true or false (e.g., $3 + 2 = 5$).

Sets of Numbers and Symbolic Conventions

• $N_0$ (Natural/Whole Numbers): $\{0, 1, 2, 3, 4, ...\}$

• $N_1$ (Natural/Whole Numbers without zero): $\{1, 2, 3, 4, 5, ...\}$

• $Z$ (Integers): $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

• $Q$ (Rational Numbers): $\{x \mid x = \frac{a}{b}, a, b \in Z, b \neq 0\}$

• $R$ (Real Numbers): \{x \mid -\infty < x < \infty\}

• $C$ (Complex Numbers): $\{z \mid z = a + bi\}$

• Logical Connectives: Conjunction ($\wedge$), Disjunction ($\vee$), Negation ($\sim$), Implication ($\rightarrow$), and Universal Quantifier ($\forall$).

• Set Theory: Union ($\cup$), Intersection ($\cap$), Element ($\in$), and Subset ($\subset$).

Relations and Functions

• Relation: A rule pairing every element in the domain with one or more elements in the range, creating ordered pairs $(x, y)$.

• Function: A specific relation where each input from the domain maps to exactly one unique output in the range.

• Core Distinction: "All functions are relations, but not all relations are functions."

• Operations on Functions:

  • Addition: $(f + g)(x) = f(x) + g(x)$

  • Subtraction: $(f - g)(x) = f(x) - g(x)$

  • Multiplication: $(fg)(x) = f(x) \times g(x)$

  • Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$)

  • Composition: $(f \circ g)(x) = f(g(x))$