Patterns & Math in Nature and Transactions

• Patterns and numbers are appreciated in houses, flowers, trees, and colors.
• Patterns organize and arrange things, making them understandable and creating appreciation.

Mathematics

• Study of numbers and operations and involves logical reasoning.
• Considered an art that studies patterns for prediction.
• Sometimes referred to as a specialized language.

Patterns in Nature

• Regularities and irregularities are observed through careful observation.
• Natural patterns like spirals, symmetry, and stripes exist.
• Harmonies in music arise from numbers.
• Objects in nature are imperfect copies.
• Waves provide clues to water flow.
• Examples of natural patterns: tiger stripes, mirror symmetry in animals, leaves, and flowers.

Numbers in Nature

• A year has 365 days.
• Cats have four legs, while spiders have eight.
• The driver changes correctly using math.

Routine Transactions

• Common, everyday actions, and transactions rely on mathematical principles.
• Examples include attendance, waiting in line, following medical instructions, budgeting, and traveling.

Pattern Recognition

• Observers anticipate future events.
• Studying patterns is important in observation, hypothesis, discovery, and creation.
• Patterns help understand, predict, imagine, and estimate.

Logic Patterns

• Sequences of pictures are presented, and the observer selects the next figure in the sequence.

Number Patterns

• Identifying and continuing numerical sequences.

Geometric Patterns

• Recognizing and predicting patterns in shapes and spatial arrangements.

Word Patterns

• Identifying patterns in language and text.

Fibonacci Numbers

• Leonardo Pisano (Fibonacci) was born in Pisa, Italy, in 1175 AD.
• Fibonacci is derived from "filius bonacci," meaning son of Bonaccio.

Fibonacci Numbers Defined

• Introduced in "The Book of Calculating".
• Series starts with 0 and 1.
• Next number is the sum of the previous two.
• Pattern repeats.
• Described by the formula: F(n + 2) = F(n + 1) + Fn.

Fibonacci Sequence

• Sequence obtained by adding the two preceding numbers.
• Starts with 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...
• Can be graphed and found in waves, flowers, humans, bodies, and snails.
• Evident in snail shells.

Golden Ratio

• Related to Fibonacci numbers.
• Special number found by dividing a line into two parts, where the longer part divided by the smaller part equals the whole length divided by the longer part.
• Approximately 1.6180339887.
• Can be calculated as a/b = (a+b)/a.
• Identified in design, art, architecture, advertising, packaging, engineering, etc.
• A golden mean gauge is used for further understanding.

Nature of Mathematics

• A systematic communication using sounds or standard symbols.
• Bridges different backgrounds.
• Study to comprehend ideas and logic.
• To understand more about the logic of mathematics

Characteristics of Math Language

• Precise.
• Concise.
• Powerful.
• Mathematics uses symbols.

Mathematical Symbols

• Σ: Sum of.
• ∃: There exist.
• ∀: For every/for any.
• ∈: Element of/member of.
• ∉: Not an element of.
• Common symbols: digits (0-9), operations (+, -, ×, ÷), unknowns (x, y, A), special symbols (=, <, >).

Expressions vs. Sentences

• Sentences: have thought, noun, predicate; can be true/false.
• Expressions: contain numbers, variables, operations.
• Expressions are translated into word phrases and vice versa.

Kinds of Sets

• Equal set: same elements.
• Equivalent set: same number of elements.
• Disjoint set: no common elements.
• Joint set: common elements.
• Finite set: countable element with a last element.
• Infinite set: uncountable elements.

Basic Set Theory

• Empty set: no elements.
• Universal set: totality within discussion.
• Subset: elements from another set.
• Combining sets: union (all), intersection (common).
• Functions define element correspondence.

Function

• A rule that connects elements from one set to another.

Relations

• Sets of pairs to represent relationships.

Elementary Logic

• Determines argument validity using reasoning.
• Proposition: is true or false, not both.
• Simple: conveys one idea.
• Compound: conveys multiple ideas

Logical Connectives

• Used to connect propositions.
• Examples include and, or.
• Symbols are used to represent statements, connectives, and types of statements

Negation

• If P is a proposition, negation of P is "not P".
• Truth tables show value relationships.

Conjunction

• "P and Q"; true when both are true.

Disjunction

• "P or Q"; true if one is true.

Conditional

• "If P, then Q"; false if P is true, Q is false, else true.

Biconditional

• "P if and only if Q"; true when values match.
• Connectives can change the meaning.
• Propositions can be combined.

Quantifiers in Mathematics

• Symbols introducing quantities, being finite or infinite.

Universal Quantifier

• "For all" or "for every."

Existential Quantifier

• "There exists."

Tautology

• Always true proposition.

Contradiction

• Always false proposition.

Conditional Proposition Properties

• Converse: switches premise/conclusion.
• Inverse: negates premise/conclusion.
• Contrapositive: negates/switches.

Problem Solving

• Requires mathematical/geometric operations.
• Relies on techniques to derive answers.
• Requires understanding, planning, and looking back.

Problem

• A situation that requires a solution not immediately known.

Exercise

• Requires resolution, but the method is clear.

Inductive Reasoning

• Reaching general conclusions by examining specific cases.

Deductive Reasoning

• Applying assumptions, procedures, principles to conclude.

Key Terms

• Argument: the reason or reasons.
• Premises: serve as its basis, and assumption.
• Syllogism, conclusion.
• Every multiple of 10 is by 4 for instance.

Inductive Reasoning

• It is a true but counterexample is false.

Deductive Reasoning

• conclusion to principles.

Polya's Strategy

• Problem Analysis or four step problem solving; deceptively simple.
• A strategy/sample test that includes understanding the goal, time, inside and the verification.

Data in statistics

Result of evaluating to figure means means and forms

• animal exhibit such as pandas, gorillas each favor people.

Statistics

• Study with the data to decision.

Descriptive/Quantitative in Statistics

• Important to describe the value.
• The collection that helps you to describe the data.

Descriptive & Inferential Statistics

• descriptive deals with the collection and presentation of data
• inferential deals with the predictions and inferences based on the analysis and interpretation

Data classification.

• Divided in to Quantities with qualities and numerical quantities.

Nominal Data

• responses with different categories and no numerical difference between.

Ordinal values in Data

• Categories from the list in order and no numerical distance.

Measurement Scales

• Subdivided and chosen upon interference.

Explore the beauty and utility of patterns and numbers in nature, from the arrangement of petals in a flower to the stripes on a tiger. Discover how mathematics, as a study of patterns, underpins our understanding of the world and facilitates routine transactions.