Mathematics in the Modern World 2ndGrading PDF

Preview

Summary

Warning: TT: undefined function: 32Modern MathThe Nature of Mathematics -Set of problem-solving tools -Process of thinking -Study of patterns -Language -ArtWhat is Mathematics -Developed by human mind and culture; a formal system of thought for recognizing, classifying and exploring patters. -Percei...

Description

Modern Math The Nature of Mathematics -Set of problem-solving tools -Process of thinking -Study of patterns -Language -Art

such that each number is the sum of the two preceding ones starting from 0 and 1 (Fn= Fn-1 + Fn-2) Other Patterns- geometric shapes; symmetries

What is Mathematics -Developed by human mind and culture; a formal system of thought for recognizing, classifying and exploring patters. -Perceived as the study of numbers, symbols & equations; an art of geometric shapes & patters; a universal language; a tool in decision-making and problem solving; a way of life to be exact and precise -The language of pattern, measurement & logical rules -The art of interpreting, quantifying, & working with error and uncertainty What is Mathematics For -making conclusions and predictions of the events of the world -describe the natural order and occurrences of the universe -organize patterns and regularities as well as irregularities -control weather & epidemics -provide tools for calculations -provides new questions to think about What is Mathematics All About -numbers, symbols, equations, operations, calculations, abstractions and devising proofs

functions,

How is Mathematics Done -With curiosity -With a penchant for seeking patterns and generalities -With a desire to know the truth -With trial and error -Without fear of facing more questions and patterns to solve Who Uses Mathematics -Applied and Pure Mathematicians -Scientists -Engineers -Data Analysts -Statisticians -Finance Analysts -Medical Practitioners -Technically, EVERYONE Importance of Mathematics -Puts order in disorder -Helps make the world a better place to live in -Helps us become better persons

Math Language Galileo- Philosophy cannot be understood unless one first learns to comprehend the language of mathematics without which it is humanly impossible to understand a single word of it; without theses one is wandering around in a dark labyrinth. Components of a Language Vocabulary- symbols and words Grammar- rules on the use of such symbols Community of People- users and receivers of language Range of Meanings- can be communicated using symbols Language of Mathematics -System used by mathematicians to communicate mathematical ideas among themselves -Consists of a substrate of some natural language using technical term and grammatical conventions that are peculiar to mathematical discourse supplemented by a highly specialized symbolic notation for mathematical formulas Characteristics of Math Language Precise- able to make fine distinctions or definitions Concise- long sentences are made brief Powerful- express complex thoughts with relative ease Mathematical Statements Axiom- basic assumptions of mathematical situations; statements accepted to be true even without proof Conjecture- a statement proposed to be a true statement; believed to be true but have no proof Theorem- statements that have been proven true Lemma- statements used in proving other true statements; a less important theorem that is helpful in the proof of other results Corollary- true statements that are a simple deduction from a theorem or proposition Definition- an explanation of the mathematical meaning of a word Postulate- same as axiom NOTATIONS and SYMBOLS SYMBOL MEANING = Is equal to ≠ Is not equal to ≈ approximately Positive-negative/ plus-minus ± > ≥

Greater than Greater than or equal to

Patterns in Nature Fractals- exhibit similar patters at increasingly small scales called self similarity (or expanding symmetry or unfolding symmetry) Fibonacci Sequence- commonly denoted Fn form a sequence

< ≤ () [] {}

Less than Less than or equal to groupings groupings sets

Modern Math N Z Q R Q’ or Q# No C ∞

U ∩

Natural Numbers Integers Rational Numbers Real Numbers Irrational Numbers Whole Numbers Complex numbers Infinity Is an element of Proper subset subset Union of two sets Intersection of two sets There exists There exists a unique For all implies conditional negation therefore Null set If and only if or and Such that where summation Product Congruent to

F F

T F

F F

Disjunction- propositions is true if at least one disjunct is true P T T F F

Q T F T F

PVQ T T T F

Implication- true except when premise is true and conclusion is false P Q P→Q T T T F T F F F

T F

T F

Biconditional- true if both have the same truth value P Q P↔Q T T T T F F F T F F F T

Similar to Because Not subset Superset Propositions Declarative sentence; either true or false-not both; provable Simple Proposition- conveys a single idea Compound Proposition- conveys two or more ideas Connectives & Truth Values Truth Value of Simple Proposition- either true or false Truth Value of Compound Proposition- depends on components and connectives Connectives Used to form compound propositions from simple propositions Unary- can be used even to only one proposition Binary- requires two propositions Negation- changes truth value of proposition P

Conjunction- proposition is true if both conjuncts are true P Q P^Q T T T T F F

¬P

Truth Table -Shows the truth value of compound proposition/statements for all possible truth values of simple statements -Number of permutations of all possible truth values of simple statements of a compound: 2n, where n= no. of statements Conditional Proposition Converse- Q → P Inverse- ¬P → ¬Q Contrapositive- ¬Q → ¬P -Truth value of contrapositive is the same as truth value of implication -Truth value of converse is the same as the truth value of inverse Quantifiers Existential-a prefix used to assert existence of something Nonmathematical form: Some x are y Some x are not y Universal-either denies existence or asserts that all objects of a given set quantify some condition...