Mathematics in the Modern World Midterms PDF

Mathematics in the Modern World MATHEMATICS study of numbers and arithmetic operations answers questions like "how many" or "how much" about patterns, language, problem-solving, and art PATTERNS way to predict what happens next based on arrangement Types of Patterns...

Mathematics in the Modern World MATHEMATICS study of numbers and arithmetic operations answers questions like "how many" or "how much" about patterns, language, problem-solving, and art PATTERNS way to predict what happens next based on arrangement Types of Patterns Logic Pattern o involves classifying or enumerating things o first type we usually notice Geometric Pattern o deals with shapes (lines, circles, polygons) o like wallpapers Number Pattern o first pattern kids learn in school Word Pattern o found in language (like rules for plurals or verb tenses) o focuses on form and syntax MATHEMATICS IN NATURE Vitruvian Man by Leonardo da Vinci: Shows proportion and symmetry in the human body. Types of Symmetry Three-fold Four-fold Five-fold Six-fold FIBONACCI SEQUENCE numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21… found in nature (e.g., pine cones with spirals). created by Leonardo of Pisa GOLDEN RATIO About 1.618 Ratios of Fibonacci numbers get closer and closer to this value LANGUAGE system for communication using sounds or symbols. code humans use to express themselves and communicate with others. LANGUAGE COMPONENTS Vocabulary – Symbols or words. Grammar – Rules for using these symbols. Community – People who use and understand the symbols. Meaning – Range of ideas that can be expressed. MATHEMATICS AS A LANGUAGE A way of communicating about numbers, variables, sets, and operations. Uses symbols and their meanings understood globally, regardless of ethnicity. Characteristics: o Precise – Clear and accurate. o Concise – Simple and straight to the point. o Powerful – Expresses complex ideas effectively. KEY TERMS Expression: Name for a mathematical object (e.g., number, set, or function). Sentence: A statement that conveys a complete thought. ENGLISH vs. MATHEMATICS ENGLISH MATHEMATICS Noun: Person, place, thing Expression: Number, set, function Sentence: True (T), False (F), Sometimes T/F Sentence: True (T), False (F), Sometimes T/F LOGIC Helps determine if arguments are valid. Highlights the need for precision and conciseness in math. PROPOSITION A declarative sentence that’s either true or false. Paradox: A self-contradictory proposition. TYPES OF PROPOSITION Simple Proposition – Conveys one thought with no connecting words. Compound Proposition – Combines two or more propositions with connective words. TYPES OF COMPOUND PROPOSITION Conjunction – Uses “and”. Disjunction – Uses “or”. Conditional – “If... then...” statements. Biconditional – “If and only if”. Negation – Opposite of the original statement.