Mathematics in the Modern World PDF

**Mathematics In the Modern World** **Mathematic as a Language** **Characteristics of Mathematical Language** - Precise (able to make very fine distinctions) - Concise (able to say things briefly) - Powerful (able to express complex thoughts with relative ease) **Vocabulary Vs. Sentences...

**Mathematics In the Modern World** **Mathematic as a Language** **Characteristics of Mathematical Language** - Precise (able to make very fine distinctions) - Concise (able to say things briefly) - Powerful (able to express complex thoughts with relative ease) **Vocabulary Vs. Sentences** **English: Nouns vs. Sentences** - **Nouns** are used to name things we want to talk about people, places, and things - **Sentences** are used to state a complete thought. **Mathematics: Expressions vs. Sentences** - **Expression** is the mathematical analogue of a noun; name given to a mathematical object of interest. - **Sentences** are used to state a complete thought **Expressions (Simple)** - Fewer Symbols - Fewer operations - Better suited for current use - Preferred style/ format **Mathematical Sentences** - Read - Always ask for the truth **Synonyms and Verbs** - For both English and mathematics different names for the same object - **English sentences** have **Verbs.** So do mathematical sentences. The "**=**" is one of the most popular mathematical verbs. **Numbers** - **Natural Numbers** - are the numbers that start from 1 and end at infinity. - **Whole Numbers** - are the numbers without fractions and it is a collection of positive integers and zero. - **Integer** - is a whole number (not a fractional number) that can be positive, negative, or zero. - **Rational numbers** - Can be written as a fraction p/q; decimals either terminate or repeat. - **Irrational numbers** - Cannot be written as a fraction; decimals neither terminate nor repeat. - **Real Numbers** - rational numbers like positive and negative integers, fractions, and irrational numbers. - **Sets** - well defined collection of distinct objects - **Elements** -- Objects that belong in a set.\ **N** -- Set of Natural Numbers\ **W** -- Set of Whole Numbers\ **Z** -- Set of Integers\ **Q** -- Set of Rational Numbers\ **Q\'** -- Set of Irrational Numbers\ **R** -- Set of Real Numbers **Ways of writing a set** - **Tabular of Roster Form** -- Listing elements between braces - **Set Builders Notation** -- Stating a property verified exactly by its elements A= {a\| a ∈ N, a \< 11} - **Finite Set** - a set is finite if it contains a countable number of elements. The number of elements in the set is a whole number. - **Infinite set** - a set is infinite if the counting elements have no end. There is no particular whole number that gives the number of elements in a set. - **Universal Set** -- The set of all elements that are being considered. **U** is used to denote the universal set. - **Empty Set/Null Set** -- Set that contains no elements. {} or ∅ is used to represent the empty set. - **Unit Set** -- Set with only one element. - **Note** -- The set {∅} and {O} are not empty since each contains one element - **Equal Set** -- A set is said to be equal if and only if two set have exactly the same elements. A=B - **Equivalent Set** -- A set is equivalent to another set if and only if these two sets have the same number of elements. A\~B\ Sample A = {1,9,7,5}, B = {2,6,0,1} - **Joint Set** -- Sets that have at least one common element - **Disjoint Set** -- Set that have no common element - **Improper Subset** -- Set A is a subset of set B if every element of A belongs to B.\ Sample A = {1,9,7,5}, B = {1,9,7,5} - **Proper Subset** -- Set A is a proper subset of set B if every element of A belongs to but there is at least one element of B that doesn't belong to A.\ Sample A = {1,9,7,5}, B = {1,9,7,5,2} - **Superset** -- A is a superset of B if A contains all the elements of B. Sample A = {1,9,7,5,2}, B = {9,7,5} - **Power Set** -- The power set of A's denoted by P(A) read as P of A is defined as the set of all subsets of A. The number of subsets of a given set is 2\^n.