MATHEMATICS IN THE MODERN WORLD

CHAPTER 1
Patterns and Numbers
Mathematics
 Formal system on recognizing, classifying, and exploiting patterns.
 Organize and systematize ideas about patterns in nature.
Fibonacci Sequence
 The sequence f1,f2,f3,f4, … which has its first two terms f1 and f2 both equal to 1 and satisfies
thereafter the recursion formula fn = fn-1 + fn-2*
 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …
 Pizano or Leonardo of Pisa, published the Liber Abacci, or “Book of Calculation.”
Golden Ratio
 First called as the Divine Proportion in the early 1500s
 Also known as Golden Section, or Golden Proportion, or Divine Proportion denoted by Phi (ϕ)
 Phi is the initial letter of Phidias’ ( He widely used the golden ratio in his works of sculpture.)
a+b a
 = =1.618034 …=ϕ
a b
Golden Rectangle
 Known as one of the most visually satisfying of all geometric forms
 Fibonacci numbers can be applied to the proportions of a rectangle
Types of Patterns
 Symmetry, a sense of harmonious and beautiful proportion of balance.
o Bilateral Symmetry, can be divided into approximately mirror images of each other
o Radial Symmetry, around a fixed point known as the center (either cyclic or dihedral)
 Spiral, more evident in plants
 Fractals, curve or geometric figure, which has the same statistical character as the whole.
CHAPTER 2
Mathematical Language
 The system used to communicate mathematical ideas.
 Efficient and powerful tool for mathematical expression, exploration, reconstruction after
exploration, and communication.
 Mathematics is a spoken and written natural language for expressing mathematical language.
Expression (Mathematical Expression)
 A finite combination of symbols that is well-defined according to rules
 Correct arrangement of mathematical symbols to represent the object of interest
 Does not contain a complete thought, and cannot be determined if it is true or false
 Some types of expressions are numbers, sets, and functions.
Sentence (mathematical sentence)
 A statement about two expressions, either using numbers, variables, or a combination of both.
 Uses symbols or words like equals, greater than, or less than.
 It is a correct arrangement of mathematical symbols.
 States a complete thought and can be determined whether it’s true or false.
Mathematics Convention
 Fact, name, notation, or usage which is generally agreed upon by mathematicians.
 Latin alphabet is commonly used for simple variables and parameters.

Four Basic Concepts

 Language Sets
 Language of Functions
 Language of Relations
 Language of Binary Operations
Set and Elements
 Set is a well-defined collection of objects (elements or members)

Methods of Writing Sets

 Roster Method, elements of the set are enumerated and separated by a comma
 Rule Method, descriptive phrase is used to describe the elements or members of the set
Some Terms on Sets
 Finite Set, whose elements are limited or countable.
 Infinite Set, whose elements are unlimited or uncountable.
 Unit Set, with only one element. Also called singleton.
  Empty Set, unique set with no elements (null set). Denoted by ∅ or { }
 Universal Set, assumed to be contained in some large fixed set. Denoted by U.

Cardinality
 The number of elements or members in the set, the cardinality of set A is denoted by n(A)
Kinds of Sets
 Subset, if every element of A is also an element of B.
 Proper Subset, if every element of A is in B but there is at least one element of B
that is not in A.

Power Set, s the collection (or sets) of all subsets of S. Denoted by ℘

 Equal Sets, if every element of A is in B and every element of B is in A.

Operations on Sets
 Union, the set of all elements x in U such that x is in A or x is in B.
 Intersection, set of all elements x in U such that x is in A and x is in B.
 Complement, set of all elements x in U such that x is not in A.
 Difference, set of all elements x in U such that x is in A and x is not in B.
 Symmetric Difference, set consisting of all elements that belong to A or to B, but
not to both A and B.
 Disjoint Sets, if they have no elements in common.
Function
 A relation in which, for each value of the first component of the ordered pairs, there is exactly
one value of the second component
Group
 Set of elements, with one operation, that satisfies the following properties:
 The set is closed with respect to the operation.
 The operation satisfies the associative property.
 There is an identity element.
 Each element has an inverse.
 An ordered pair (G, ) where G is a set and is a binary operation on G satisfying the four
properties:
 Closure Property
 Associative Property
 Identity Property
 Inverse Property
Statement (proposition)
 A declarative sentence which is
either true or false, but not both.
 The true value of the statements is the
truth and falsity of the statement.
Formal Propositional
 Written using propositional logic notation, p, q, and r are used to represent statements.
Conjunction

p ∧ q, where ∧ is the symbol for “and.”

 The compound statement “p and q.”


Disjuction

 p ∨ q, where ∨ is the symbol for “or.”

Negation
 Denoted by ~p, where ~ is the symbol for “not.”

Conditional
 The statement p and q is the compound statement “if p then q.”
Biconditional
 Statement p and q is the compound statement “p if and only if q.”

Exclusive-or
 Statement p and q is the compound statement “p exclusive or q.”

Existential Quantifiers

 The statement “there exists an x such that P(x),” is symbolized by ∃x P(x).

 The statement “∃x P(x)”is true if there is at least one value of x for which P(x) is true.
Universal Quantifiers

The statement “for all x, P(x),” is symbolized by ∀x P(x).

The symbol ∀ is called the universal quantifier.


 The statement “∀x P(x)”is true if only if P(x) is true for every
value of x.
CHAPTER 3
Inductive Reasoning
 Drawing a general conclusion from a repeated observation or limited sets of observations of
specific examples.
Conjecture
 Drawing conclusions using inductive reasoning
 May be true or false depending on the truthfulness of the argument
 An unproved proposition that is believed to be true is known as a conjecture

Counterexample
 Statement is a true statement provided that it is true in all cases and it only takes one example to
prove the conjecture is false.

Deductive Reasoning
 Drawing general to specific examples or simply from general case to specific case.
 Starts with a general statement (or hypothesis) and examines to reach a specific conclusion.

Mathematical Intuition
  A reliable mathematical belief without being formalized and proven directly
 Serves as an essential part of mathematics
 Carries a heavy load of mystery and ambiguity and it is not legitimate substitute for a formal
proof.

Proof (example of mathematical logical certainty)

 An inferential argument for a mathematical statement

 Example of exhaustive deductive and inductive reasoning
 Demonstrates that a certain statement is always true in all possible cases

Polya’s Four-Steps in Problem Solving

 George Polya, a mathematics educator who strongly
believed that the skill of problem solving can be taught.
 Step 1: : Understand the problem.
 Step 2: Devise a plan.
 Step 3: Carry out the plan.
 Step 4: Look back

Two Ways to Present the Proof

Outline Form Paragraph Form

Kinds of Proof
  Direct Proof, a mathematical argument that uses rules of inference to derive the conclusion
from the premises. (If P then Q)
1. Assume P is true.
2. Conclusion is true.

 Indirect Proof, a statement to be proved is assumed false if the assumption leads to an

impossibility, then the statement assumed false has been proved to be true. (If not Q then not P)
1. Assume conclusion is true
2. Therefore, P is true
 Proof by Counter Example (Disproving Universal Statement),
 Proof by Contradiction

General Sequence

 Having the first term a1 , the second term is a2 , the third term is a3 , and the nth term, also
called the general term of the sequence, is an

Difference Table

 Shows the differences between successive terms of the sequence