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Mathematics

Deals with numbers, operations, measurement, quantity, space, and patterns.

Known as precise, logical, and systematic subject.

No single universally accepted definition.

Snowflakes

Six fold symmetry; no two are the same

Fishes

Spots and stripes

Mammals

Spots, stripes, and blotches

Insects

Stripes and hexagonal patterns

Flying Geese

Flies in formation (V pattern)

Starlings

Tornado-shaped

Mackerels

Ball-shaped pattern

Spiderweb

Logarithmic spiral (innermost to outermost)

Beehive / Honeycomb

Repeated hexagon pattern

Desert Sand

Small waves

Symmetry

Sense of harmonies and beautiful proportion of balance

Bilateral Symmetry

Left and right sides can be divided into two; mirror image along the midline

Examples: Humans, Insects

Radial Symmetry

Fixed point known as center and be either cyclic or dihedral (rotational balance)

Examples: Fruits, Jellyfish

Spirals (Logarithmic Spiral)

Growth spiral or self-similar curve.

First described by Rene Descartes, later investigated by Jacob Bernoulli.

Examples: Cabbage, Vegetables, Typhoons

Fibonacci Sequence

Formed by adding the preceding two numbers beginning with 0 and 1.

Pattern should be consistent

The Man Behind the Fibonacci Sequence

Named Leonardo Pisano Bogollo.

Fibonacci was his nickname meaning “Son of Bonacci.”

Introduced Hindu-Arabic numerals and Roman numerals in his book Liber Abaci

Golden Ratio

Two quantities are in golden ratio if their ratio is the ratio of their sum to the larger of the two quantities.

Successive Fibonacci numbers will give you the golden ratio

Golden Rectangle

1:1.618 (should be equal to this).

Pleasing shape and frequently found in art and architecture that is “right” to the eye.

Involves first five Fibonacci numbers

Mathematics language is non-temporal

No past, present, or future; statements are timeless.

All is just as is

Mathematics language is emotionless

Lacks emotional content; only logic and precision

Mathematics language is precise

Exact meaning, no hidden agenda, or unspoken cultural conclusions

Mathematics language is concise

Expresses ideas briefly and clearly

Mathematics language is powerful

Can describe complex concepts with ease

Expression

Numbers are the most common type of mathematical expression.

A particular number may have several names.

Like a noun.

Cannot be true or false.

Includes numbers, sets, and functions

Sentences

English sentences have verbs; it also applies in mathematics.

Object of interest.

Can be true or false.

Example: “3 + 5 = 8.”

Set

Well-defined group of objects.

Relations

Subset of Cartesian product (M × N)

Function

Relation with exactly one output per input

Binary Operation

Rule combining two elements → one element (e.g., +, ×)

Logic

Science of correct reasoning (from Greek word logos)

Statements

Declarative sentence that is either true or false (not both).

Example: “0 is an even number” = True.

“Is it raining?” = Not a statement (false)

Negation (~ or ¬)

Opposite of truth value.

Example: “0 is an even number” → “0 is not an even number.”

Simple statements

One/single idea

Compound statements

Two or more ideas joined by connectives

~ (not): Negation

false

∧ (and): Conjunction

true if both true

∨ (or): Disjunction

true if at least one is true

(if…then): Conditional

false only if the first is true and the second is false

(if and only if): Biconditional

true only if both have the same truth value

Truth Table

Tool to determine truth values of compound statements.

Example:

p ∧ q → only true if p = T and q = T.

p → q → false if p = T and q = F

Inductive reasoning

Specific examples → generalization (conjecture)

Deductive reasoning

General law → specific conclusion

Counterexample

One false case disproves a statement

George Polya

Born (1887–1985).

Born in Hungary and moved to the United States in 1940

Polya’s Four-Step Problem-Solving Strategy

1. Understand the problem → What is asked, what is given.

2. Devise a plan → List, diagram, pattern, work backward, equations.

3. Carry out the plan → Solve carefully.

4. Review the solution → Check and generalize

Problem-Solving Strategies

Organized list.

Draw a diagram.

Simplify problem.

Guess and check.

Look for pattern.

Use equation.

Applications

Patterns (number sequence).

Logic puzzles (roles, schedules, assignments).

Recreational math (games, riddles)

Conventions in Mathematics

The mathematical counterpart of a “noun” is called an expression.

An expression is a label given to a mathematical object of interest.

The object of interest in mathematics may be numbers, sets, etc.

The mathematical counterpart of a “sentence” is also called a sentence

Importance of Mathematics

Used in finances, money, adapt new things, experimenting.

Used to add, to subtract, multiply, divide.

Helps to develop an analytical mind.

Better organization of ideas and accurate expression of thoughts.

Helps recreational activities (sports).

Reveals secret of nature (symmetry).

Mathematics is around us (all human activities).

Importance of mathematics for a common man is underpinned

Universal Quantifiers

All, every, none, no.

Deny the existence of something (no/none).

Assert that every element satisfies some condition (all/every)

Existential Quantifiers

Some, there exists, at least one.

Assert the existence of something

Negation of Quantified Statements

“All X are Y” → “Some X are not Y.”

“No X are Y” → “Some X are Y.”

“Some X are Y” → “No X are Y.”

“Some X are not Y” → “All X are Y.”

Four Basic Concepts

1. Set

2. Relations

3. Function

4. Binary Operation

Operations

Union (∪), Intersection (∩), Difference (−), Complement (A’).

Subsets

Proper (⊂), Improper (∅, itself)

Power Set

P(A) = all subsets (2ⁿ elements)