Fundamental Concepts of Math

Numbers
- Natural Numbers: {1, 2, 3, ...}
- Whole Numbers: {0, 1, 2, ...}
- Integers: {..., -2, -1, 0, 1, 2, ...}
- Rational Numbers: Fractions that can be expressed as a/b where a and b are integers and b ≠ 0.
- Irrational Numbers: Cannot be expressed as a fraction (e.g. √2, π).
Operations
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)
Order of Operations
- Parentheses
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
- Acronym: PEMDAS

Variables: Symbols representing numbers (e.g. x, y).
Expressions: Combinations of numbers and variables (e.g. 2x + 3).
Equations: Statements of equality (e.g. 2x + 3 = 7).
Functions: Relations that assign exactly one output for each input (e.g. f(x) = 2x + 1).

Points, Lines, and Planes
- Point: A location with no dimensions.
- Line: A straight path extending infinitely in both directions.
- Plane: A flat surface extending infinitely.
Shapes and Properties
- Triangles: Types include equilateral, isosceles, and scalene.
- Quadrilaterals: Types include squares, rectangles, and trapezoids.
- Circles: Properties include radius, diameter, and circumference (C = πd).

Data Types
- Qualitative (categorical) and Quantitative (numerical).
Measures of Central Tendency
- Mean: Average of data.
- Median: Middle value when data is arranged.
- Mode: Most frequently occurring value.
Data Representation
- Charts: Bar graphs, line graphs, pie charts.

Basic Concepts
- Probability: Likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).
- Formula: P(Event) = Number of favorable outcomes / Total number of outcomes.
Types of Events
- Independent Events: The outcome of one does not affect another.
- Dependent Events: The outcome of one affects the other.

Limits: The value that a function approaches as the input approaches some value.
Derivatives: Measures the rate at which a function changes (slope of the tangent line).
Integrals: Represents the area under a curve; can be definite or indefinite.

Logical Statements
- Proposition: A declarative statement that can be true or false.
- Logical Connectives: AND, OR, NOT, implication (if-then).
Proof Techniques
- Direct Proof: Proving the statement directly.
- Indirect Proof: Assuming the opposite to reach a contradiction.
- Proof by Induction: Proving a statement for all natural numbers.

Natural numbers are positive whole numbers starting from 1: {1, 2, 3,...}
Whole numbers include zero and all natural numbers: {0, 1, 2,...}
Integers encompass all positive and negative whole numbers, including zero: {..., -2, -1, 0, 1, 2,...}
Rational numbers can be expressed as fractions where the numerator and denominator are integers and the denominator is not zero.
Irrational numbers cannot be expressed as fractions, for example, the square root of 2 (√2) and pi (π).
Basic mathematical operations include addition (+), subtraction (-), multiplication (×), and division (÷).

The order of operations acronym PEMDAS helps remember the correct sequence:
- Parentheses
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)

Variables represent unknown numbers using letters like x or y.
Expressions combine numbers and variables using operations.
Equations state that two expressions are equal.
Functions relate inputs to outputs, assigning exactly one output for each input.

Triangles can be classified into equilateral, isosceles, and scalene based on side lengths.
Quadrilaterals include shapes like squares, rectangles, and trapezoids.
Circles have properties like radius, diameter, and circumference (C = πd).

Data can be categorized as qualitative (categorical) or quantitative (numerical).
Measures of central tendency include the mean (average), median (middle value), and mode (most frequent value).

Probability refers to the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).
The probability formula calculates the probability of an event by dividing the number of favorable outcomes by the total number of outcomes.

Limits describe the value a function approaches as its input gets closer to a specific value.
Derivatives measure how fast a function changes, representing the slope of the tangent line.
Integrals calculate the area under a curve and can be definite or indefinite.

Propositions are declarative statements that can be either true or false.
Logical connectives like AND, OR, NOT, and implication (if-then) combine propositions.

Direct proof proves a statement directly.
Indirect proof assumes the opposite of the statement and derives a contradiction.
Proof by induction proves a statement for all natural numbers by proving its truth for the first case and then assuming its truth for a general case.