Fundamental Concepts in Mathematics

Numbers and Types:
- Natural Numbers: Positive integers (1, 2, 3, ...)
- Whole Numbers: Natural numbers plus zero (0, 1, 2, 3, ...)
- Integers: Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
- Rational Numbers: Numbers that can be expressed as a fraction of two integers (e.g., 1/2, -3)
- Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π)
- Real Numbers: All rational and irrational numbers
Basic Operations:
- Addition (+): Combining quantities
- Subtraction (-): Finding the difference between quantities
- Multiplication (×): Repeated addition
- Division (÷): Splitting into equal parts or groups
Order of Operations:
- Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) - acronym: PEMDAS.
Algebra:
- Variables: Symbols representing numbers (e.g., x, y)
- Expressions: Combinations of variables and constants (e.g., 3x + 2)
- Equations: Statements of equality (e.g., 3x + 2 = 11)
- Functions: Relations between inputs and outputs (e.g., f(x) = x^2)
Geometry:
- Shapes: Two-dimensional (2D) and three-dimensional (3D)
  - 2D examples: triangles, circles, rectangles
  - 3D examples: spheres, cubes, pyramids
- Angles: Measured in degrees; types include acute (< 90°), right (90°), obtuse (> 90°)
- Perimeter: Total distance around a shape
- Area: Amount of space inside a shape (e.g., A = length × width for rectangles)
- Volume: Amount of space inside a 3D shape (e.g., V = length × width × height for cubes)
Statistics:
- Mean: Average of a set of numbers
- Median: Middle value of a sorted set
- Mode: Most frequently occurring value in a set
- Range: Difference between the highest and lowest values
Probability:
- Measures the likelihood of an event occurring
- Calculated as: P(Event) = Number of favorable outcomes / Total outcomes
Calculus:
- Differentiation: Finding the rate of change of a function
- Integration: Finding the total accumulation (area under the curve)

Key Theorems and Formulas

Pythagorean Theorem: a² + b² = c² (relationship between sides of a right triangle)
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a (solutions to ax² + bx + c = 0)
Area Formulas:
- Triangle: A = 1/2 × base × height
- Circle: A = πr²
Volume Formulas:
- Sphere: V = 4/3πr³
- Cylinder: V = πr²h

Problem-Solving Techniques

Identify the Problem: Clearly define what needs to be solved.
Develop a Plan: Choose appropriate strategies and methods.
Execute the Plan: Carry out calculations and logical steps.
Review/Reflect: Check the results and revise if necessary.

Numbers and Types

Natural Numbers: Positive whole numbers beginning with 1 (e.g., 1, 2, 3, ...).
Whole Numbers: Includes all natural numbers and zero (e.g., 0, 1, 2, 3, ...).
Integers: Encompass all whole numbers and their negative counterparts (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers: Can be expressed as a fraction of two integers (e.g., 1/2, -3).
Irrational Numbers: Cannot be expressed as a fraction (e.g., √2, π).
Real Numbers: Include both rational and irrational numbers.

Basic Operations

Addition (+): Combining quantities to find a sum.
Subtraction (-): Finding the difference between two quantities.
Multiplication (×): Repeated addition of a quantity, often represented by a symbol (e.g., × or *).
Division (÷): Splitting a quantity into equal parts or groups.

Order of Operations

PEMDAS: An acronym used to remember the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Algebra

Variables: Symbols representing unknown numbers, typically letters (e.g., x, y).
Expressions: Combinations of variables, constants, and mathematical operations (e.g., 3x + 2).
Equations: Statements of equality between two expressions (e.g., 3x + 2 = 11).
Functions: Relationships that define an output for every given input, often represented by symbols like f(x) (e.g., f(x) = x²).

Geometry

Shapes: Two-dimensional (2D) figures and three-dimensional (3D) objects.
- 2D Examples: Triangles, squares, circles.
- 3D Examples: Cubes, spheres, pyramids.
Angles: Measured in degrees and classified as acute (< 90°), right (90°), or obtuse (> 90°).
Perimeter: The total distance around a two-dimensional shape.
Area: The amount of space enclosed within a two-dimensional shape, using formulas like A = length × width for rectangles.
Volume: The amount of space occupied by a three-dimensional object, using formulas like V = length × width × height for cubes.

Statistics

Mean: The average of a set of numbers, calculated by summing all values and dividing by their total count.
Median: The middle value in a sorted set of numbers.
Mode: The most frequently occurring value in a set of data.
Range: The difference between the highest and lowest values in a set.

Probability

Probability: A measure of the likelihood that an event will occur, calculated as: P(event) = Number of favorable outcomes / Total number of possible outcomes.

Calculus

Differentiation: A process in calculus used to find the rate of change of a function.
Integration: A process used to find the accumulation of a function, often interpreted as the area under its curve.

Key Theorems and Formulas

Pythagorean Theorem: Relates the sides of a right triangle: a² + b² = c² (where c is the hypotenuse and a and b are the other two sides).
Quadratic Formula: Solves equations of the form ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a.
Area Formulas:
- Triangle: A = 1/2 × base × height.
- Circle: A = πr² (where r is the radius).
Volume Formulas:
- Sphere: V = 4/3πr³.
- Cylinder: V = πr²h (where h is the height).

Problem-Solving Techniques

Identify the Problem: Clearly define the question or situation to be addressed.
Develop a Plan: Choose appropriate strategies, formulas, or methods to solve the problem.
Execute the Plan: Perform the necessary calculations, logical steps, or problem-solving techniques.
Review/Reflect: Check the results for accuracy, revise if necessary, and consider alternative approaches or generalizations.