Mathematics Basics and Concepts

Numbers:
- Natural Numbers: {1, 2, 3, ...}
- Whole Numbers: {0, 1, 2, 3, ...}
- Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Rational Numbers: Can be expressed as a fraction (e.g., 1/2, 3/4).
- Irrational Numbers: Cannot be expressed as a fraction (e.g., π, √2).
Operations:
- Addition (+), Subtraction (−), Multiplication (×), Division (÷).
- Order of Operations (PEMDAS/BODMAS):
  1. Parentheses/Brackets
  2. Exponents/Orders
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

Variables: Symbols (e.g., x, y) used to represent numbers.
Expressions: Combinations of numbers and variables (e.g., 2x + 3).
Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: Relationships where each input has a single output, often written as f(x).

Shapes:
- 2D: Circle, Triangle, Square, Rectangle.
- 3D: Cube, Sphere, Cylinder, Cone.
Key Properties:
- Area: Measurement of surface (e.g., A = length × width for rectangles).
- Perimeter: Length around a shape (e.g., P = 2(length + width) for rectangles).
- Volume: Space within a 3D object (e.g., V = length × width × height for cubes).

Basic Functions: Sine (sin), Cosine (cos), Tangent (tan).
Relationships in a Right Triangle:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent

Limits: Value that a function approaches as the input approaches some point.
Derivatives: Measure of how a function changes as its input changes (e.g., slope of the tangent line).
Integrals: Represent the area under a curve; reverse process of differentiation.

Descriptive Statistics:
- Mean: Average value.
- Median: Middle value in a sorted data set.
- Mode: Most frequently occurring value.
Probability: Measure of the likelihood that an event will occur; ranges from 0 to 1.

Prime Numbers: Numbers greater than 1 that have no positive divisors other than 1 and themselves (e.g., 2, 3, 5, 7).
Greatest Common Divisor (GCD): Largest number that divides two or more integers.
Least Common Multiple (LCM): Smallest multiple that is exactly divisible by two or more integers.

Mathematical Induction: A technique to prove statements for all integers by proving a base case and an inductive step.
Graphs: Visual presentations of data or functions, frequently used in various branches of mathematics.
Matrices: Rectangular arrays of numbers used for solving systems of equations, among other applications.

Numbers: Different categories of numbers with distinct properties
- Natural Numbers: Positive whole numbers starting with 1
- Whole Numbers: Include 0 and all natural numbers
- Integers: All positive and negative whole numbers, including zero
- Rational Numbers: Can be expressed as a fraction, where numerator and denominator are integers
- Irrational Numbers: Cannot be expressed as a fraction (e.g., pi, square root of 2)
Operations: Fundamental mathematical operations
- Addition (+), Subtraction (-): Combining or removing quantities
- Multiplication (×), Division (÷): Repeated addition or splitting a quantity into equal parts
Order of Operations: A set of rules to solve expressions in a specific order (PEMDAS/BODMAS)
- Parentheses/Brackets (prioritized)
- Exponents/Orders (prioritized)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)

Variables: Symbols (like "x" or "y") used to represent unknown or changing values
Expressions: Combinations of variables, numbers, and operations (e.g., "2x + 3" or "y² - 5")
Equations: Statements that two expressions are equal, showing a relationship (e.g., "2x + 3 = 7")
Functions: Mathematical relationships where each input value has a unique output value, often represented by "f(x)"

Shapes: Geometric figures with specific properties
- 2D Shapes: Flat shapes, such as circles, triangles, squares, and rectangles
- 3D Shapes: Solid shapes, such as cubes, spheres, cylinders, and cones
Key Properties: Characteristics that define shapes
- Area: The measurement of the surface enclosed by a shape (e.g., "length × width" for rectangles)
- Perimeter: The total length of the boundary of a shape (e.g., "2(length + width)" for rectangles)
- Volume: The amount of space a 3D object occupies (e.g., "length × width × height" for cubes)

Basic Functions: Sine (sin), Cosine (cos), and Tangent (tan) relate angles and sides in right-angled triangles
Relationships in a Right Triangle: How trigonometric functions connect angles and sides
- sin(θ) = opposite side / hypotenuse
- cos(θ) = adjacent side / hypotenuse
- tan(θ) = opposite side / adjacent side

Limits: The value a function approaches as its input value gets closer to a specific number
Derivatives: Measure of how a function changes as its input changes (e.g., slope of a tangent line)
Integrals: Represent the area under a curve and are the reverse process of differentiation

Descriptive Statistics: Numerical summaries of data
- Mean: Average value, calculated by summing all values and dividing by the number of values
- Median: Middle value when a data set is sorted
- Mode: The most frequently occurring value in a data set
Probability: The likelihood that an event will occur, expressed as a number between 0 and 1

Prime Numbers: Whole numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7)
Greatest Common Divisor (GCD): The largest number that divides two or more integers without leaving a remainder
Least Common Multiple (LCM): The smallest number that is a multiple of two or more integers

Statements: Sentences that are either true or false
Logical Connectives: Symbols connecting statements to form complex logical expressions
- AND (∧): True only when both connected statements are true
- OR (∨): True if at least one connected statement is true
- NOT (¬): Reverses the truth value of a statement
Truth Tables: Tables that systematically show the truth values of logical expressions for all possible combinations of truth values of the statements involved

Mathematical Induction: A proof technique used to establish statements for all integers by proving a base case and an inductive step
Graphs: Visual representations of data or functions, used in various branches of mathematics
Matrices: Rectangular arrays of numbers, used for solving systems of equations, among other applications.