Basic Concepts of Mathematics

Numbers:
- Natural numbers: 1, 2, 3, ...
- Whole numbers: 0, 1, 2, 3, ...
- Integers: ..., -2, -1, 0, 1, 2, ...
- Rational numbers: Fractions (e.g., 1/2, -3/4)
- Irrational numbers: Non-repeating, non-terminating decimals (e.g., √2, π)
Operations:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)

Shapes:
- 2D: Circle, triangle, square, rectangle.
- 3D: Sphere, cube, cylinder, cone.
Properties:
- Perimeter: Total distance around a shape.
- Area: Measure of space within a shape (e.g., A = l × w for rectangles).
- Volume: Measure of space within a 3D object (e.g., V = l × w × h for cubes).

Definitions:
- Sine (sin), Cosine (cos), Tangent (tan) relate angles to ratios of sides in right triangles.
Key Ratios:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent

Limits: The value that a function approaches as the input approaches a point.
Derivatives: Measure of how a function changes as its input changes (slope of the tangent).
Integrals: Measure of the area under a curve (accumulation of quantities).

Basic Concepts:
- Experiment: An action or process that leads to results.
- Sample Space: All possible outcomes of an experiment.
- Event: A subset of the sample space.
Probability Formula: P(E) = Number of favorable outcomes / Total number of outcomes.

Prime Numbers: Natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
Factors: Numbers that divide another number exactly.
Multiples: The product of a number and an integer.

Statements: Declarative sentences that are either true or false.
Logical Connectives: AND, OR, NOT; used to form compound statements.
Quantifiers: Universal (for all) and existential (there exists) quantifiers.

Numbers:
- Natural numbers include all positive integers starting from 1.
- Whole numbers expand natural numbers by including zero.
- Integers encompass whole numbers and their negative counterparts.
- Rational numbers can be expressed as fractions, including both positive and negative values (e.g., 1/2, -3/4).
- Irrational numbers are those which cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions (e.g., √2, π).
Operations:
- Basic mathematical operations consist of addition, subtraction, multiplication, and division, fundamental for performing calculations.

Variables: Symbols (often x and y) that represent unknown or variable quantities.
Expressions: Combinations of numbers and variables, such as 2x + 3, which can be simplified or evaluated.
Equations: Statements asserting that two expressions are equal, such as 2x + 3 = 7, which can be solved for variable values.
Functions: Relationships defining how each input is related to a specific output, typically expressed as f(x) = x^2.

Shapes:
- Two-dimensional (2D) shapes include circles, triangles, squares, and rectangles.
- Three-dimensional (3D) shapes include spheres, cubes, cylinders, and cones.
Properties:
- Perimeter measures the total distance surrounding a geometric shape.
- Area quantifies the space within a shape, calculated differently based on the shape (e.g., area of rectangles A = l × w).
- Volume calculates the space within a 3D object (e.g., V = l × w × h for cubes).

Definitions:
- Sine (sin), Cosine (cos), and Tangent (tan) are trigonometric functions relating the angles of right triangles to the ratios of their sides.
Key Ratios:
- sin(θ) = opposite side / hypotenuse
- cos(θ) = adjacent side / hypotenuse
- tan(θ) = opposite side / adjacent side

Limits: Determine the value that a function approaches as its input approaches a specific point.
Derivatives: Represent the rate at which a function changes, correlating to the slope of the tangent line at any given point.
Integrals: Calculate the area beneath a curve, providing a method for accumulating quantities over an interval.

Mean: The average value computed by summing all data points and dividing by the total number of points.
Median: The middle value of a data set when organized in ascending order, representing the 50th percentile.
Mode: The value that appears most frequently in a data set, highlighting common occurrences.
Standard Deviation: A measure quantifying the dispersion of data points relative to the mean, indicating how spread out the values are.

Basic Concepts:
- An experiment refers to any process or action that results in distinct outcomes.
- Sample Space encompasses all potential outcomes of an experiment.
- An Event is a subset of the sample space, representing specific outcomes.
Probability Formula: The probability of an event occurring is calculated as P(E) = Number of favorable outcomes / Total number of outcomes.

Prime Numbers: Defined as natural numbers greater than 1 with no divisors other than 1 and themselves, essential for factorization.
Factors: Numbers that can divide another number evenly, leading to integers.
Multiples: Represent the products of a number when multiplied by integers, highlighting repeated addition.

Statements: Declarative sentences that can be categorized as true or false.
Logical Connectives: Words like AND, OR, and NOT are used to build compound statements in logic.
Quantifiers: Universal quantifiers (for all) and existential quantifiers (there exists) help indicate the scope of statements in logic and mathematics.