Fundamental Concepts in Mathematics

Natural Numbers (N): Positive integers (1, 2, 3, …).
Whole Numbers (W): Natural numbers plus zero (0, 1, 2, …).
Integers (Z): Whole numbers including negatives (… -3, -2, -1, 0, 1, 2, 3 …).
Rational Numbers (Q): Numbers that can be expressed as a fraction of integers (1/2, 3, 0.75).
Irrational Numbers: Numbers that cannot be expressed as a simple fraction (π, √2).
Real Numbers (R): All rational and irrational numbers.

Addition (+): Combining numbers; associative, commutative.
Subtraction (−): Taking one number away from another; non-commutative.
Multiplication (×): Repeated addition; associative, commutative.
Division (÷): Splitting into equal parts; non-commutative, undefined for division by zero.

Variables: Symbols representing unknown values (e.g., x, y).
Expressions: Combinations of numbers, variables, and operations (e.g., 3x + 5).
Equations: Statements of equality (e.g., 2x + 3 = 7).
Functions: Relationships where each input has one output (e.g., f(x) = x²).

Shapes:
- 2D: Circles, triangles, rectangles, polygons.
- 3D: Spheres, cubes, cones, cylinders.
Properties:
- Perimeter: Total distance around a shape.
- Area: Measure of space within a shape.
- Volume: Measure of space within a 3D object.

Right Triangle: Key focus; examines relationships between angles and sides.
Functions:
- Sine (sin): Opposite/Hypotenuse.
- Cosine (cos): Adjacent/Hypotenuse.
- Tangent (tan): Opposite/Adjacent.
Theorems:
- Pythagorean Theorem: a² + b² = c² for right triangles.

Descriptive Statistics: Summarizing data (mean, median, mode).
Inferential Statistics: Making predictions or inferences about a population based on a sample.
Probability: Measure of likelihood; expressed between 0 and 1.

Natural Numbers (N): Positive whole numbers beginning with 1.
Whole Numbers (W): Natural numbers along with zero.
Integers (Z): All Whole numbers, including negative whole numbers.
Rational Numbers (Q): Numbers representable as fractions of integers.
Irrational Numbers: Numbers that cannot be represented as simple fractions, such as pi and the square root of 2.
Real Numbers (R): Encompasses both rational and irrational numbers.

Addition (+): Combining numbers to find their sum. It is associative and commutative.
Subtraction (−): Taking one number away from another. This is non-commutative.
Multiplication (×): Repeated addition of a number by itself. This is associative and commutative.
Division (÷): Splitting a number into equal parts. It is non-commutative and undefined when dividing by zero.

2D Shapes: Basic flat geometric shapes including circles, triangles, rectangles, and more complex polygons.
3D Shapes: Three-dimensional shapes like spheres, cubes, cones, and cylinders.
Perimeter: The total distance around a two-dimensional shape.
Area: The measurement of the space enclosed within a two-dimensional shape.
Volume: The measurement of the space enclosed within a three-dimensional object.

Right Triangle: A triangle where one angle measures 90 degrees.
Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite an angle to the length of the side adjacent to that angle.
Pythagorean Theorem: In a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c².

Limits: Examining the behavior of a function as its input approaches a specific value.
Derivatives: Measure the instantaneous rate of change of a function. In essence, they determine the slope of a function at any given point.
Integrals: Used to calculate the area under the curve of a function. This concept helps find the accumulation of quantities.

Descriptive Statistics: Summarizing data using measures like mean, median, and mode to understand the main characteristics of a dataset.
Inferential Statistics: Making predictions or generalizations about a population based on a sample of observations.
Probability: The likelihood of an event occurring, expressed as a number between 0 and 1.

Inductive Reasoning: Drawing general conclusions from specific observations or patterns.
Deductive Reasoning: Applying general principles to reach specific conclusions.

Coordinate System: Typically, the Cartesian plane, utilizing two axes (x and y) to represent points in a two-dimensional space.
Linear Equations: Equations represented by straight lines on a graph, often in the form y = mx + b, where m is the slope and b is the y-intercept.
Quadratic Functions: Functions whose graphs are parabolas, represented as y = ax² + bx + c.

Understand the Problem: Carefully read and interpret the problem to ensure a clear understanding.
Devise a Plan: Outline the steps needed to solve the problem.
Carry Out the Plan: Execute the steps outlined in the plan.
Review and Extend: Check the results for accuracy and explore any potential extensions or further implications of the solution.