Key Concepts in Mathematics

Natural Numbers (N): Counting numbers (1, 2, 3, ...)
Whole Numbers (W): Natural numbers plus zero (0, 1, 2, ...)
Integers (Z): Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
Rational Numbers (Q): Numbers that can be expressed as a fraction (a/b, where a and b are integers, b ≠ 0)
Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π)
Real Numbers (R): All rational and irrational numbers
Complex Numbers (C): Numbers of the form a + bi (where i is the imaginary unit)

Variables: Symbols (e.g., x, y) representing unknown values.
Expressions: Combinations of numbers, variables, and operations (e.g., 2x + 3).
Equations: Mathematical statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: Relationships between inputs and outputs (e.g., f(x) = 2x + 1).

Shapes and Properties: Points, lines, planes, angles, polygons, circles, etc.
Area: The amount of space inside a shape (e.g., A = lw for rectangles).
Perimeter: The distance around a shape.
Volume: The space occupied by a 3D object (e.g., V = lwh for rectangular prisms).

Sine (sin), Cosine (cos), Tangent (tan): Ratios in right triangles.
Pythagorean Theorem: a² + b² = c² for right triangles.
Unit Circle: A circle with a radius of 1 used to define trigonometric functions.

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

Factoring: Breaking down numbers or expressions into their constituent parts.
Expanding: Distributing terms in algebraic expressions.
Graphing: Visual representation of equations or inequalities on a coordinate system.

Natural Numbers consist of positive integers starting from 1.
Whole Numbers include natural numbers and zero, starting from 0.
Integers encompass all whole numbers and their negative counterparts.
Rational Numbers can be expressed as the quotient of two integers, with a non-zero denominator.
Irrational Numbers cannot be represented as fractions; common examples include √2 and π.
Real Numbers include both rational and irrational numbers, covering all points on the number line.
Complex Numbers comprise real and imaginary parts, represented as a + bi.

Variables symbolize unknown amounts and are often represented by letters (e.g., x, y).
Expressions are combinations of numbers, variables, and operations (e.g., 2x + 3).
Equations assert that two expressions are equal (e.g., 2x + 3 = 7).
Functions express a relation between input and output, commonly noted as f(x) = expression.

Geometry involves the study of shapes, properties, and relationships such as points, lines, and angles.
Area quantifies the space within a two-dimensional shape using formulas (e.g., A = lw for rectangles).
Perimeter measures the total distance around a shape.
Volume assesses the space occupied by three-dimensional objects (e.g., V = lwh for rectangular prisms).

Key ratios in right triangles include sine (sin), cosine (cos), and tangent (tan).
The Pythagorean Theorem states a² + b² = c², establishing a relationship between the sides of right triangles.
The Unit Circle, with a radius of 1, is essential for defining trigonometric functions.

Limits describe the value a function approaches as the input approaches a specific point.
Derivatives signify the rate of change of a function, analogous to the slope of a graph.
Integrals represent the accumulation of quantities, often associated with the area under a curve.

Mean represents the average value within a data set.
Median is the central value when data is ordered.
Mode is the value that appears most frequently in a data set.
Probability quantifies the likelihood of a given event, ranging from 0 (impossible) to 1 (certain).

Identifying the problem lays the groundwork for understanding the query.
Devise a plan to select the most effective strategies or formulas.
Carrying out the plan means implementing chosen methods step by step.
Reviewing and extending verify the solution's accuracy and broaden its applicability.