Key Concepts in Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division).
Geometry: Study of shapes, sizes, and properties of space.
Algebra: Use of symbols and letters to represent numbers in equations.
Calculus: Study of change (derivatives and integrals).
Statistics: Collection, analysis, interpretation, presentation of data.
Probability: Study of randomness and uncertainty.
Discrete Mathematics: Study of mathematical structures that are fundamentally discrete rather than continuous.

Natural Numbers: Positive integers (1, 2, 3, ...).
Whole Numbers: Natural numbers including zero (0, 1, 2, ...).
Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).
Rational Numbers: Numbers that can be expressed as a fraction (a/b, where b ≠ 0).
Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
Real Numbers: All rational and irrational numbers.

Pythagorean Theorem: In a right triangle, ( a^2 + b^2 = c^2 ) (where c is the hypotenuse).
Fundamental Theorem of Algebra: Every non-constant polynomial equation has at least one complex root.
Central Limit Theorem: Distribution of sample means approximates a normal distribution as sample size increases.

Perimeter: Total distance around a shape.
Area: Size of a surface (e.g., rectangle: length × width).
Volume: Space occupied by a 3D object (e.g., cube: side³).
Angles: Measured in degrees; types include acute (< 90°), right (= 90°), obtuse (> 90°).

Equations: Mathematical statements asserting equality (e.g., ( ax + b = c )).
Functions: Relationships between inputs and outputs (e.g., ( f(x) = mx + b )).
Polynomials: Expressions of variables raised to non-negative integer powers.

Limits: Value that a function approaches as the input approaches a certain point.
Derivatives: Measure of how a function changes as its input changes.
Integrals: Measure of the area under a curve.

Experiment: An action or process that leads to one or more outcomes.
Event: A set of outcomes of an experiment.
Probability Formula: ( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} ).

These notes provide a structured overview of essential mathematics concepts and principles, serving as a useful reference for study and review.

Arithmetic encompasses basic operations: addition, subtraction, multiplication, and division.
Geometry involves the study of shapes, sizes, and spatial properties.
Algebra uses symbols and letters to create and solve equations.
Calculus focuses on the study of change, specifically through derivatives and integrals.
Statistics involves collecting, analyzing, interpreting, and presenting data.
Probability studies randomness, uncertainty, and the likelihood of events.
Discrete Mathematics examines mathematical structures that are fundamentally distinct and separate rather than continuous.

Natural Numbers are positive integers starting from 1 (e.g., 1, 2, 3).
Whole Numbers include natural numbers and zero (e.g., 0, 1, 2).
Integers encompass whole numbers and their negative counterparts (..., -2, -1, 0, 1, 2).
Rational Numbers can be expressed as the ratio of two integers (a/b, b ≠ 0).
Irrational Numbers cannot be expressed as simple fractions (examples include √2 and π).
Real Numbers combine all rational and irrational numbers.

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).
The Fundamental Theorem of Algebra asserts every non-constant polynomial equation has at least one complex root.
The Central Limit Theorem indicates that as the sample size increases, the distribution of sample means approximates a normal distribution.

Perimeter is the total distance around a geometric shape.
Area refers to the quantity of space within a two-dimensional shape, such as length multiplied by width for rectangles.
Volume measures the space occupied by a three-dimensional object, as calculated for a cube by side³.
Angles are measured in degrees; include types such as acute (< 90°), right (= 90°), and obtuse (> 90°).

Equations represent mathematical statements of equality, such as ( ax + b = c ).
Functions define specific relationships between inputs and outputs, exemplified by ( f(x) = mx + b ).
Polynomials consist of terms with variables raised to non-negative integer powers.

Limits refer to the value a function approaches as the input nears a specific point.
Derivatives quantify how a function's output changes when its input varies.
Integrals calculate the area under a curve, emphasizing accumulation and total values.

An Experiment is an action or process that results in one or more outcomes.
An Event consists of a specific set of outcomes derived from an experiment.
The Probability Formula calculates the likelihood of an event occurring: ( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} ).