Overview of Mathematics

Definition: The study of numbers, quantities, shapes, and patterns.
Branches:
- Arithmetic: Basic operations (addition, subtraction, multiplication, division).
- Algebra: Use of symbols and letters to represent numbers in equations.
- Geometry: Study of shapes, sizes, and properties of space.
- Trigonometry: Relationship between angles and sides of triangles.
- Calculus: Study of change and motion; includes derivatives and integrals.
- Statistics: Collection, analysis, interpretation, and presentation of data.
- Probability: Study of randomness and uncertainty.

Numbers:
- Natural Numbers: Positive integers (1, 2, 3, ...).
- Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...).
- 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).
Basic Operations:
- Addition (+): Combining quantities.
- Subtraction (−): Finding the difference between quantities.
- Multiplication (×): Repeated addition of a quantity.
- Division (÷): Splitting a quantity into equal parts.

Expressions: Combinations of numbers, variables, and operations (e.g., 3x + 2).
Equations: Mathematical statements asserting equality (e.g., 2x + 3 = 7).
Inequalities: Statements about the relative size of two values (e.g., x > 3).

Basic Shapes:
- Circle: All points equidistant from a center point.
- Triangle: Three-sided polygon. Types include equilateral, isosceles, and scalene.
- Square: Four equal sides and right angles.
- Rectangle: Opposite sides equal and right angles.
Key Formulas:
- Area of a rectangle: A = length × width
- Area of a triangle: A = 1/2 × base × height
- Circumference of a circle: C = 2πr (r = radius)
- Area of a circle: A = πr²

Key Functions: Sine (sin), Cosine (cos), Tangent (tan).
Relationships:
- sin²θ + cos²θ = 1 (Pythagorean identity).
Applications: Used in modeling periodic phenomena, navigation, and engineering.

Descriptive Statistics: Summarizes data (mean, median, mode, range).
Inferential Statistics: Makes predictions or inferences about a population based on a sample.
Probability: Quantitative measure of the likelihood of an event occurring.
- Basic formula: P(A) = Number of favorable outcomes / Total number of outcomes.

Strategies:
- Understand the problem.
- Devise a plan.
- Carry out the plan.
- Review/extend the solution.

Real-world Uses: Science, engineering, finance, computer science, economics, and everyday decision making.
Critical Thinking: Develops problem-solving skills and logical reasoning.

Numbers are categorized into different types:
- Natural Numbers: Positive integers starting from 1.
- Whole Numbers: Natural numbers that include zero.
- Integers: Whole numbers along with their negatives.
- Rational Numbers: Can be represented as fractions (a/b).
- Irrational Numbers: Cannot be expressed as simple fractions (e.g., √2, π).
Basic Operations define how numbers interact:
- Addition (+): Combines quantities.
- Subtraction (−): Determines the difference between quantities.
- Multiplication (×): Represents repeated addition.
- Division (÷): Divides a quantity into equal sections.

Expressions: Combinations of numbers, variables, and operations (e.g., 3x + 2).
Equations: Mathematical statements asserting equality (e.g., 2x + 3 = 7).
Inequalities: Indicate the relative sizes of two values (e.g., x > 3).

Key shapes studied include:
- Circle: Defined by all points equidistant from a center.
- Triangle: A three-sided polygon categorized as equilateral, isosceles, or scalene.
- Square: Four equal sides with right angles.
- Rectangle: Opposite sides are equal with right angles.
Important formulas:
- Area of a rectangle: A = length × width
- Area of a triangle: A = 1/2 × base × height
- Circumference of a circle: C = 2πr (r = radius)
- Area of a circle: A = πr²

Key trigonometric functions include sine (sin), cosine (cos), and tangent (tan).
Notable relationship: sin²θ + cos²θ = 1 (Pythagorean identity).
Applications extend to modeling periodic phenomena, navigation, and various engineering tasks.

Derivatives: Assess the rate of change of a function.
Integrals: Represent the area under curves or the accumulation of quantities.
Fundamental Theorem of Calculus: Establishes the connection between differentiation and integration.

Descriptive Statistics: Summarizes data using measures like mean, median, mode, and range.
Inferential Statistics: Allows for predictions about a population based on sample data.
Probability: Quantifies the likelihood of an event using the formula P(A) = Number of favorable outcomes / Total number of outcomes.

Effective strategies involve:
- Comprehending the problem.
- Formulating a plan.
- Implementing the plan.
- Reviewing and possibly extending the solution.

Mathematics applies broadly in fields like science, engineering, finance, computer science, economics, and everyday decision making.
Promotes critical thinking by enhancing problem-solving skills and logical reasoning.