Overview of Mathematics

Definition: The study of numbers, shapes, patterns, and their relationships.
Branches:
- Arithmetic: Basic operations (addition, subtraction, multiplication, division).
- Algebra: Use of symbols and letters to represent numbers and quantities in formulas and equations.
- Geometry: Study of shapes, sizes, and properties of space.
- Calculus: Study of change (differentiation and integration).
- Statistics: Collection, analysis, interpretation, and presentation of data.
- Probability: Study of randomness and uncertainty.

Numbers:
- Natural Numbers: Counting numbers (1, 2, 3,...).
- Integers: Whole numbers, including negatives (..., -3, -2, -1, 0, 1, 2, 3,...).
- Rational Numbers: Fractions and ratios of integers.
- Irrational Numbers: Non-repeating, non-terminating decimals (e.g., √2, π).
- Real Numbers: All rational and irrational numbers.
- Complex Numbers: Numbers that include a real part and an imaginary part (a + bi).
Operations:
- Addition (+): Combining quantities.
- Subtraction (−): Finding the difference between quantities.
- Multiplication (×): Scaling one number by another.
- Division (÷): Splitting into equal parts.

Pythagorean Theorem: a² + b² = c² for right triangles.
Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely factored into prime numbers.
Binomial Theorem: Describes the algebraic expansion of powers of a binomial.
Central Limit Theorem: States that the distribution of sample means approximates a normal distribution as sample size increases.

Area:
- Rectangle: A = length × width
- Circle: A = πr²
- Triangle: A = 1/2 × base × height
Volume:
- Cube: V = side³
- Sphere: V = 4/3πr³
Speed: Speed = Distance / Time

Problem Solving: Analyzing and finding solutions to a variety of problems.
Critical Thinking: Evaluating and synthesizing information to make decisions.
Proofs: Logical arguments that validate mathematical statements.
Modeling: Using mathematical concepts to represent real-world scenarios.

Science and Engineering: For modeling phenomena and solving engineering problems.
Finance: In budgeting, investing, and economic analysis.
Technology: Algorithms and data analysis in computer science.
Everyday Life: Budgeting, cooking, shopping, and time management.

Types of Numbers:
- Natural Numbers: The set of positive counting numbers starting from 1.
- Integers: Whole numbers that include both positive and negative values, as well as zero.
- Rational Numbers: Represented as fractions or ratios of integers.
- Irrational Numbers: Numbers with non-repeating, non-terminating decimal expansions (e.g., √2, π).
- Real Numbers: The complete set including both rational and irrational numbers.
- Complex Numbers: Consist of a real part and an imaginary part, expressed in the form a + bi.
Basic Operations:
- Addition (+): Combines two or more quantities into a total.
- Subtraction (−): Computes the difference between quantities.
- Multiplication (×): Involves scaling one number by another.
- Division (÷): Distributes a quantity into equal segments.

Pythagorean Theorem: Establishes the relationship in right triangles as a² + b² = c².
Fundamental Theorem of Arithmetic: Asserts that every integer greater than 1 can be distinctively factored into primes.
Binomial Theorem: Outlines the algebraic expansion for powers of a binomial expression.
Central Limit Theorem: Indicates that sample means form a normal distribution as sample size increases.

Area Calculations:
- Rectangle: A = length × width
- Circle: A = πr²
- Triangle: A = 1/2 × base × height
Volume Measurements:
- Cube: V = side³
- Sphere: V = 4/3πr³
Speed Formula: Speed = Distance / Time

Problem Solving: Involves assessing and devising solutions for various mathematical challenges.
Critical Thinking: Engages in evaluation and synthesis of information for informed decision-making.
Proofs: Structured logical arguments validating mathematical propositions.
Modeling: Applies mathematical concepts to represent and analyze real-world situations.

Science and Engineering: Essential in modeling scientific phenomena and solving engineering challenges.
Finance: Integral for budgeting, investment decisions, and economic strategies.
Technology: Underlies algorithms and data analysis practices in computer science.
Everyday Life: Utilized in managing budgets, cooking recipes, shopping calculations, and scheduling.

Euclid: Known as the "Father of Geometry" for his work in the field, notably in "Elements".
Isaac Newton: Co-founder of calculus, contributing to mathematical analysis of motion and change.
Carl Friedrich Gauss: Made significant contributions to number theory and statistics.
Alan Turing: Pioneer in computational mathematics and theoretical computer science.