Overview of Mathematics

Definition: Mathematics is the study of numbers, shapes, patterns, and their relationships.
Branches:
- Arithmetic: Basic operations like addition, subtraction, multiplication, and division.
- Algebra: Study of symbols and rules for manipulating those symbols; solving equations.
- Geometry: Study of shapes, sizes, and properties of space.
- Trigonometry: Study of relationships between angles and sides of triangles.
- Calculus: Study of change, involving derivatives and integrals.
- Statistics: Study of data collection, analysis, interpretation, and presentation.
- Discrete Mathematics: Study of mathematical structures that are fundamentally discrete rather than continuous.

Numbers:
- Natural Numbers: Counting numbers (1, 2, 3, ...).
- Integers: Whole numbers that can be negative, zero, or positive.
- Rational Numbers: Numbers that can be expressed as a fraction of two integers.
- Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., π, √2).
- Real Numbers: All rational and irrational numbers.
- Complex Numbers: Numbers that have a real part and an imaginary part (e.g., a + bi).
Functions:
- Definition: A relation between a set of inputs and a set of outputs.
- Types:
  - Linear Functions: Functions that graph as a straight line.
  - Quadratic Functions: Functions that graph as a parabola.
  - Exponential Functions: Functions that involve exponents.
  - Logarithmic Functions: Inverse of exponential functions.

Basic Operations: Addition (+), Subtraction (−), Multiplication (×), Division (÷).
Order of Operations: PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Pythagorean Theorem: In a right triangle, a² + b² = c², where c is the hypotenuse.
Fundamental Theorem of Algebra: Every non-constant polynomial has as many roots as its degree.
Central Limit Theorem: The distribution of sample means approaches a normal distribution as the sample size increases.

Numbers include various types:
- Natural Numbers: Counting numbers beginning from 1.
- Integers: Whole numbers that can be negative, zero, or positive.
- Rational Numbers: Numbers expressible as a fraction of two integers.
- Irrational Numbers: Numbers that cannot be written as a simple fraction, such as π or √2.
- Real Numbers: The combination of rational and irrational numbers.
- Complex Numbers: Include a real part and an imaginary part, represented as a + bi.
Functions:
- Definition: A relationship between sets of inputs and outputs.
- Common types include:
  - Linear Functions: Represented graphically by a straight line.
  - Quadratic Functions: Form parabolas when graphed.
  - Exponential Functions: Involve exponents, showing rapid growth or decay.
  - Logarithmic Functions: Serve as the inverse of exponential functions.

Basic operations in mathematics are addition (+), subtraction (−), multiplication (×), and division (÷).
Order of Operations: Follow PEMDAS/BODMAS rules to determine computation sequence:
- Parentheses/Brackets
- Exponents/Orders
- Multiplication and Division
- Addition and Subtraction

Pythagorean Theorem: In a right triangle, the relationship a² + b² = c² holds true, where c is the hypotenuse.
Fundamental Theorem of Algebra: Asserts that every non-constant polynomial will have as many roots as its degree.
Central Limit Theorem: States that as sample size increases, the distribution of sample means approaches a normal distribution.

Essential in various fields such as:
- Science and Engineering: Utilized in physics, chemistry, and design engineering.
- Economics: Employed in financial modeling and market analysis statistics.
- Computer Science: Important for algorithms, data structures, and cryptography.
- Daily Life: Applied in budgeting, cooking measurements, and construction planning.