Key Branches of Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division) involving numbers.
Algebra: Study of symbols and rules for manipulating those symbols; includes solving equations.
Geometry: Focuses on properties and relationships of points, lines, surfaces, and solids.
Trigonometry: Deals with the relationships between the angles and sides of triangles.
Calculus: Study of change through derivatives and integrals; includes limits, functions, and continuity.
Statistics: Involves data collection, analysis, interpretation, presentation, and organization.
Probability: Measures the likelihood of events occurring; foundational for statistics.

Numbers:
- Natural numbers: 1, 2, 3, ...
- Whole numbers: 0, 1, 2, 3, ...
- Integers: ..., -2, -1, 0, 1, 2, ...
- Rational numbers: Can be expressed as a fraction (e.g., 1/2).
- Irrational numbers: Cannot be expressed as a simple fraction (e.g., √2).
- Real numbers: All rational and irrational numbers.
Operations:
- Commutative Property: a + b = b + a; ab = ba.
- Associative Property: (a + b) + c = a + (b + c); (ab)c = a(bc).
- Distributive Property: a(b + c) = ab + ac.

Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Pythagorean Theorem: In right triangles, a² + b² = c², where c is the hypotenuse.
Properties of Shapes:
- Triangles: Sum of angles = 180 degrees.
- Quadrilaterals: Sum of angles = 360 degrees.
- Circles: Circumference = 2πr; Area = πr².

Binomial Theorem: (a + b)ⁿ = Σ (nCk)(a^(n-k))(b^k).
Fundamental Theorem of Algebra: Every non-constant polynomial equation has at least one complex root.
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) for triangle ABC.
Law of Cosines: c² = a² + b² - 2ab*cos(C) for triangle sides a, b, c.

Understand the problem: Read carefully and identify what is being asked.
Devise a plan: Choose strategies such as drawing a diagram or creating equations.
Carry out the plan: Solve using chosen methods and calculations.
Review/Extend: Check results and think about how the solution can be applied or extended.

Science and Engineering: Quantitative analysis, modeling phenomena, and designing structures.
Finance: Calculating interest rates, investments, and risk assessment.
Data Analysis: Using statistical methods for decision making and predictions.
Computer Science: Algorithms, programming, and data structures reliant on mathematical concepts.

Arithmetic is the foundation of mathematics, dealing with fundamental operations like addition, subtraction, multiplication, and division.
Algebra introduces symbols to represent numbers and focuses on manipulating those symbols to solve equations.
Geometry explores the properties and relationships of geometric shapes, including lines, surfaces, and solids.
Trigonometry focuses on angles and their relationship to the sides of triangles.
Calculus studies change, using tools like derivatives and integrals to understand rates of change, limits, and functions.
Statistics encompasses data collection, analysis, interpretation, and presentation, and is vital for extracting meaning from data sets.
Probability measures the likelihood of events happening and provides a foundation for statistical analysis.

Numbers form the building blocks of mathematics:
- Natural numbers: 1, 2, 3... represent counting objects.
- Whole numbers: 0, 1, 2, 3... include zero and are used for counting and ordering.
- Integers: ..., -2, -1, 0, 1, 2... include both positive and negative whole numbers.
- Rational numbers: can be expressed as a fraction (e.g. 1/2) and include all integers.
- Irrational numbers: cannot be expressed as a simple fraction.
- Real numbers: encompass all rational and irrational numbers.
Operations represent how numbers interact:
- Commutative property: Order doesn't matter. (a + b = b + a)
- Associative property: Grouping doesn't matter. (a + b) + c = a + (b + c)
- Distributive property: a(b + c) = ab + ac

Order of operations: Dictates the order in which operations are performed within an equation, usually using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
Pythagorean theorem: a² + b² = c² specifically applies to right-angled triangles, where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.
Properties of shapes:
- Triangles: The sum of internal angles in a triangle is always 180 degrees.
- Quadrilaterals: The sum of internal angles in a quadrilateral is always 360 degrees.
- Circles: Circumference = 2πr and Area = πr²

Binomial theorem: Provides a formula for expanding (a + b)ⁿ, where 'n' is a positive integer, and represents the sum of terms.
Fundamental theorem of algebra: States that every polynomial equation with a degree of 1 or more has at least one complex root.
Law of Sines: Establishes a relationship between the sides and angles of a triangle, a/sin(A) = b/sin(B) = c/sin(C).
Law of Cosines: Relates the sides and angles of a triangle using: c² = a² + b² - 2ab*cos(C).

Understand the problem: Carefully read and identify the core question or task.
Devise a plan: Choose appropriate strategies, such as drawing diagrams, formulating equations, or breaking down the problem into smaller parts.
Carry out the plan: Execute chosen methods and perform necessary calculations.
Review/Extend: Check and interpret the solution, consider how the solution can be applied to similar problems.

Science and Engineering: Mathematics provides the foundation for quantitative analysis, modeling various phenomena, and designing structures.
Finance: Essential for calculations involving interest rates, investments, and risk assessments.
Data Analysis: Statistical methods are used for decision making and predicting outcomes.
Computer Science: Algorithms, programming, and data structures rely on mathematical ideas.