Key Concepts in Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division).
Algebra: Symbols and rules for manipulating them; solving equations and inequalities.
Geometry: Study of shapes, sizes, and properties of space; involves points, lines, angles, and surfaces.
Trigonometry: Relationships between angles and lengths in triangles; sine, cosine, tangent functions.
Calculus: Study of change and motion; involves limits, derivatives, and integrals.
Statistics: Collection, analysis, interpretation, presentation, and organization of data.
Probability: Study of randomness and uncertainty; likelihood of events occurring.

Order of Operations: PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication & Division, Addition & Subtraction).
Properties of Numbers:
- 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.

Pythagorean Theorem: In a right triangle, a² + b² = c².
Fundamental Theorem of Algebra: Every non-constant polynomial equation has at least one complex root.
Prime Number Theorem: Describes the asymptotic distribution of prime numbers.

Linear Algebra: Study of vectors, matrices, and linear transformations.
Abstract Algebra: Study of algebraic structures such as groups, rings, and fields.
Real Analysis: Examination of real numbers and real-valued functions; focuses on limits, continuity, and convergence.
Complex Analysis: Study of functions that operate on complex numbers; includes concepts like analytic functions and residues.

In Science: Used for modeling and solving scientific problems (e.g., physics, chemistry).
In Finance: Application of algorithms and formulas for calculations in economics and investment.
In Technology: Algorithms and computational mathematics are fundamental in computer science.

Arithmetic involves basic operations such as addition, subtraction, multiplication, and division.
Algebra utilizes symbols and rules to manipulate them, primarily used for solving equations and inequalities.
Geometry focuses on shapes, sizes, and properties of space, including points, lines, angles, and surfaces.
Trigonometry explores relationships between angles and lengths in triangles, using functions such as sine, cosine, and tangent.
Calculus studies change and motion using limits, derivatives, and integrals.
Statistics involves collecting, analyzing, interpreting, presenting, and organizing data.
Probability deals with randomness and uncertainty, focusing on the likelihood of events occurring.

Order of Operations: PEMDAS/BODMAS is used to determine the order in which calculations are done.
Properties of Numbers:
- Commutative Property: Order doesn't matter in addition and multiplication, a + b = b + a and ab = ba.
- Associative Property: Grouping doesn't matter when adding or multiplying, (a + b) + c = a + (b + c) and (ab)c = a(bc).
- Distributive Property: Multiplication distributes over addition, a(b + c) = ab + ac.

Pythagorean Theorem: In a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b), a² + b² = c².
Fundamental Theorem of Algebra: Every non-constant polynomial equation has at least one complex root.
Prime Number Theorem: Describes the asymptotic distribution of prime numbers.

Linear Algebra: Studies vectors, matrices, and linear transformations.
Abstract Algebra: Focuses on algebraic structures such as groups, rings, and fields.
Real Analysis: Examines real numbers and real-valued functions, focusing on limits, continuity, and convergence.
Complex Analysis: Studies complex functions and their properties, including analytic functions and residues.

Science: Mathematics is essential for modeling and solving scientific problems in fields like physics and chemistry.
Finance: Algorithms and mathematical formulas are used extensively in economics and investment.
Technology: Computational mathematics and algorithms are fundamental to computer science.