Basic Concepts of Mathematics

Operations: Addition, subtraction, multiplication, division
Properties:
- Commutative: a + b = b + a; a * b = b * a
- Associative: (a + b) + c = a + (b + c); (a * b) * c = a * (b * c)
- Distributive: a * (b + c) = a * b + a * c

Shapes: Points, lines, angles, polygons, circles
Formulas:
- Area: A = l * w (rectangle), A = πr² (circle)
- Perimeter: P = 2(l + w) (rectangle), P = 2πr (circle)
Theorems: Pythagorean theorem (a² + b² = c² in right triangles)

Functions: Sine (sin), cosine (cos), tangent (tan)
Relationships:
- sinθ = opposite/hypotenuse
- cosθ = adjacent/hypotenuse
- tanθ = opposite/adjacent
Applications: Used in calculating angles and distances in various fields

Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves
Factors and Multiples: Factorization, least common multiple (LCM), greatest common divisor (GCD)
Modular Arithmetic: System for integers where numbers "wrap around" upon reaching a certain value (modulus)

Logic: Use of formal reasoning to derive conclusions
Proofs: Techniques to establish the validity of mathematical statements (direct, indirect, contradiction)
Sets: Collections of distinct objects, operations like union, intersection, and difference

Science and Engineering: Modeling natural phenomena, solving practical problems
Finance: Interest calculations, risk assessments, and market predictions
Computer Science: Algorithms, data structures, and cryptography

Calculators: For complex computations
Software: MATLAB, Python, R for advanced calculations and simulations
Graphs: Visual representation of data and functions to analyze relationships

Arithmetic deals with the basic operations of addition, subtraction, multiplication, and division.
Key properties of arithmetic operations include commutativity (order doesn't matter), associativity (grouping doesn't matter), and distributivity (distributing multiplication over addition).

Algebra uses variables (symbols like 'x' or 'y') to represent unknown values.
Expressions are combinations of variables and constants (e.g., 2x + 3).
Equations are statements of equality (e.g., 2x + 3 = 7).
Factoring involves breaking down expressions into simpler factors.

Geometry focuses on shapes, including points, lines, angles, polygons, and circles.
Area and perimeter formulas are used to calculate the size and boundary of shapes. For example, the area of a rectangle is length times width (A = l * w), and the perimeter is twice the sum of length and width (P = 2(l + w)).
Theorems like the Pythagorean Theorem (a² + b² = c² in right triangles) provide fundamental relationships between sides of triangles.

Trigonometry uses functions like sine (sin), cosine (cos), and tangent (tan) to relate angles and sides of right triangles.
Relationships between these functions include: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, and tanθ = opposite/adjacent.
Trigonometry has applications in various fields, including calculating angles and distances in engineering and astronomy.

Calculus focuses on how functions change.
Limits define a function's behavior as inputs approach a specific value.
Derivatives measure the rate of change of a function.
Integrals calculate the accumulation of quantities (e.g., areas under curves).
The Fundamental Theorem of Calculus establishes a fundamental connection between differentiation and integration.

Descriptive Statistics provides summary measures like mean, median, mode, and range to understand data.
Probability deals with uncertainty, focusing on the likelihood of events.
Different distributions (like normal and binomial) model how data is spread out.
Inferential Statistics allows us to draw conclusions from data and perform hypothesis testing.

Prime Numbers are greater than 1 and only divisible by 1 and themselves.
Factors and Multiples involve finding divisors and multiples of numbers.
Modular Arithmetic is a system where numbers "wrap around" after reaching a particular value (the modulus).

Logic is a formal system of reasoning used to derive conclusions.
Proofs provide rigorous methods for establishing the validity of mathematical statements using various techniques like direct proof, indirect proof, and contradiction.
Sets are collections of distinct objects, and operations like union, intersection, and difference are used to manipulate them.

Mathematics has extensive applications in various fields, including:
- Science and engineering: Modeling natural phenomena and solving practical problems.
- Finance: Calculating interest, assessing risk, and predicting market trends.
- Computer science: Developing algorithms, data structures, and cryptography.

Calculators assist in complex computations.
Software like MATLAB, Python, and R provide capabilities for advanced calculations and simulations.
Graphs are visual representations of data and functions, helping to analyze relationships and trends.