Key Concepts in Mathematics

Arithmetic
- Basic operations: addition, subtraction, multiplication, division
- Order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)
Algebra
- Variables and constants
- Expressions, equations, and inequalities
- Functions: linear, quadratic, polynomial
- Solving for x: methods include substitution, elimination, and factoring
Geometry
- Basic shapes: triangles, rectangles, circles, polygons
- Properties of angles: complementary, supplementary, vertical, adjacent
- Theorems: Pythagorean theorem, properties of congruence and similarity
- Area and perimeter formulas for various shapes
Trigonometry
- Functions: sine, cosine, tangent and their reciprocals
- Right triangle properties
- Unit circle and radians
- Trigonometric identities: Pythagorean, angle sum, double angle
Statistics
- Data collection methods: surveys, experiments, observational studies
- Measures of central tendency: mean, median, mode
- Measures of variability: range, variance, standard deviation
- Probability concepts: independent and dependent events, combinations, and permutations
Calculus
- Limits and continuity
- Derivatives: definitions, applications (tangent lines, optimization)
- Integrals: definite and indefinite integrals, area under a curve
- Fundamental Theorem: relationship between differentiation and integration
Number Theory
- Prime numbers and composite numbers
- Divisibility rules and factors
- Greatest common divisor (GCD) and least common multiple (LCM)
- Modular arithmetic and congruences
Mathematical Logic
- Statements, truth values, and logical connectives
- Conditional statements and truth tables
- Mathematical proof: direct, indirect, contradiction, and induction

These concepts form the foundation of mathematics and can be further expanded with specific theories, methods, and applications in various fields. Each area has its own set of principles and problems to solve, contributing to a deeper understanding of math as a whole.

Arithmetic

Basic operations: addition, subtraction, multiplication, division
Order of operations: PEMDAS - Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)

Algebra

Variables represent unknown values, constants are fixed values.
Expressions combine variables and constants using operations.
Equations set two expressions equal, inequalities compare them.
Functions express relationships between inputs and outputs, linear, quadratic, and polynomial functions are common types.
Solving for x involves finding the unknown value that satisfies the equation.
- Methods include:
  - Substitution: replacing one variable with its equivalent expression
  - Elimination: adding or subtracting equations to eliminate a variable
  - Factoring: breaking down expressions into simpler factors

Geometry

Basic shapes: triangles, rectangles, circles, polygons
Properties of angles:
- Complementary: add up to 90 degrees
- Supplementary: add up to 180 degrees
- Vertical: opposite angles formed by intersecting lines are equal
- Adjacent: share a common side and vertex
Theorems:
- Pythagorean theorem: relates the sides of a right triangle (a² + b² = c²)
- Congruence: shapes with same size and shape
- Similarity: shapes with the same shape but different size
Formulas for area and perimeter for different shapes

Trigonometry

Functions: sine, cosine, tangent - ratios of sides in a right triangle
Reciprocal functions: cosecant, secant, cotangent
Unit circle: circle with radius 1 used to visualize trigonometric values for different angles
Radians: another way to measure angles (2π radians = 360 degrees)
Trigonometric identities: equations that hold true for all angles
- Pythagorean identity: sin²θ + cos²θ = 1
- Angle sum: sin(α+β) = sinαcosβ + cosαsinβ
- Double-angle: sin(2θ) = 2sinθcosθ

Statistics

Data collection methods: surveys, experiments, observational studies
Measures of central tendency:
- Mean: average value
- Median: middle value when data is ordered
- Mode: most frequent value
Measures of variability:
- Range: difference between highest and lowest values
- Variance: average squared deviation from the mean
- Standard deviation: square root of variance, measures spread
Probability concepts:
- Independent events: outcome of one event does not affect the other
- Dependent events: outcome of one event affects the other
- Combinations: number of ways to choose a subset from a set, order doesn't matter
- Permutations: number of ways to order elements in a set

Calculus

Limits: value a function approaches as its input approaches a certain value
Continuity: a function is continuous if it can be drawn without lifting the pen
Derivatives: measure the instantaneous rate of change of a function
- Applications: finding tangent lines to curves, optimizing values
Integrals: measure the area under a curve
- Definite integral: area between a curve and the x-axis over a specific interval
- Indefinite integral: family of functions whose derivative is the given function
Fundamental Theorem of Calculus: relates differentiation and integration

Number Theory

Prime numbers: divisible only by 1 and themselves
Composite numbers: have more than two factors
Divisibility rules: easy ways to determine if a number is divisible by another
- Example: a number is divisible by 3 if the sum of its digits is divisible by 3
Greatest common divisor (GCD): largest number that divides into two or more numbers
Least common multiple (LCM): smallest number that is a multiple of two or more numbers
Modular arithmetic: working with remainders after division
Congruences: stating that two numbers have the same remainder when divided by a specific number

Mathematical Logic

Statements: sentences that can be true or false
Truth values: truth or falsehood of a statement
Logical connectives: combine statements
- AND: both statements must be true
- OR: at least one statement must be true
- NOT: negates the truth value of a statement
Conditional statements: use "if... then..." to express a relationship
- Truth tables: show truth values of a conditional statement given different truth values for its parts
Mathematical proof: demonstrating the truth of a statement
- Direct proof: uses logical steps to reach the conclusion
- Indirect proof: proves the statement by assuming its negation is true and showing a contradiction
- Proof by contradiction: assumes the statement is false and then derives a contradiction
- Proof by induction: proves a statement for all natural numbers by showing it's true for the first case and then assuming it's true for any case and proving it's also true for the next case