Fundamental Concepts in Mathematics

Arithmetic: Basics of numbers, operations (addition, subtraction, multiplication, division).
Algebra: Variables, equations, functions, polynomials, and algebraic structures.
Geometry: Shapes, sizes, spatial reasoning, properties of figures, angles, and theorems.
Calculus: Limits, derivatives, integrals, and their applications in real-world problems.
Statistics: Data collection, analysis, interpretation, measures of central tendency (mean, median, mode), probability.

Types of Numbers:
- Natural Numbers: ( {1, 2, 3, \ldots} )
- Whole Numbers: ( {0, 1, 2, 3, \ldots} )
- Integers: ( {\ldots, -2, -1, 0, 1, 2, \ldots} )
- Rational Numbers: Fractions of integers, can be expressed as ( \frac{a}{b} ).
- Irrational Numbers: Cannot be expressed as fractions (e.g., ( \sqrt{2}, \pi )).
Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves.

Expressions and Equations: An expression combines numbers and variables; an equation states that two expressions are equal.
Linear Equations: Equations of the first degree (e.g. ( ax + b = 0 )).
Quadratic Equations: Equations of the second degree (e.g. ( ax^2 + bx + c = 0 )).
Functions: Relationships between sets of inputs and outputs, typically expressed as ( f(x) ).

Basic Shapes:
- Triangles (types: equilateral, isosceles, scalene)
- Quadrilaterals (types: squares, rectangles, parallelograms)
- Circles (radius, diameter, circumference, area).
Key Theorems:
- Pythagorean Theorem: In a right triangle, ( a^2 + b^2 = c^2 ).
- The sum of interior angles in a triangle is ( 180^\circ ).

Differentiation: Finding the rate at which a function is changing at any point (derivative).
Integration: Finding the area under a curve (integral).
Fundamental Theorem of Calculus: Links differentiation and integration.

Descriptive Statistics: Summarizes data using measures like mean, median, and mode.
Inferential Statistics: Draws conclusions and makes predictions based on data samples.
Probability: Measure of likelihood of an event occurring, ranges from 0 to 1.

Statements: Any declarative sentence that is either true or false.
Logical Operators: AND, OR, NOT; used to form compound statements.
Quantifiers:
- Universal quantifier (∀): "for all"
- Existential quantifier (∃): "there exists"

Understand the Problem: Analyze and reread the question.
Devise a Plan: Create a strategy or formula to tackle the problem.
Carry Out the Plan: Execute the method chosen.
Review/Extend: Check the solution and consider its implications or extensions.

Arithmetic involves basic operations like addition, subtraction, multiplication, and division.
Algebra introduces variables, equations, functions, and polynomials, allowing for abstract problem-solving.
Geometry deals with shapes, sizes, and spatial reasoning, exploring properties of figures, angles, and theorems.
Calculus focuses on limits, derivatives, and integrals, with applications in real-world scenarios.
Statistics analyzes data, interpreting trends, and using measures like mean, median, and mode to understand probability.

Natural numbers include 1, 2, 3, and so on.
Whole numbers encompass natural numbers along with zero.
Integers include both positive and negative whole numbers.
Rational numbers are fractions formed by dividing two integers, which can be expressed as ( \frac{a}{b}).
Irrational numbers cannot be expressed as simple fractions; examples include ( \sqrt{2}) and ( \pi ).
Prime numbers are greater than 1 and only divisible by 1 and themselves.

Expressions combine variables and numbers; an equation equates two expressions.
Linear equations are first-degree equations with a general form of ( ax + b = 0 ).
Quadratic equations are second-degree equations with a general form of ( ax^2 + bx + c = 0 ).
Functions establish relations between inputs and outputs, typically represented as ( f(x) ).

Basic Shapes include triangles (equilateral, isosceles, scalene), quadrilaterals (squares, rectangles, parallelograms), and circles (with properties like radius, diameter, circumference, and area).
Key Theorems such as the Pythagorean Theorem (in a right triangle, ( a^2 + b^2 = c^2 )) and the sum of interior angles in a triangle being ( 180^\circ ) are fundamental concepts in geometry.

Differentiation reveals the rate of change of a function at any given point, referred to as its derivative.
Integration calculates the area under a curve, represented by the integral.
The Fundamental Theorem of Calculus establishes a connection between differentiation and integration, allowing us to solve complex problems using both concepts.

Descriptive Statistics provides summaries of data through measures like mean, median, and mode, providing insights into data distribution.
Inferential Statistics draws conclusions and makes predictions based on data samples, allowing us to generalize findings.
Probability quantifies the likelihood of an event occurring, using values ranging from 0 to 1.

Statements are declarative sentences that can be classified as true or false.
Logical Operators like AND, OR, and NOT combine statements to form compound statements.
Quantifiers generalize statements:
- Universal quantifier (∀): "for all," applies to all elements within a set.
- Existential quantifier (∃): "there exists," indicates that at least one element within a set satisfies a condition.

Understanding the problem involves analyzing and comprehending the question before attempting a solution.
Devising a plan consists of crafting a strategy or using relevant formulas to approach the problem.
Carrying out the plan means implementing the chosen strategy or method to solve the problem.
Reviewing and extending involves verifying the solution and exploring its implications or potential extensions.