Fundamental Concepts in Math

Numbers:
- Natural Numbers: Counting numbers (1, 2, 3, ...)
- Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...)
- Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...)
- Rational Numbers: Numbers that can be expressed as a fraction (1/2, 3, -4)
- Irrational Numbers: Numbers that cannot be expressed as fractions (√2, π)
Operations:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)
Algebra:
- Variables: Symbols that represent numbers (e.g., x, y)
- Expressions: Combinations of numbers and variables (3x + 2)
- Equations: Statements that two expressions are equal (2x = 10)
- Functions: Relationships between inputs and outputs (f(x) = x^2)
Geometry:
- Points, Lines, and Planes: Fundamental concepts defining shapes
- Angles: Measured in degrees (acute, right, obtuse)
- Shapes:
  - 2D (circles, triangles, squares)
  - 3D (cubes, spheres, cylinders)
- Area and Perimeter:
  - Area: Space inside a shape
  - Perimeter: Distance around a shape
Trigonometry:
- Sine, Cosine, Tangent: Ratios of sides in right-angled triangles
- The Pythagorean Theorem: a² + b² = c² (relationship in right triangles)
Calculus:
- Limits: The value a function approaches as the input approaches a point
- Derivatives: Measure of how a function changes as its input changes
- Integrals: Measure of the area under a curve
Statistics:
- Mean: Average of a set of numbers
- Median: Middle value when numbers are arranged in order
- Mode: Most frequently occurring number in a set
- Standard Deviation: Measure of the amount of variation or dispersion in a set
Mathematical Reasoning:
- Inductive Reasoning: Making generalizations based on specific observations
- Deductive Reasoning: Drawing specific conclusions from general principles
Mathematical Notation:
- Symbols used to represent numbers, operations, and relationships (e.g., ∑ for summation, ∞ for infinity)
Problem-Solving Strategies:
- Understand the problem
- Devise a plan
- Carry out the plan
- Review/extend findings

Applications of Math

Real World:
- Finance: Budgeting, interest calculations
- Engineering: Design and analysis of structures
- Data Science: Statistics and probability in data analysis
Technology:
- Algorithms: Mathematical procedures for calculations
- Cryptography: Secure communication through mathematical techniques
Science:
- Physics: Mathematical models represent physical phenomena
- Biology: Statistics in research and population studies

Fundamental Concepts in Math

Numbers:
- Natural Numbers: The set of positive counting numbers, starting from 1.
- Whole Numbers: This set includes all natural numbers plus zero.
- Integers: Comprises whole numbers and their negative counterparts.
- Rational Numbers: Numbers that can be expressed as fractions, including integers and decimals.
- Irrational Numbers: Numbers that cannot be expressed as simple fractions, such as √2 and π.
Operations:
- Mathematical operations for combining or manipulating numbers include addition, subtraction, multiplication, and division.
Algebra:
- Variables represent unknown values, commonly denoted as x and y.
- Expressions consist of numbers and variables combined through operations.
- Equations assert equality between two expressions, allowing for the solution of variables.
- Functions describe the relationship between inputs and outputs, often expressed in functional notation such as f(x).
Geometry:
- Fundamental elements such as points, lines, and planes form the basis for defining shapes.
- Angles are categorized by their degree measures: acute (< 90°), right (90°), obtuse (> 90°).
- Shapes are classified into 2D (like circles and triangles) and 3D (like cubes and spheres).
- Area quantifies the space within a shape, while perimeter measures the distance around it.
Trigonometry:
- Key functions include sine, cosine, and tangent, which relate angles to the ratios of side lengths in right triangles.
- The Pythagorean Theorem (a² + b² = c²) describes the relation of the sides in a right triangle.
Calculus:
- Limits describe the behavior of functions as inputs approach a certain value.
- Derivatives represent the rate of change of a function with respect to its variable.
- Integrals calculate the total area under a curve, serving as a fundamental concept in calculus.
Statistics:
- Mean calculates the average value of a data set, while median identifies the middle value.
- Mode is the number that appears most frequently in a data set.
- Standard Deviation measures the variation or dispersion within a set of values.
Mathematical Reasoning:
- Inductive Reasoning involves making generalizations based on specific observations or patterns.
- Deductive Reasoning entails applying general principles to reach specific conclusions.
Mathematical Notation:
- Uses symbols to succinctly represent numbers and operations, such as ∑ for summation and ∞ for infinity.
Problem-Solving Strategies:
- A systematic approach involves understanding the problem, devising a plan, executing the plan, and reviewing findings for insights.

Applications of Math

Real World:
- Finance utilizes math for budgeting and calculating interest.
- Engineering applies mathematical principles in the design and analysis of structures.
- Data Science employs statistics and probability for analyzing data trends.
Technology:
- Algorithms are step-by-step mathematical procedures essential for solving calculations efficiently.
- Cryptography relies on mathematical techniques to secure communication and protect information.
Science:
- Physics uses mathematical models to abstractly represent physical phenomena and their interactions.
- Biology applies statistical methods in research to study populations and their characteristics.