Basic Concepts in Mathematics

Numbers
- Natural numbers: 1, 2, 3, ...
- Whole numbers: 0, 1, 2, 3, ...
- Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Rational numbers: Fractions and decimals (e.g., 1/2, 0.75)
- Irrational numbers: Non-repeating, non-terminating decimals (e.g., π, √2)
Arithmetic
- Basic operations: Addition, subtraction, multiplication, division
- Order of operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right) - PEMDAS
Algebra
- Variables: Symbols that represent numbers (e.g., x, y)
- Expressions: Combinations of numbers and variables (e.g., 3x + 4)
- Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7)
- Functions: A relationship that assigns a single output for each input (e.g., f(x) = 2x + 1)
Geometry
- Basic shapes: Circle, square, triangle, rectangle
- Area formulas:
  - Square: A = side²
  - Rectangle: A = length × width
  - Triangle: A = (base × height) / 2
  - Circle: A = πr²
- Volume formulas:
  - Cube: V = side³
  - Rectangular prism: V = length × width × height
  - Cylinder: V = πr²h
Trigonometry
- Fundamental ratios: Sine (sin), Cosine (cos), Tangent (tan)
- Key identities:
  - sin²θ + cos²θ = 1
  - tanθ = sinθ / cosθ
- Pythagorean theorem: a² + b² = c² (for right triangles)
Statistics and Probability
- Measures of central tendency: Mean (average), Median (middle value), Mode (most frequent value)
- Probability: Likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain)
- Fundamental counting principle: If one event can occur in m ways and another can occur independently in n ways, then the total number of ways the two events can occur is m × n.
Calculus
- Derivative: Measures the rate of change of a function.
- Integral: Measures the area under a curve.
- Fundamental theorem of calculus: Connects differentiation and integration.
Number Theory
- Prime numbers: Numbers greater than 1 that have no divisors other than 1 and themselves.
- Factors and multiples:
  - Factor: A number that divides another without leaving a remainder
  - Multiple: The product of a number and an integer

These foundational concepts serve as the basis for advanced mathematical study and applications in various fields.

Numbers

Natural numbers, also known as counting numbers, start at 1 and continue infinitely – 1,2,3, and so on.
Whole numbers include all natural numbers but also include zero – 0, 1, 2, 3, and so on.
Integers are whole numbers and their negatives – … -3, -2, -1, 0, 1, 2, 3, …
Rational numbers can be expressed as fractions or terminating or repeating decimals, like 1/2 or 0.75.
Irrational numbers cannot be expressed as fractions and have decimal representations that are non-repeating and non-terminating, like pi (π) or the square root of 2.

Arithmetic Operations

Addition combines numbers, subtraction finds the difference, multiplication involves repeated addition, and division separates a number into equal parts.
The order of operations PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) dictates the sequence in which operations are performed, from left to right.

Algebra

Variables, represented by letters like x or y, can stand in for any number.
Expressions combine numbers and variables using operations, like 3x + 4.
Equations establish equality between two expressions, like 2x + 3 = 7.
Functions establish a relationship between input and output values, where each input corresponds to one unique output, like f(x) = 2x + 1.

Geometry

Basic shapes include polygons like squares, triangles, and rectangles, as well as circles.
Area is the amount of surface covered by a two-dimensional shape. There are formulas for calculating the area of various shapes, including squares (A = side²), rectangles (A = length × width), triangles (A = (base × height) / 2), and circles (A = πr²).
Volume measures the amount of space occupied by a three-dimensional object. Common volume formulas include the cube (V = side³), rectangular prism (V = length × width × height), and cylinder (V = πr²h).

Trigonometry

Fundamental ratios in trigonometry are sine (sin), cosine (cos), and tangent (tan), which relate angles and sides of right triangles.
Key trigonometric identities, such as sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ, help simplify calculations and solve problems.
The Pythagorean theorem relates the lengths of the sides of a right triangle: a² + b² = c², where c is the hypotenuse.

Statistics and Probability

Measures of central tendency help characterize a dataset based on its typical value: mean (average), median (middle value), and mode (most frequent value).
Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).
The fundamental counting principle calculates the total number of possible outcomes when multiple events occur independently: if one event has m possibilities and another has n possibilities, then there are m × n total possibilities.

Calculus

Derivative measures the rate of change of a function at a specific point.
Integral measures the area under a curve, representing the accumulation of a quantity.
The fundamental theorem of calculus links derivatives and integrals, establishing a fundamental relationship in calculus.