Fundamental Concepts in Mathematics

Numbers and Operations
- Types of numbers: Natural, Whole, Integers, Rational, Irrational, Real, Complex.
- Basic operations: Addition, Subtraction, Multiplication, Division.
- Order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Algebra
- Expressions: Combinations of numbers, variables, and operations.
- Equations: Mathematical statements asserting equality; includes solving for unknowns.
- Functions: Relationships between sets of data; commonly expressed as f(x).
Geometry
- Shapes: Points, lines, angles, polygons, circles, solids.
- Properties: Area, perimeter, volume, surface area.
- Theorems: Pythagorean theorem, properties of triangles and quadrilaterals.
Trigonometry
- Functions: Sine, cosine, tangent and their reciprocals (cosecant, secant, cotangent).
- Relationships: SOH-CAH-TOA for right triangles.
- Applications: Angles of elevation and depression, periodic phenomena.
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: Represents area under a curve or accumulation of quantities.
Statistics
- Descriptive statistics: Mean, median, mode, range.
- Inferential statistics: Hypothesis testing, confidence intervals, p-values.
- Probability: Measure of the likelihood of an event occurring.
Discrete Mathematics
- Structures: Sets, graphs, trees, algorithms.
- Combinatorics: Counting principles, permutations, combinations.
- Logic: Propositional and predicate logic, truth tables.

Key Mathematical Principles

The Fundamental Theorem of Arithmetic: Every integer greater than 1 can be expressed uniquely as a product of prime numbers.
The Multiplicative Inverse: For any non-zero number, there exists a number that, when multiplied together, yields one.
Euclidean Geometry: The study of shapes, sizes, and properties of space based on postulates set forth by Euclid.
Non-Euclidean Geometry: Systems that relax or replace Euclid's postulates, such as spherical and hyperbolic geometry.

Important Formulas

Area of a rectangle: A = l × w
Circumference of a circle: C = 2πr
Volume of a cylinder: V = πr²h
Pythagorean theorem: a² + b² = c² (in right triangles)
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Problem-Solving Strategies

Understand the problem: Read carefully and identify what is being asked.
Plan: Consider how to approach the problem, what methods or formulas may apply.
Execute: Carry out the plan and perform calculations accurately.
Review: Check the solution for errors and ensure it answers the original question.

Applications of Mathematics

Finance: Interest calculations, budgeting, investment analysis.
Engineering: Structural calculations, modeling, optimization problems.
Natural Sciences: Data analysis, modeling phenomena, experimental design.

Numbers and Operations

Types of numbers:
- Natural numbers: Counting numbers (1, 2, 3, ...)
- Whole numbers: Natural numbers including zero (0, 1, 2, 3, ...)
- Integers: Positive and negative whole numbers, including zero (... -3, -2, -1, 0, 1, 2, 3, ...)
- Rational numbers: Numbers that can be expressed as a fraction (p/q) where p and q are integers and q is not zero (e.g., 1/2, -3/4, 2)
- Irrational numbers: Numbers that cannot be expressed as a fraction (e.g., pi, square root of 2)
- Real numbers: All rational and irrational numbers
- Complex numbers: Numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1)
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

Expressions: Combinations of numbers, variables, and operations
- Example: 2x + 3y - 5
Equations: Mathematical statements asserting equality. The goal is to find the value of the unknown variables.
- Example: 2x + 5 = 11
Functions: Relationships between sets of data. Represent how one value changes based on another.
- Commonly expressed as f(x) where 'x' is the input and 'f(x)' is the output.
- Example: f(x) = 2x + 1

Geometry

Shapes: Points, lines, angles, polygons (triangles, quadrilaterals, etc.), circles, solids
Properties:
- Area: The amount of space a two-dimensional shape occupies.
- Perimeter: The total length of all sides of a two-dimensional shape.
- Volume: The amount of space a three-dimensional shape occupies.
- Surface Area: The total area of all surfaces of a three-dimensional shape.
Theorems:
- Pythagorean theorem: a² + b² = c² (in a right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse).

Trigonometry

Functions: sine (sin), cosine (cos), tangent (tan) and their reciprocals: cosecant (csc), secant (sec), cotangent (cot).
Relationships: SOH-CAH-TOA (for right triangles)
- Sine = Opposite / Hypotenuse
- Cosine = Adjacent / Hypotenuse
- Tangent = Opposite / Adjacent
Applications: Angles of elevation, angles of depression, periodic phenomena (e.g., wave patterns)

Calculus

Limits: The value a function approaches as the input approaches a specific point.
Derivatives: Measure how fast a function changes as its input changes (slope of the line tangent to a curve).
Integrals: Represent the area under the curve of a function, or the accumulation of quantities.

Statistics

Descriptive statistics:
- Mean: The average of a set of numbers.
- Median: The middle value in a sorted set of numbers.
- Mode: The most frequently occurring value in a set of numbers.
- Range: The difference between the largest and smallest values in a set of numbers.
Inferential statistics:
- Hypothesis testing: Used to test a claim about a population based on sample data.
- Confidence intervals: A range of values that is likely to contain the true value of a population parameter.
- p-values: The probability of obtaining the observed results if the null hypothesis is true.
Probability: The measure of the likelihood of an event occurring.

Discrete Mathematics

Structures: Sets, graphs, trees, algorithms.
Combinatorics: Counting principles, permutations (arrangements of objects), combinations (selections of objects where order doesn't matter).
Logic: Propositional logic (statements are either true or false), predicate logic (statements about variables and quantities).

Key Mathematical Principles

The Fundamental Theorem of Arithmetic: Every integer greater than 1 can be expressed uniquely as a product of prime numbers (e.g., 12 = 2 × 2 × 3).
The Multiplicative Inverse: For any non-zero number, there exists a number that, when multiplied together, yields one (e.g., the multiplicative inverse of 5 is 1/5, because 5 × 1/5 = 1).
Euclidean Geometry: The study of shapes, sizes, and properties of space based on postulates set forth by Euclid, which mainly describe how points and lines interact.
Non-Euclidean Geometry: Systems that relax or replace some of Euclid's postulates. Different geometries arise, such as spherical (geometry on a sphere) and hyperbolic (geometry on a saddle-like surface).

Important Formulas

Area of a rectangle: A = l × w (l = length, w = width)
Circumference of a circle: C = 2πr (r = radius)
Volume of a cylinder: V = πr²h (r = radius, h = height)
Pythagorean theorem: a² + b² = c² (in a right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse).
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a (used to solve quadratic equations of the form ax² + bx + c = 0).

Problem-Solving Strategies
Understand the problem: Read the question carefully and identify what is being asked.
Plan: Consider how to approach the problem and what methods or formulas may apply.
Execute: Carry out the plan and perform calculations accurately using the right mathematical tools.
Review: Check the solution for errors and ensure it answers the original question.