Introduction to Mathematics

• Math is the science and study of quantity, structure, space, and change.

Core Areas of Math

• Arithmetic studies numbers and operations like addition, subtraction, multiplication, and division.
• Algebra studies mathematical symbols and the rules for manipulating these symbols.
• Geometry studies shapes, sizes, and positions of figures, and their properties.
• Trigonometry studies relationships between angles and sides of triangles.
• Calculus studies continuous change.
• Statistics studies the collection, analysis, interpretation, presentation, and organization of data.

Numbers

• Natural numbers are positive integers (1, 2, 3...).
• Integers include natural numbers, zero, and negative integers (...-2, -1, 0, 1, 2...).
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational numbers cannot be expressed as a fraction (e.g., √2, π).
• Real numbers include all rational and irrational numbers.
• Complex numbers have a real and imaginary part, in the form a + bi, where i is the imaginary unit (√-1).

Arithmetic Operations

• Addition combines numbers to find their sum.
• Subtraction finds the difference between numbers.
• Multiplication finds the product of numbers.
• Division finds how many times one number is contained in another.
• Exponentiation raises a number to a power.

Algebra Basics

• Variables are symbols representing unknown or changeable values.
• Expressions are combinations of variables, numbers, and operations.
• Equations state the equality between two expressions.
• Solving equations involves finding the values of variables that make the equation true.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Geometry Fundamentals

• Points are locations in space.
• Lines are straight paths extending infinitely in both directions.
• Planes are flat surfaces extending infinitely in all directions.
• Angles are formed by two rays sharing an endpoint.
• Triangles are three sided polygons.
• Quadrilaterals are four sided polygons.
• Circles are set of points equidistant from a center.
• Volume is the amount of space occupied by a 3D object.
• Area is the measure of a 2D surface.
• Perimeter is the distance around a 2D shape.

Trigonometry Functions

• Sine (sin) of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
• Cosecant (csc) is the reciprocal of sine.
• Secant (sec) is the reciprocal of cosine.
• Cotangent (cot) is the reciprocal of tangent.

Calculus Concepts

• Limits describe the behavior of a function as its input approaches a certain value.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals accumulate quantities, finding the area under a curve.
• Differentiation is the process of finding the derivative of a function.
• Integration is the process of finding the integral of a function.
• Fundamental Theorem of Calculus links differentiation and integration.

Statistics Principles

• Descriptive statistics summarize and describe data.
• Inferential statistics make inferences and generalizations about a population based on a sample.
• Mean is the average of a set of numbers.
• Median is the middle value in a sorted set of numbers.
• Mode is the value that appears most frequently in a set of numbers.
• Standard deviation measures the spread of data around the mean.
• Probability quantifies the likelihood of an event occurring.
• Distributions describe the probability of different outcomes (e.g., normal distribution).
• Hypothesis testing is a method for testing a claim about a population.
• Regression analysis examines the relationship between variables.

Mathematical Logic

• Statements are declarative sentences that are either true or false.
• Propositional logic deals with logical relationships between statements.
• Predicate logic extends propositional logic to include predicates and quantifiers.
• Set theory studies sets, which are collections of objects.

Mathematical Proofs

• Direct proofs start with assumptions and use logical steps to reach a conclusion.
• Indirect proofs (proof by contradiction) assume the negation of the statement and derive a contradiction.
• Mathematical induction proves a statement for all natural numbers by showing it holds for the base case and proving the inductive step.

Applications of Math

• Physics uses math to describe the laws of nature.
• Engineering uses math to design and analyze structures, systems, and devices.
• Computer science uses math for algorithms, data structures, and cryptography.
• Economics uses math for modeling and analysis.
• Finance uses math for investment, risk management, and pricing.
• Biology uses math for modeling population dynamics and biological processes.
• Cryptography relies heavily on number theory and algebra to secure communications.

Mathematical Constants

• Pi (π) is the ratio of a circle's circumference to its diameter, approximately 3.14159.
• Euler's number (e) is the base of the natural logarithm, approximately 2.71828.
• The square root of 2 (√2) is an irrational number, approximately 1.41421.

Mathematical Notation

• Symbols are used to represent mathematical operations, quantities, and relationships.
• Parentheses, brackets, and braces are used to group expressions.
• Common notation includes +, -, ×, ÷, =, <, >, ≤, ≥.
• Summation notation (Σ) represents the sum of a series.
• Product notation (Π) represents the product of a series.

Types of Equations

• Linear equations have the form ax + b = 0.
• Quadratic equations have the form ax² + bx + c = 0.
• Polynomial equations involve higher powers of variables.
• Differential equations involve derivatives of functions and describe rates of change.
• Integral equations involve integrals of functions.

Coordinate Systems

• Cartesian coordinate system uses x and y axes to locate points in a plane.
• Polar coordinate system uses a distance and an angle to locate points in a plane.
• Three-dimensional coordinate system extends the Cartesian system with a z-axis.

Sequences and Series

• A sequence is an ordered list of numbers.
• An arithmetic sequence has a constant difference between terms.
• A geometric sequence has a constant ratio between terms.
• A series is the sum of the terms in a sequence.

Matrices and Linear Algebra

• A matrix is a rectangular array of numbers.
• Linear algebra studies matrices, vectors, and linear transformations.
• Matrix operations include addition, subtraction, multiplication, and inversion.
• Eigenvalues and eigenvectors describe the behavior of linear transformations.

Mathematical Modeling

• Mathematical modeling creates mathematical representations of real-world phenomena.
• Models can be used to predict behavior, optimize processes, and gain insights.
• Examples include population models, economic models, and climate models.

Overview of mathematics, including arithmetic, algebra, geometry and statistics. Mathematics is the study of numbers, shapes, patterns, and change. Key concepts include number systems, algebraic manipulation, geometric shapes, and data analysis.