Statistics

• Mathematics is the study of numbers, shapes, quantities, and patterns
• A fundamental tool for understanding the world
• Provides a framework for problem-solving in various fields

Core Branches of Mathematics

• Arithmetic involves basic operations on numbers, including addition, subtraction, multiplication, and division
• Algebra deals with symbols and the rules for manipulating those symbols and is a generalization of arithmetic
• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• Calculus is the study of continuous change, covering topics like limits, derivatives, integrals, and functions
• Trigonometry studies the relationships between angles and sides of triangles
• Statistics involves collection, analysis, interpretation, presentation, and organization of data
• Probability is the analysis of random phenomena

Arithmetic

• The foundation of numerical computation
• Involves operations like addition (+), subtraction (-), multiplication (*), and division (/)
• Includes concepts like fractions, decimals, percentages, and ratios
• Order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction
• Integers are whole numbers, either positive, negative, or zero
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero
• Real numbers includes all rational and irrational numbers (cannot be expressed as a fraction)
• Complex numbers have a real and imaginary part, denoted as a + bi, where i is the imaginary unit (sqrt(-1))

Algebra

• Uses symbols (variables) to represent unknown quantities in equations and expressions
• Includes solving equations, simplifying expressions, and working with functions
• Linear equations have the highest power of the variable as 1
• Quadratic equations have the highest power of the variable as 2
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Systems of equations are sets of two or more equations that are solved simultaneously
• Inequalities are mathematical statements that compare two expressions using inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to)
• Functions are mathematical relationships that map inputs to outputs

Geometry

• Deals with shapes, sizes, and positions of figures in two and three dimensions
• Euclidean geometry is based on axioms and postulates by Euclid, focusing on points, lines, angles, and shapes like triangles, circles, and polygons
• Coordinates geometry uses algebra to describe geometric objects with coordinate systems (e.g., Cartesian coordinates)
• Trigonometry relates angles and sides of triangles, particularly right triangles, using trigonometric functions like sine, cosine, and tangent
• Transformations are operations that change the position, size, or shape of geometric figures (e.g., translation, rotation, reflection, dilation)
• Solid geometry deals with three-dimensional shapes like cubes, spheres, cylinders, and cones

Calculus

• Focuses on the study of continuous change
• Differential calculus deals with the rate of change of functions, finding derivatives, and determining slopes of curves
• Integral calculus deals with the accumulation of quantities, finding integrals, and determining areas under curves
• Limits define the value that a function "approaches" as the input approaches some value
• Derivatives measure the instantaneous rate of change of a function
• Integrals represent the area under a curve; the reverse operation of differentiation
• The Fundamental Theorem of Calculus connects differentiation and integration
• Applications: Optimization problems, related rates, and finding areas and volumes

Trigonometry

• Studies the relationships between angles and sides of triangles
• Trigonometric functions: Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)
• The Unit circle, with a radius of 1, is used to define trigonometric functions for all real numbers
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables (e.g., sin²θ + cos²θ = 1)
• The Law of Sines and Law of Cosines are used to solve triangles that are not right triangles
• Applications in navigation, physics, and engineering

Statistics

• Involves collecting, analyzing, interpreting, and presenting data
• Descriptive statistics summarizes and describes the main features of a dataset
• Inferential statistics makes inferences and generalizations about a population based on a sample
• Measures of central tendency: Mean, median, and mode describe the center of a dataset
• Measures of dispersion: Variance, standard deviation, and range describe the spread of data
• Probability distributions describe the likelihood of different outcomes in a random experiment
• Hypothesis testing is used to make decisions based on data, determining whether there is enough evidence to reject a null hypothesis
• Regression analysis is used to model the relationship between variables

Probability

• Deals with the analysis of random phenomena
• Probability measures the likelihood of an event occurring
• The Sample space is the set of all possible outcomes of an experiment
• Events are subsets of the sample space
• Probability distributions describe the probabilities of different outcomes
• Conditional probability: The probability of an event occurring given that another event has already occurred
• Independent events: Events where the occurrence of one does not affect the probability of the other
• Expected value: The average outcome of an experiment if it is repeated many times

Mathematical Reasoning and Proofs

• Mathematical reasoning involves using logic to solve problems and prove statements
• Deductive reasoning starts with general principles and applies them to specific cases
• Inductive reasoning starts with specific observations and generalizes to broader principles
• Proofs are logical arguments that establish the truth of a mathematical statement
• Direct proof starts with assumptions and uses logical steps to reach the conclusion
• Indirect proof (proof by contradiction) assumes the negation of the statement and shows that this leads to a contradiction
• Mathematical induction is used to prove statements that hold for all natural numbers

Applications of Mathematics

• Physics: Mathematics is essential for describing physical phenomena and building models
• Engineering: Used in design, analysis, and optimization of systems and structures
• Computer Science: Algorithms, data structures, and cryptography rely heavily on mathematical principles
• Economics: Mathematical models are used to analyze markets, predict economic trends, and make decisions
• Finance: Used for investment analysis, risk management, and pricing derivatives
• Biology: Mathematical models are used to study population dynamics, genetics, and epidemiology
• Cryptography: Number theory and algebra form the basis for encryption and decryption
• Data Science: Statistics, probability, and linear algebra are fundamental tools

Mathematical Notation

• Symbols are used to represent mathematical objects and operations
• Numbers: 0, 1, 2, 3, …, π, e, i
• Variables: x, y, z, θ
• Operators: +, -, *, /, √, Σ, ∫
• Sets: { }, ∈, ∪, ∩
• Functions: f(x), g(x)
• Logical symbols: ∧ (and), ∨ (or), ¬ (not), ⇒ (implies), ⇔ (if and only if)