Arithmetic, Number Systems and Algebra Basics

Arithmetic Operations

• Addition, subtraction, multiplication, and division are fundamental arithmetic operations.
• Addition combines numbers to find their sum.
• Subtraction finds the difference between two numbers.
• Multiplication involves repeated addition of a number.
• Division splits a number into equal parts.
• These operations form the basis for more complex math calculations.

Number Systems

• Number systems include natural numbers, integers, rational numbers, and real numbers.
• Natural numbers are positive whole numbers (1, 2, 3,...).
• Integers include positive and negative whole numbers and zero (... -2, -1, 0, 1, 2,...).
• Rational numbers can be expressed as a fraction of two integers (e.g., 1/2, -3/4).
• Real numbers include all rational and irrational numbers (e.g., √2, π).
• Complex numbers include a real and imaginary part, expressed as a+bi, where i is the square root of -1.

Algebra Basics

• Algebra uses variables to represent unknown quantities in equations.
• Algebraic expressions combine variables, constants, and arithmetic operations.
• Equations state the equality between two expressions.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Basic algebraic operations include simplifying expressions, factoring, and solving linear and quadratic equations.

Geometry Fundamentals

• Geometry deals with the properties and relations of points, lines, surfaces, and solids.
• Basic geometric shapes include points, lines, angles, triangles, squares, circles, and cubes.
• Key concepts include area, perimeter, volume, and surface area.
• Geometric theorems, such as the Pythagorean theorem, describe relationships between sides and angles of shapes.
• Coordinate geometry uses coordinates to represent geometric shapes and solve geometric problems.

Calculus Overview

• Calculus studies continuous change, including rates of change and accumulation.
• Differential calculus deals with derivatives, measuring a function's output change relative to its input.
• Integral calculus deals with integrals, which measure the accumulation of a function's values over an interval.
• The Fundamental Theorem of Calculus connects differentiation and integration.
• Applications of calculus include optimization, modeling, and solving differential equations.

Statistics Essentials

• Statistics involves collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics summarize data set features using measures of central tendency (mean, median, mode) and variability (range, variance, standard deviation).
• Inferential statistics uses samples to make inferences about populations.
• Probability theory quantifies uncertainty and makes predictions.
• Common statistical techniques include hypothesis testing, regression analysis, and analysis of variance (ANOVA).

Trigonometry Principles

• Trigonometry studies relationships between angles and sides of triangles.
• Trigonometric functions (sine, cosine, tangent) relate angles to ratios of sides in right triangles.
• Trigonometric identities are equations that are true for all values of the variables.
• Inverse trigonometric functions find the angle corresponding to a given trigonometric ratio.
• Applications of trigonometry include solving triangles, modeling periodic phenomena, and navigation.

Discrete Mathematics

• Discrete mathematics deals with discrete, not continuous, mathematical structures.
• Logic is the study of reasoning and argumentation.
• Set theory deals with collections of objects.
• Combinatorics involves counting and arranging objects.
• Graph theory studies relationships between objects using networks of nodes and edges.
• Discrete mathematics is used in computer science, cryptography, and operations research.

Mathematical Logic

• Mathematical logic applies formal logic to mathematical reasoning.
• Propositional logic deals with true or false propositions (declarative statements).
  • Logical connectives (AND, OR, NOT, implication, equivalence) form compound statements.
  • Truth tables define the truth values of compound statements.
• Predicate logic extends propositional logic with quantifiers (universal and existential) over a domain.
  • Predicates are statements dependent on variables, becoming true or false with assigned values.
  • Quantifiers, like "for all" and "there exists," make statements about object properties in a domain.
• Proof theory studies the structure of mathematical proofs.
  • Axioms are fundamental assumptions accepted as true without proof.
  • Inference rules derive new statements from existing ones.
  • Proofs start from axioms and apply inference rules to demonstrate a statement's truth.
• Model theory relates formal languages to their interpretations.
  • A model satisfies a set of axioms or formulas.
  • Model theory explores relationships between formal theories and their models.

Set Theory

• Set theory studies sets, which are collections of objects.
• A set is a well-defined collection of distinct objects, called elements or members.
• Sets are defined by listing elements (e.g., {1, 2, 3}) or specifying element properties (e.g., {x | x is an even number}).
• Basic set operations include:
  • Union: A ∪ B is the set of elements in A or B (or both).
  • Intersection: A ∩ B is the set of elements in both A and B.
  • Difference: A - B is the set of elements in A but not in B.
  • Complement: A' is the set of elements not in A (relative to a universal set).
• Relations describe how set elements relate.
  • A relation is a set of ordered pairs (a, b), where a relates to b in some way.
  • Properties of relations include reflexivity, symmetry, transitivity, and antisymmetry.
• Functions are relations mapping each domain element to a unique range element.
  • A function f: A → B assigns each element a in A a unique element f(a) in B.
  • Properties of functions include injectivity (one-to-one), surjectivity (onto), and bijectivity (one-to-one correspondence).
• Cardinality measures a set's "size".
  • Finite sets have a finite number of elements, while infinite sets have an infinite number of elements.
  • Countable sets have a one-to-one correspondence with natural numbers, while uncountable sets do not.

Number Theory

• Number theory studies the properties of integers.
• Divisibility: Integer a divides integer b if there exists an integer k such that b = ak.
• Prime numbers are integers greater than 1 divisible only by 1 and themselves.
• The Fundamental Theorem of Arithmetic states that every integer greater than 1 is uniquely expressed as a product of prime numbers.
• Modular arithmetic deals with congruences and remainders.
  • Integers a and b are congruent modulo n if they share the same remainder when divided by n, denoted a ≡ b (mod n).
  • Modular arithmetic has applications in cryptography, computer science, and coding theory.
• Euclidean algorithm finds the greatest common divisor (GCD) of two integers.
  • The GCD of a and b is the largest integer that divides both a and b.
  • The Euclidean algorithm follows the principle that GCD(a, b) = GCD(b, a mod b).
• Diophantine equations are polynomial equations seeking only integer solutions.
  • Linear Diophantine equations have the form ax + by = c, where a, b, and c are integers.
  • The solvability of Diophantine equations depends on coefficient relationships.

Combinatorics

• Combinatorics counts, arranges, and selects objects.
• Counting techniques determine the number of ways to perform a task.
  • Permutations arrange objects in a specific order.
  • Combinations select objects without regard to order.
  • The binomial theorem expands expressions of the form (a + b)^n.
• Graph theory studies graphs, which model pairwise relations between objects.
  • A graph consists of vertices (nodes) and edges (connections between vertices).
  • Graph algorithms solve problems like finding the shortest path between two vertices.
• Recurrence relations define a sequence using previous terms.
  • The Fibonacci sequence is a classic recurrence relation: F(n) = F(n-1) + F(n-2).
  • Recurrence relations model phenomena in computer science, biology, and economics.