Introduction to Mathematics

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be used as a tool for quantifying and measuring attributes in the real world.
• It uses abstraction and logical reasoning to study these topics.
• Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics concerns itself with the use of mathematical tools to solve problems in natural science, engineering, medicine, finance, and social sciences.
• Pure mathematics concerns itself with the study of mathematical concepts, regardless of applications.

Major Areas of Mathematics

• Arithmetic studies numbers and the operations between them (addition, subtraction, multiplication, division).
• Algebra studies general relationships, using symbols to represent numbers and quantities.
• Geometry studies shapes, sizes, and positions of figures, and properties of space.
• Trigonometry deals with relationships between angles and sides of triangles.
• Calculus studies continuous change, rates of change, accumulation, and motion.
• Statistics is the practice or science of collecting and analyzing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample.

Key Concepts in Arithmetic

• Natural numbers: The set of positive integers (1, 2, 3...).
• Integers: The set of whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).
• Rational numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Real numbers: Includes all rational and irrational numbers.
• Operations: Addition, subtraction, multiplication, and division are fundamental arithmetic operations.
• Order of operations: PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) dictates the order in which operations are performed.

Key Concepts in Algebra

• Variables: Symbols representing unknown quantities or values.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Statements that two expressions are equal.
• Polynomials: Expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Linear equations: Equations where the highest power of the variable is 1.
• Quadratic equations: Equations where the highest power of the variable is 2.
• Functions: Relationships that map each input value to a unique output value.

Key Concepts in Geometry

• Points, lines, and planes: Fundamental elements of Euclidean geometry.
• Angles: Formed by two rays sharing a common endpoint (vertex).
• Triangles: Three-sided polygons with three angles; classified by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse).
• Quadrilaterals: Four-sided polygons (squares, rectangles, parallelograms, trapezoids).
• Circles: Set of points equidistant from a center point.
• Area and perimeter: Measures of the two-dimensional space enclosed by a shape and the length of its boundary, respectively.
• Volume and surface area: Measures of the three-dimensional space contained within a solid and the total area of its surface, respectively.
• Coordinate geometry: Using a coordinate system to define and analyze geometric shapes.

Key Concepts in Trigonometry

• Trigonometric functions: Sine, cosine, and tangent are primary functions relating angles of a right triangle to ratios of its sides.
• Unit circle: A circle with a radius of 1, used to define trigonometric functions for all angles.
• Trigonometric identities: Equations that are true for all values of the variables.
• Solving triangles: Using trigonometric functions to find unknown sides or angles of a triangle.
• Applications: Navigation, surveying, and physics applications.

Key Concepts in Calculus

• Limits: 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 accumulation of a quantity.
• Fundamental Theorem of Calculus: Relates differentiation and integration.
• Applications: Optimization problems, physics, engineering.

Key Concepts in Statistics

• Descriptive statistics: Methods for summarizing and presenting data (mean, median, mode, standard deviation).
• Probability: The measure of the likelihood that an event will occur.
• Distributions: Describe the probability of different outcomes in a population (normal distribution, binomial distribution).
• Hypothesis testing: A method for testing claims or hypotheses about a population based on a sample of data.
• Regression analysis: Examines the relationship between a dependent variable and one or more independent variables.

Mathematical Notation

• Symbols: +, -, ×, ÷, =, <, >, ≤, ≥, √, ∑, ∫
• Conventions: Mathematical expressions are written following specific rules for order of operations and symbol usage.
• Greek letters: Commonly used to represent variables or constants (e.g., α, β, θ, π, Σ, Δ).

Mathematical Proofs

• Direct proof: Show that a statement is true by logically following from known facts.
• Indirect proof: (Proof by contradiction) Assume the statement is false and show that this leads to a contradiction.
• Proof by induction: Establish a base case and then show that if the statement is true for one case, it is true for the next case.

Mathematical Problem Solving

• Understanding the problem: Identify what is given and what needs to be found.
• Devising a plan: Choose an appropriate strategy or method to solve the problem.
• Carrying out the plan: Execute the chosen method, showing all steps.
• Looking back: Verify the solution, check for errors, and consider alternative approaches.

Importance of Mathematics

• Critical thinking: Mathematics develops logical reasoning and problem-solving skills.
• Scientific advancements: Mathematics is fundamental to advancements in science and technology.
• Financial literacy: Understanding mathematical concepts is essential for managing personal finances and making informed financial decisions.
• Technological applications: Mathematics is used in computer science, engineering, and other technology-driven fields.

Sets and Set Theory

• A set is a well-defined collection of distinct objects, considered as an object in its own right
• Sets are typically denoted by uppercase letters
• Objects in a set are called elements or members of the set
• Sets can be defined by listing their elements, e.g., A = {1, 2, 3}
• Sets can be defined by a property that all their elements satisfy, e.g., B = {x | x is an even integer}
• Common sets include: Natural numbers (N), Integers (Z), Rational numbers (Q), Real numbers (R), Complex numbers (C)
• The null set (empty set) is a set with no elements, denoted by ∅ or {}
• A is a subset of B (A ⊆ B) if every element of A is also an element of B
• A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B
• The union of sets A and B (A ∪ B) is the set of elements which are in A, in B, or in both.
• The intersection of sets A and B (A ∩ B) is the set of elements which are in both A and B.
• The difference of sets A and B (A - B) is the set of elements which are in A but not in B.
• The complement of set A (A') is the set of elements not in A, with respect to a universal set.

Logic

• Logic is the study of reasoning and argumentation
• A statement (or proposition) is a declarative sentence that is either true or false, but not both
• Logical operators are used to form compound statements
• Conjunction (∧) is true if both statements are true; otherwise, it is false (AND)
• Disjunction (∨) is true if at least one of the statements is true; it is false only if both are false (OR)
• Negation (¬) reverses the truth value of a statement: if a statement is true, its negation is false, and vice versa (NOT)
• Conditional (→) is true unless the first statement is true and the second is false (IF...THEN...)
• Biconditional (↔) is true if both statements have the same truth value; otherwise, it is false (IF AND ONLY IF)
• A truth table is a table that shows the truth value of a compound statement for all possible truth values of its components
• A tautology is a statement that is always true, regardless of the truth values of its components
• A contradiction is a statement that is always false, regardless of the truth values of its components
• Logical equivalence occurs when two statements always have the same truth value
• De Morgan’s Laws provide rules for how to negate conjunctions and disjunctions: ¬(A ∧ B) is equivalent to (¬A ∨ ¬B), and ¬(A ∨ B) is equivalent to (¬A ∧ ¬B)
• Mathematical induction is a method of proving a statement that asserts something about all natural numbers