Introduction to Core Mathematics

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be studied in its own right (pure mathematics) or as it is applied to other disciplines such as physics and engineering (applied mathematics).

Core Areas of Mathematics

• Arithmetic: Basic operations on numbers including addition, subtraction, multiplication, and division.
• Algebra: Study of mathematical symbols and the rules for manipulating these symbols.
• Geometry: Deals with the shape, size, relative position of figures, and the properties of space.
• Calculus: Studies continuous change, including differentiation and integration.
• Trigonometry: Branch of mathematics that deals with relationships between the sides and angles of triangles.
• Statistics: Collection, analysis, interpretation, presentation, and organization of data.
• Probability: Analysis of random phenomena.

Arithmetic

• Deals with operations on numbers and their properties.
• Includes whole numbers, integers, rational numbers, and real numbers.
• Basic operations are addition, subtraction, multiplication, and division.
• Order of operations (PEMDAS/BODMAS) is critical for evaluating expressions: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction.
• Properties of operations: Commutative, associative, distributive, identity, and inverse.

Algebra

• Uses symbols to represent numbers and quantities.
• Simplifies complex mathematical relationships.
• Variables are symbols (usually letters) that represent unknown values.
• Expressions are combinations of variables, numbers, and operations.
• Equations are statements that two expressions are equal.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations involve variables raised to the power of 1.
• Quadratic equations involve variables raised to the power of 2.
• Systems of equations are sets of two or more equations with the same variables.

Geometry

• Concerned with shapes, sizes, and spatial relationships.
• Euclidean geometry deals with flat spaces.
• Non-Euclidean geometries include spherical and hyperbolic geometries.
• Key concepts include points, lines, angles, surfaces, and solids.
• Shapes include triangles, quadrilaterals, circles, polygons, and polyhedra.
• Theorems and postulates provide the basis for geometric proofs.
• Pythagorean theorem relates the sides of a right triangle: a² + b² = c².
• Area and volume calculations are important in geometry.

Calculus

• Deals with continuous change.
• Differentiation finds the rate of change of a function.
• Integration finds the area under a curve.
• Limits are fundamental to calculus, describing the behavior of a function as its input approaches a certain value.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals measure the accumulation of a quantity over an interval.
• Fundamental Theorem of Calculus relates differentiation and integration.

Trigonometry

• Studies relationships between angles and sides of triangles.
• Trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent.
• Unit circle provides a visual representation of trigonometric functions.
• Trigonometric identities are equations that are true for all values of the variables.
• Law of Sines and Law of Cosines relate the sides and angles of any triangle.
• Used in navigation, surveying, and physics.

Statistics

• Involves collecting, analyzing, and interpreting data.
• Descriptive statistics summarize and describe data.
• Inferential statistics makes inferences and generalizations about a population based on a sample.
• Measures of central tendency include mean, median, and mode.
• Measures of variability include range, variance, and standard deviation.
• Probability distributions describe the likelihood of different outcomes.
• Hypothesis testing is used to determine whether there is enough evidence to support a claim.

Probability

• Quantifies the likelihood of an event occurring.
• Probability values range from 0 to 1.
• Theoretical probability is based on reasoning.
• Experimental probability is based on observations.
• Independent events: the occurrence of one event does not affect the probability of the other.
• Dependent events: the occurrence of one event affects the probability of the other.
• Conditional probability: the probability of an event given that another event has occurred.

Number Theory

• Branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions.
• Prime numbers, divisibility, and congruences are central topics.
• Fermat's Last Theorem and the Riemann Hypothesis are famous problems in number theory.

Mathematical Logic

• Explores the applications of formal logic to mathematics.
• Focuses on proof theory, model theory, recursion theory, and set theory.
• Used in computer science for algorithm design and verification.

Discrete Mathematics

• Studies mathematical structures that are discrete rather than continuous.
• Includes combinatorics, graph theory, and cryptography.
• Essential for computer science and related fields.

Topology

• Deals with properties that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending.
• Studies concepts like connectedness, continuity, and boundaries.
• Important in analysis, geometry, and physics.

Applied Mathematics

• Applies mathematical methods to solve problems in other disciplines.
• Includes numerical analysis, optimization, and mathematical modeling.
• Used in physics, engineering, economics, and finance.

Numerical Analysis

• Develops and analyzes algorithms for solving mathematical problems numerically.
• Focuses on approximation methods and error analysis.
• Used extensively in scientific computing.

Optimization

• Finds the best solution to a problem, subject to constraints.
• Linear programming, nonlinear programming, and dynamic programming are common techniques.
• Used in engineering, economics, and operations research.

Mathematical Modeling

• Creates mathematical representations of real-world systems.
• Uses equations, simulations, and other tools to analyze and predict behavior.
• Applied in various fields, including biology, ecology, and climate science.

Importance of Mathematics

• Basis for scientific and technological advancements.
• Develops critical thinking and problem-solving skills.
• Enhances logical reasoning and analytical abilities.
• Used in everyday life for making informed decisions.
• Provides a foundation for further study in many different fields.