Introduction to Mathematics

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be used as a pure science, or it may be applied to other disciplines.
• Applied mathematics is used in engineering, computing, physics, and other natural sciences.

History of Mathematics

• Mathematical study began well before the Common Era, across the world, and initially focused on practical sciences.
• Mathematical development became increasingly abstracted over time.
• Mathematical thought became rigorous and axiomatic in ancient Greece.
• Mathematics flourished in the Islamic world during the Arab empires.
• Many Arabic and Greek texts were translated into Latin during the medieval period, leading to further development of mathematics in Europe.
• Mathematical study has been occurring worldwide since the Renaissance.

Subdisciplines of Mathematics

• Mathematics is a broad field, and is typically broken into subdisciplines.
• These subdisciplines concern quantity, structure, space, and change.
• There are also subdisciplines focused on connecting mathematics to other fields.

Quantity

• The study of quantity begins with numbers, first the familiar natural numbers (1, 2, 3, ...).
• Natural numbers relate to counting and ordering.
• The system of natural numbers can be extended to include zero.
• Natural numbers can be manipulated via addition and multiplication.
• The integers extend the natural numbers to include negative values (-1, -2, -3, ...).
• The rational numbers extend integers to include ratios between integers (1/2, 3/4, 5/8, ...).
• Rational numbers can be represented with a finite decimal expansion.
• The real numbers extend the rational numbers to include all numbers that can be represented by a decimal expansion.
• Real numbers include the irrational numbers, which cannot be represented as a ratio of integers.
• Irrational numbers have an infinite, non-repeating decimal expansion.(e.g. the square root of 2, pi)
• The complex numbers extend the real numbers to include numbers with an imaginary part.
• Complex numbers include the imaginary unit i, defined as a square root of -1.
• The study of quantity includes arithmetic, which concerns basic operations on numbers.
• Number theory concerns the properties of numbers, especially integers.

Structure

• The study of structure considers patterns, organization, and relationships.
• Algebra is the study of structure.
• Algebra includes the study of equations, relations, and formulas.
• Abstract algebra involves the study of algebraic structures such as groups, rings, and fields.
• The study of structure includes order theory, lattices, and database theory.

Space

• Space is one of the central mathematical objects of study.
• Geometry is the branch of mathematics concerned with spatial relationships.
• Trigonometry concerns relationships between angles and sides of triangles.
• Differential geometry studies more general curves and surfaces.
• Topology studies properties of objects that are unchanged by continuous deformations (such as stretching or bending).

Change

• Understanding and describing change is important in natural sciences.
• Calculus was developed to examine change.
• Calculus includes differential calculus, which studies instantaneous rates of change.
• Calculus includes integral calculus, which studies the accumulation of quantities.
• Differential equations relate quantities to their rates of change.
• Chaos theory describes systems that exhibit complex, unpredictable behavior.
• Numerical analysis is used to approximate solutions to problems in continuous mathematics.

Foundations and Philosophy

• To clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed.
• Mathematical logic concerns itself with the application of formal logic to mathematics.
• Set theory is the branch of mathematics that studies sets.
• A set is an abstract collection of objects.
• Category theory is an abstract approach to mathematics that focuses on relationships between mathematical structures.
• There are varied philosophies of mathematics.
• Mathematical Platonism posits that mathematical objects exist independently of human thought.
• Mathematical empiricism suggests that mathematical knowledge is based on empirical observation.
• Formalism states that mathematics is concerned with formal systems and rules.
• Constructivism asserts that mathematical objects must be constructed in some sense.

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete, rather than continuous.
• Discrete mathematics includes combinatorics.
• Graph theory, computability theory, and cryptography are areas of discrete mathematics.

Applied Mathematics

• Applied mathematics concerns the application of mathematical knowledge to other domains.
• Statistics is used to analyze and interpret data.
• Probability theory studies random phenomena.
• Actuarial science uses mathematical and statistical methods to assess risk in insurance and finance.
• Mathematical optimization is used to find the best solution to a problem.
• Operations research uses mathematical techniques to improve decision-making.
• Control theory is used to control the behavior of dynamical systems.
• Mathematical economics applies mathematical methods to economic theory.
• Mathematical finance uses mathematical tools to analyze financial markets
• Game theory analyzes strategic interactions between individuals.
• Mathematical biology applies mathematics to biological research.

Mathematics is the abstract science of number, quantity, and space. It began before the Common Era and has been developed across the world. It is a broad field broken into many subdisciplines.