Introduction to Mathematics

• Mathematics is the abstract study of topics such as quantity, structure, space, and change.
• There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.
• Mathematics seeks out patterns and formulates new conjectures, which are resolved by mathematicians through mathematical proofs.
• The foundation of mathematics is based upon axioms which are considered as self-evident, and upon definitions, from which new statements can be deduced.
• Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines.
• Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind. There is often a time lag between a mathematical discovery and the time when it is applied in practice.

History

• The history of mathematics stretches back thousands of years; evidence of mathematical knowledge is found in ancient civilizations.
• Mathematical activity is documented as far back as the Old Kingdom of Egypt (c. 3000–2600 BC) and Babylonian mathematics (c. 1800 BC).
• From 600 to 300 BC the Ancient Greeks systematized mathematics.
• From roughly 300 BC to 1100 AD, mathematics developed further in India.
• Islamic mathematics developed mathematics, particularly algebra, to a high standard.
• During the European Renaissance, there was a quick development of mathematics and science.

Areas of Mathematics

• Mathematics includes the study of quantity, structure, space, and change.
• The different branches of mathematics are interconnected. In exploring these connections, mathematics finds applications in diverse fields.

Quantity

• Quantity begins with numbers, first the familiar natural numbers and integers and the arithmetical operations on them, which are covered in arithmetic.
• Deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem.
• Number theory also includes the study of transcendental numbers, numbers that cannot be written as roots of polynomials with rational coefficients.
• The investigation of methods to solve equations leads to the field of algebra.
• Algebra also studies structures, such as vector spaces.
• The closely related field of mathematical analysis rigorously puts the ideas of calculus on a sound footing.
• Mathematical analysis studies such topics as sequences, series, and analytic functions.
• Quantitative study is prevalent in science, such as physics.

Structure

• Structure deals with relations, patterns, and organizations. Numbers can be arranged into structures like matrices, which are studied in linear algebra.
• Linear algebra has applications throughout computer science.
• In abstract algebra, structures are studied in the abstract, such as groups, rings, and fields.
• Abstract algebra is a crucial area for modern cryptography.

Space

• Space is the study of shapes and their properties.
• Geometry, with its roots in prehistory, deals with familiar objects.
• Trigonometry specifically deals with the relationships between sides and angles of triangles.
• Modern geometry examines these objects using calculus and abstract algebra.
• Topology examines properties shared by different shapes.

Change

• Change includes calculus and closely related fields.
• Calculus deals with rates of change in quantities and has numerous applications in science.
• Functions are central to calculus, describing how an input produces an output.
• Differential equations analyze change in functions.
• Chaos theory studies systems governed by differential equations that exhibit unpredictable behavior.

Foundations and Philosophy

• In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed.
• Mathematical logic deals with putting mathematics on a rigorous axiomatic framework and studies the results of that framework.
• Set theory studies sets, collections of objects.
• Set theory is often used as a foundational system for modern mathematics.
• Category theory deals in an abstract way with mathematical structures and relations between them.
• The philosophy of mathematics is concerned with the epistemological status of mathematical propositions.

Discrete Mathematics

• Discrete mathematics comprises areas of mathematics that are discrete, as opposed to continuous.
• Combinatorics studies ways of counting certain objects.
• Graph theory studies graphs, networks of points connected by lines.

Applied Mathematics

• Applied mathematics concerns itself with the use of mathematical tools to solve problems in natural sciences, engineering, medicine, finance, business, computer science, and social sciences.
• Operations research studies methods of optimizing certain goals, such as maximizing profit or reducing risk.
• Numerical analysis studies algorithms for solving mathematical problems on computers.

Mathematics as a Science

• Carl Friedrich Gauss referred to mathematics as the "Queen of the Sciences."
• The Russian mathematician Israel Gelfand once stated that "mathematics is the only science."
• The use of mathematics is essential for almost all sciences.
• Mathematics shares similarities with other sciences, such as exploration of consequences of assumptions.
• There is also a difference; mathematics does not accept verification by experiment but relies on logical arguments and proofs.
• Mathematics is essential in many sciences, including physics, computer science, and engineering.
• Mathematics underlies many technological advances.

Mathematics explores quantity, structure, space, and change through abstraction. Mathematicians seek patterns, formulate conjectures, and resolve them with proofs based on axioms and definitions. It's an essential tool in science, engineering, finance and inspires new discoveries and disciplines.