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 the branch of mathematics concerned with the application of mathematical knowledge to other fields
• Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social science
• Applied mathematics inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines
• Mathematicians engage in pure mathematics without having any application in mind
• The application of math to solve real-world problems has led to the development of new fields, such as statistics and game theory

History of Mathematics

• The history of mathematics is nearly as old as humanity itself
• Mathematical study first arose as a response to practical problems such as trade, land measurement, and astronomy
• The oldest mathematical texts available are from ancient Egypt and Mesopotamia
• The Rhind Mathematical Papyrus (c. 1650 BC) is an example of Egyptian mathematics
• Mesopotamian mathematics is exemplified by clay tablets such as Plimpton 322 (c. 1800 BC)
• The development of formal mathematics is attributed to the ancient Greeks
• Greek mathematics was largely based on geometry
• Euclid (c. 300 BC) introduced the axiomatic method still used in mathematics today
• Archimedes (c. 287–212 BC) is known for the method of exhaustion, a precursor to integral calculus
• The numeral system we use today has its origins in India
• Hindu-Arabic numerals and rules for the use of its operations were developed by around AD 600
• Mathematics flourished in the medieval Islamic world
• Important contributions were made by Persian and Arab mathematicians
• Al-Khwarizmi (c. 791–850) gave his name to the concept of the algorithm
• The term algebra is derived from the Arabic "al-jabr"
• During the Renaissance, there was an increased emphasis on both mathematics and science in Europe
• The development of calculus by Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716) was a major turning point
• Leonhard Euler (1707–1783) made numerous contributions to modern notation, terminology and formulas
• In the 19th century, mathematics became more abstract and self-reflective
• Important advances were made in areas such as non-Euclidean geometry, group theory, and set theory
• The first international Mathematical Congress was held in 1897
• The 20th century saw rapid growth in mathematical activity
• There was increasing abstraction and computer assistance

Subdisciplines

• Mathematics is broadly divided into several subdisciplines
• These subdisciplines can, in turn, be further divided

Quantity

• Quantity begins with numbers, first the familiar natural numbers and integers and then arithmetic operations on them
• The deeper properties of integers are studied in number theory
• Real numbers more closely describe continuous quantities
• The study of real numbers is known as real analysis
• Complex numbers permit solutions to algebraic equations not possible with real numbers alone
• Complex numbers are studied in complex analysis

Space

• Space starts with geometry, specifically Euclidean geometry
• Trigonometry combines space and number, and incorporates the well-known Pythagorean theorem
• The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries, and topology
• Calculus takes these ideas in another direction, introducing the notions of smoothness and continuity
• Topology describes the properties that are invariant under continuous deformations

Change

• Understanding and describing change in nature is a common theme in the natural sciences
• Calculus was developed as a tool to investigate change
• The central concepts are derivatives and integrals
• Differential equations relate a function to its derivatives
• In dynamical systems, iteration of a map from a space to itself is studied
• Chaos theory studies dynamical systems that exhibit highly sensitive behavior

Structure

• Mathematics involves characterizing the internal structure of various objects
• It looks at collections themselves in the study of groups, rings, fields, and other abstract systems
• These are the subjects of abstract algebra
• Vector spaces are important in linear algebra
• The vector and matrix notation is important in statistics

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
• Set theory studies collections of objects
• The search for a foundation for math led to the discovery of different schools of thought
• There is the intuitionist school, the Platonist school, and others

Discrete Mathematics

• Discrete mathematics groups together the fields of mathematics which are used to study discrete objects
• Graph theory, combinatorics, coding theory, cryptography, and game theory are considered part of discrete mathematics

Applied Mathematics

• Applied mathematics considers the application of mathematics to other disciplines such as science, engineering, and business
• Fields included are numerical analysis, optimization, and mathematical modeling
• Statistics is used to infer conclusions from data

Mathematical Notation, Terminology, and Style

• Mathematics uses its own, sometimes unique, notation and terminology
• Mathematical notation is used by mathematicians to represent mathematical concepts
• Notation can refer to a wide variety of things
• Notation may be used for the name of a function, and a theorem, among other things
• One major aspect of mathematical notation is that it is now mostly standard throughout the world
• The notation clarifies aspects of the subject that would be cumbersome to describe in ordinary written language
• Mathematical notation is more terse than natural language
• Mathematical notation helps some people grasp concepts

Mathematical rigor

• Mathematical rigor is paramount in mathematics
• A mathematical proof is a sufficient demonstration that, given certain axiomatic systems, certain statements are necessarily true
• A theorem is a statement that has been proven to be true
• A lemma is a theorem that is used to prove another theorem
• A proposition is a statement that has been proven to be true, but is not as important as a theorem
• A corollary is a theorem that follows directly from another theorem
• In order to be considered worthwhile, a mathematical statement must be proven
• Proofs can be complicated, and can take a long time to find
• Once a theorem has been proven, it can be used to build other mathematical results

Mathematics as Science

• Carl Friedrich Gauss called mathematics the "Queen of the Sciences"
• Mathematics shares similarities with the sciences in that it involves the rigorous investigation of a subject
• Mathematics also differs from the sciences in several ways
• Mathematics does not rely on experimentation or observation
• Mathematics is based on logical reasoning and deduction
• Mathematics is often used as a tool in the sciences
• Some mathematicians are also scientists and vice versa
• Physics uses mathematics extensively
• Physics is often a source of inspiration for new mathematics

Mathematics and Art

• Mathematics and art are related in a variety of ways
• Mathematics can be used to create art
• The principles of mathematics, such as ratio, proportion, and symmetry, are used in art
• Mathematical objects, such as the Mobius strip, have inspired art
• Computer art makes use of mathematical formulas and algorithms

Awards

• The Fields Medal and the Abel Prize are major awards in mathematics
• These are awarded to mathematicians for outstanding achievements in the field
• There is no Nobel Prize in mathematics