Introduction to Mathematics

• Mathematics studies quantity, structure, space, and change.
• Mathematicians and philosophers hold a range of views on the exact scope and definition of mathematics.
• Mathematics seeks patterns and creates new conjectures.
• Truth or falsity of conjectures are resolved by mathematical proofs.
• Mathematical reasoning offers insight/predictions about nature when mathematical structures model real phenomena well.
• Mathematics evolved through abstraction and logical reasoning from counting, calculation, measurement, and the systematic study of physical objects' shapes/motions.
• Practical mathematics has been a human activity throughout recorded history.
• Mathematical research is ongoing.
• Mathematics is essential in natural science, engineering, medicine, finance, and social science.
• Applied mathematics has spurred new mathematical disciplines like statistics and game theory.
• Pure mathematics is pursued for its own sake, without consideration of application.
• Mathematicians do not have a consensus on what constitutes mathematics.
• Schools of thought are united by logical rigor, abstraction, and generality.
• Mathematics is broadly classified into algebra, analysis, and geometry.
• Subdivisions investigate links from mathematics to logic, set theory (foundational mathematics), empirical sciences (applied mathematics), and uncertainty.

History

• Mathematics history is intertwined with social developments.
• Tally marks dating back 30,000 years suggest early human mathematics use.
• Early mathematics served trade, land surveying, taxation, and calendar development.

Ancient Near East (c. 3000–300 BC)

• Mesopotamian mathematics originated in Sumer from 3000 BC.
• A base-60 numeral system was developed.
• Old Babylonian period (c. 2000–1600 BC) tablets cover fractions, algebra, quadratic/cubic equations, and regular reciprocal pair calculations.
• Babylonian tablets hold the oldest known √2 approximation.
• Egyptian mathematics developed independently from Mesopotamia, starting in the predynastic period (c. 3000 BC).
• The Rhind Mathematical Papyrus (c. 1650 BC) is a key source, featuring arithmetic, algebra, geometry, and trigonometry problems.
• The Moscow Mathematical Papyrus is another important source.
• It contains geometry problems, such as finding the volume of a frustum.

Classical Antiquity (c. 800 BC – 600 AD)

• Greek mathematics began with Thales (c. 624–546 BC) and Pythagoras (c. 582–507 BC).
• Pythagoras believed "All is number" and sought to explain the world through numbers.
• Euclid (c. 300 BC) systematized mathematical knowledge in Elements.
• Elements used deductive reasoning to present geometric theorems with proofs.
• Archimedes (c. 287–212 BC) devised methods for calculating areas and volumes.
• He also approximated pi (π).
• Diophantus (c. 200–284 AD) is considered the "father of algebra".
• Greek mathematics played a role in astronomy and geography.

Middle Ages (c. 500–1500 AD)

• Mathematics thrived in the Islamic world during the Middle Ages, especially in the 9th and 10th centuries.
• Muslim mathematicians advanced Greek mathematics and created new concepts.
• Al-Khwarizmi (c. 780–850 AD) is considered the "father of algebra".
• He introduced systematic approaches to solve linear/quadratic equations in The Compendious Book on Calculation by Completion and Balancing.
• Arabic numerals, including zero, originated in India and spread to the Islamic world and Europe.
• Leonardo Fibonacci (c. 1170–1250 AD) popularized the Hindu-Arabic numeral system in Europe via Liber Abaci.
• Medieval scholars translated Greek/Arabic texts into Latin, disseminating mathematical knowledge in Europe.
• Indian mathematics contributed significantly to trigonometry, including sine and cosine functions.

Renaissance and Early Modern Period (c. 1400–1700 AD)

• European mathematicians made discoveries and developed tools during the Renaissance.
• 16th-century Italy saw solutions for cubic and quartic equations.
• Nicolaus Copernicus (1473–1543) used mathematics for his heliocentric solar system model.
• Johannes Kepler (1571–1630) used math to develop his laws of planetary motion.
• René Descartes (1596–1650) created analytic geometry, combining algebra and geometry.
• Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716) independently invented calculus.
• Calculus enabled new solutions for physics, engineering, and other fields.

Modern Period (c. 1700–present)

• The 18th and 19th centuries saw rapid mathematical development.
• Leonhard Euler (1707–1783) contributed to calculus, number theory, and graph theory.
• Joseph-Louis Lagrange (1736–1813) developed Lagrangian mechanics, used in physics/engineering.
• Carl Friedrich Gauss (1777–1855) advanced number theory, algebra, and analysis.
• Non-Euclidean geometry emerged in the 19th century.
• This challenged Euclidean geometry as the only possible geometry.
• Georg Cantor (1845–1918) developed set theory, a foundation of modern mathematics.
• The 20th century saw mathematics become more abstract and specialized.
• David Hilbert (1862–1943) proposed 23 unsolved problems that have guided mathematical research.
• Kurt Gödel (1906–1978) proved incompleteness theorems, demonstrating limits to provability in mathematics.
• Computers have revolutionized mathematics, enabling solution of complex problems and exploration of new ideas.
• Mathematics is applied across science, engineering, finance, and computer science.

Notation, terminology, and style

• Mathematics relies on notation.
• Mathematical notation has evolved over centuries.
• Mathematical notation makes mathematics efficient.
• Mathematical notation employs specific terminology.
• Mathematics typically uses a hierarchy of axioms, definitions, and theorems.
• Axioms are self-evident statements.
• Definitions define new terms using existing ones.
• Theorems are proven true using axioms, definitions, and previously proven theorems.
• Proofs are used to demonstrate the truth of theorems.

Pure and applied mathematics

• Pure mathematics explores mathematical concepts without regard to external applications.
• Applied mathematics uses mathematical concepts to address problems in other fields.
• Pure mathematics can later find applications.
• Applied mathematics can influence pure mathematics.
• Applied mathematics can lead to discovery and exploration of new mathematical areas.

Branches of mathematics

• Mathematics has a large number of different areas
• These areas can be divided in different ways

Quantity

• Quantity begins with numbers
• Numbers are familiar as the counting numbers
• The number system has been extended over the centuries to include fractions, rational numbers, irrational numbers, and complex numbers.
• Number theory studies the properties of integers.
• Number theory contains unsolved problems.

Structure

• Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of the operations or relations that are defined on the set.
• Mathematics then studies properties of these structures.
• Abstract algebra studies algebraic structures such as groups, rings, and fields.
• Analysis studies structures that involve the concept of limit, such as real numbers, complex numbers, and functions.

Space

• Space is studied through geometry.
• Trigonometry studies the relationships between the sides and angles of triangles.
• Differential geometry studies curves and surfaces.
• Topology studies properties of space that are preserved under continuous deformations.

Change

• Understanding and describing change is a common theme in the natural sciences.
• Calculus was developed to investigate change.
• Vector calculus extends calculus to functions of several variables.
• Differential equations study functions that describe how quantities change over time.
• Dynamical systems theory studies systems that evolve over time.
• Chaos theory studies systems that are highly sensitive to initial conditions.