Introduction to Mathematics

• Mathematics is the study of topics such as quantity, structure, space, and change.
• There are different views among mathematicians and philosophers as to the exact scope and definition of mathematics.
• Mathematics seeks out patterns and formulates new conjectures.
• Mathematicians resolve the truth or falsity of conjectures by mathematical proofs.
• When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature.
• Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects.
• Practical mathematics has been a human activity for as long as written records exist.
• Research in mathematics continues to this day.
• Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory.
• Mathematicians engage in pure mathematics without having any application in mind.
• There is no clear line separating pure and applied mathematics.
• Topics typically found in school mathematics are arithmetic, algebra, geometry, trigonometry, calculus, probability, and statistics.

History

• The history of mathematics is thousands of years old.
• It is strongly linked to the developments of civilization.
• Mathematics started out with counting.
• Mesopotamian/Babylonian mathematics (c. 3000–300 BC) developed arithmetic, algebra, and geometry.
• This was for purposes of taxation, commerce, finance, and construction.
• The oldest mathematical texts from Mesopotamia and Egypt date back to the 2nd millennium BC.
• Most ancient mathematical texts available to us are from 1800 to 200 BC.
• Some of the most notable are the Rhind Mathematical Papyrus, Moscow Mathematical Papyrus, Sulba Sutras, and the Plimpton 322.
• In ancient Egypt, mathematics was used for surveying, construction, and astronomy.
• A well-established numeral system was used.
• The ancient Sulba Sutras in India document the beginnings of algebra.
• This included rules for finding Pythagorean triples.
• Greek mathematics began to be systematically organized as a deductive science around 600 to 300 BC.
• Early Greek mathematics includes notable figures such as Euclid, Archimedes, and Pythagoras.
• Euclid introduced mathematical rigor to geometry and established the axiomatic method.
• Archimedes is credited with using the method of exhaustion to calculate the area under a parabola.
• The book The Nine Chapters on the Mathematical Art focused on solving practical problems.
• The book provided methods for finding square roots and cube roots.
• There was significant development of mathematics in the medieval Islamic world.
• Islamic scholars preserved and translated Greek mathematical manuscripts.
• Islamic scholars developed mathematics further.
• Islamic mathematics included people such as Al-Khwarizmi, Omar Khayyam, and Al-Kashi.
• Al-Khwarizmi developed algebra in the 9th century.
• Omar Khayyam found geometric solutions to cubic equations.
• Al-Kashi discovered the law of cosines.
• Many Greek and Arabic texts on mathematics were translated into Latin during the medieval period.
• From the 12th century onwards, mathematics developed at an accelerating pace in Europe.
• The development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century revolutionized mathematics.
• Leonhard Euler made discoveries in mathematical analysis, number theory, and graph theory in the 18th century.
• Carl Friedrich Gauss contributed significantly to number theory, algebra, statistics, and analysis in the 19th century.
• The International Mathematical Union was founded in 1920.

Subdisciplines

• Mathematics includes the subdisciplines of arithmetic, algebra, geometry, and calculus.
• In addition, mathematics includes trigonometry, mathematical analysis, number theory, discrete mathematics, numerical analysis, topology, and statistics.

Arithmetic

• Arithmetic is the study of numbers and the basic operations on them.
• Basic arithmetic operations include addition, subtraction, multiplication, and division.
• More advanced topics in arithmetic include exponents, roots, percentages, and logarithms.
• Arithmetic is the foundation for all other branches of mathematics.

Algebra

• Algebra is the study of mathematical symbols and the rules for manipulating these symbols.
• Algebra includes solving equations, simplifying expressions, and studying abstract structures.
• Elementary algebra is essential for understanding more advanced mathematical concepts.

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Geometry includes Euclidean, non-Euclidean, analytic, differential, and topological geometries.
• Geometry has practical applications in fields.
• These include art, architecture, engineering, and physics.

Trigonometry

• Trigonometry focuses on relationships between angles and sides of triangles.
• Trigonometry is used in navigation, surveying, and engineering.
• Trigonometry relates angles to distances.

Calculus

• Calculus is concerned with the study of continuous change.
• The two main branches of calculus are differential calculus and integral calculus.
• Differential calculus deals with rates of change and slopes of curves.
• Integral calculus deals with the accumulation of quantities and the areas under and between curves.
• Calculus is used in physics, engineering, economics, and computer science.

Mathematical Analysis

• Mathematical analysis is a branch of mathematics that deals with the rigorous study of calculus, real and complex numbers, and related topics.
• Mathematical analysis provides a solid foundation for advanced topics in mathematics.
• It includes topics such as sequences, series, limits, and continuity.

Number Theory

• Number theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers.
• Number theory includes prime numbers, divisibility, and congruences.
• Number theory concepts are used in cryptography and computer science.

Discrete Mathematics

• Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
• Discrete mathematics includes combinatorics, graph theory, logic, and set theory.
• Discrete mathematics is essential for computer science and information technology.

Numerical Analysis

• Numerical analysis is concerned with developing and analyzing algorithms for solving mathematical problems numerically.
• These problems include those arising in calculus, linear algebra, and differential equations.
• Numerical analysis is widely used in engineering, physics, finance, and other scientific disciplines.

Topology

• Topology is concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending.
• Topology includes point-set topology, algebraic topology, and differential topology.
• Topology has applications in physics, computer science, and data analysis.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Statistics includes descriptive statistics, inferential statistics, probability, and regression analysis.
• Statistics is used in almost every field, including natural sciences, social sciences, engineering, and business.

Mathematical Notation

• Mathematical notation is a system used to record mathematical concepts.
• Notation uses symbols to represent numbers, operations, relations, and other mathematical objects.
• Common symbols include +, −, ×, ÷, =, <, >, and π.
• Correct and consistent notation is essential for understanding and communicating mathematical ideas.

Proofs

• Mathematical proofs are rigorous logical arguments that demonstrate the truth of a statement.
• Proofs rely on axioms, definitions, and previously proven theorems.
• Different types of proofs include direct proofs, indirect proofs, proofs by contradiction, and proofs by induction.
• Proofs are essential for establishing mathematical knowledge and ensuring its reliability.

Abstraction

• Abstraction is a fundamental concept of mathematics.
• Abstraction involves generalizing from particular instances to more general concepts or principles.
• Abstraction allows mathematicians to identify common structures and patterns across different areas of mathematics.
• Abstraction leads to the development of more powerful and versatile mathematical tools.

Open Problems

• There are many unsolved problems in mathematics.
• These unresolved problems drive ongoing research and discovery.
• Some famous open problems include the Riemann Hypothesis and Fermat's Last Theorem, the latter of which was solved in 1994 by Andrew Wiles.
• Unsolved problems often lead to new insights and techniques.