Intro to Mathematics: History and Application

• Mathematics encompasses the study of quantity, structure, space, and change.
• The field lacks a universally accepted definition.
• Mathematicians explore patterns and create new conjectures.
• They use mathematical proofs to determine the truth or falsity of these conjectures.
• Mathematical concepts are applicable to phenomena in the real world.
• Studying mathematics enhances analytical thinking and problem-solving abilities.
• It is used globally across various fields as an essential tool.
• These fields include natural science, engineering, medicine, finance, and the social sciences.

History

• Mathematics has a history spanning thousands of years.
• Crucial to the advancement of numerous early civilizations.
• Civilizations that relied on trade and agriculture saw the most frequent mathematical advancements.
• Essential for tasks like dividing fields and assessing crop yields.
• The earliest mathematical evidence comes from the Sumerians.
• They created a sophisticated metrology system around 3000 BC.
• Multiplication tables and geometric exercises also began around 3000 BC.
• Babylonian mathematics surpassed Sumerian mathematics.
• The Babylonians created an advanced numeral system.
• This allowed them to create algebraic solutions and tackle difficult problems.
• The Egyptians used mathematics for surveying and building.
• Their own numeral system was also developed.
• Egyptian mathematics incorporated geometric understanding, including area and volume formulas.
• Greek mathematics started in the 6th century BC with the Pythagoreans.
• The term "mathematics" was coined by them.
• Greeks revolutionized mathematics.
• They formalized mathematical rigor and proof.
• Euclid created a very influential geometry system.
• Archimedes is known for using the method of exhaustion to determine the area under a parabola.
• Hindu mathematics was developed around 1500 BC.
• A modern numeral system was also developed.
• Hindu mathematicians made significant early contributions to trigonometry.
• Chinese mathematics independently emerged by the 11th century BC.
• Contributions were made to algebra and geometry.
• During the Golden Age of Islam, Islamic mathematics flourished.
• Islamic mathematicians greatly improved algebra.
• The Renaissance saw increased interest in both mathematics and science.
• Isaac Newton and Gottfried Wilhelm Leibniz developed infinitesimal calculus in the 17th century.
• Leonhard Euler contributed significantly to mathematics.
• This includes analytic geometry, trigonometry, geometry, and number theory.
• Mathematics became more abstract and focused on rigor in the 19th century.
• Traditional geometry was challenged by the development of Non-Euclidean geometry in the 19th century.
• Georg Cantor created set theory.
• Modern mathematics is based on set theory.
• In the 20th century, the first electronic digital computer was developed.
• This led to greater advancements in mathematics, especially in numerical methods.

Subdisciplines

• Mathematics can be divided into several subdisciplines.
• These reflect historical branches and areas of application.
• Different subdisciplines often exhibit unexpected connections.
• This allows mathematics to progress and solve long standing problems.

Quantity

• The study of quantity starts with numbers
• Whole numbers are the most basic numbers
• They are represented by the natural numbers 1, 2, 3...
• Operations with these numbers include addition, subtraction, multiplication, and division
• Further study of numbers includes integers which are negative whole numbers
• Rational numbers are fractions of two integers
• Real numbers can be represented as decimals
• Some decimals continue indefinitely and are non repeating
• These are irrational numbers which can't be expressed as a fraction of two integers
• The study of numbers goes beyond the real numbers to include complex numbers
• These include the imaginary unit i, which is the square root of negative one
• Many number problems are best studied with complex numbers
• Number theory deals with the properties of integers
• It contains many open problems, even though the problems can be easily understood by non-mathematicians

Structure

• Many mathematical objects, such as sets of numbers or functions, posses internal structure
• This structure can be studied using group theory, ring theory, field theory, and module theory
• These are defined using algebraic structures
• An algebraic structure consists of a set and one or more operations
• Abstract algebra is a mathematical discipline that studies algebraic structures
• Vector spaces are important algebraic structures
• Linear algebra studies these structures
• Linear algebra is applicable to both pure mathematics and applied mathematics
• For example, functional analysis studies vector spaces of functions

Space

• The study of space originated with geometry.
• Trigonometry studies the relationships between the sides and angles of triangles.
• Modern geometry examines more complicated spaces than Euclidean space.
• Differential geometry studies smooth spaces, such as manifolds.
• Topology studies properties of spaces that are invariant under continuous deformations.
• This includes properties such as connectedness and compactness.

Change

• Understanding and describing change in nature is a common theme in the natural sciences
• Calculus was developed to study change
• Functions are central to the description of change
• Differential calculus studies the instantaneous rate of change of functions
• Integral calculus studies the accumulation of quantities
• Differential equations relate a function to its derivatives
• They are essential in describing many processes in physics, engineering, and other disciplines
• Numerical analysis studies algorithms for solving mathematical problems
• Most of which are related to change
• Chaos theory describes systems that exhibit unpredictable behavior
• Dynamical systems theory studies the long-term behavior of systems
• Real analysis rigorously defines the concepts of calculus

Foundations and Philosophy

• To clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed
• Mathematical logic studies the formal systems used to prove theorems
• Set theory studies sets, which are collections of objects
• Category theory is an abstract way of formalizing mathematical theories and their relationships

Discrete Mathematics

• Discrete mathematics groups together the fields of mathematics used to study discrete objects
• These include set theory, logic, and combinatorics
• Theoretical computer science includes computability theory, complexity theory, and information theory

Applied Mathematics

• Applied mathematics concerns itself with the application of mathematical knowledge to other fields
• These fields include science, engineering, and computer science
• Game theory studies strategic situations
• Operations research optimizes complex processes
• Control theory controls the behavior of dynamical systems
• Mathematical biology applies mathematical tools to study biological systems

Notation, Terminology, and Style

• Mathematics relies on notation.
• It is more concise than natural language.
• This allows mathematicians to manipulate mathematical ideas easily.
• Mathematical notation includes symbols for numbers, variables, operations, functions, and other objects.
• Mathematical expression is a sequence of symbols that can be evaluated.
• A formula is an expression that expresses a fact or relationship.
• Theorems and proofs are used to establish mathematical truths.
• A theorem is a statement that has been proven to be true based on axioms.
• A proof is a logical argument that demonstrates the truth of a theorem.
• Axioms are statements that are assumed to be true without proof.
• Definitions are used to give precise meanings to mathematical terms.
• Conventions define commonly used notations and terminology.