Introduction to Mathematics

• Mathematics is the abstract study of quantity, structure, space, and change.
• Mathematicians and philosophers hold varied opinions regarding the precise scope and definition of mathematics.
• Mathematics aims to identify patterns and create new conjectures.
• Mathematicians use mathematical proofs to determine whether conjectures are true or false.
• Mathematical reasoning offers insights and predictions about nature when mathematical structures effectively model real-world phenomena.
• Mathematics evolved through abstraction and logic from counting, calculation, measurement, and the systematic study of physical object shapes and motions.

Areas of Mathematics

• Arithmetic involves studying numbers and their operations.
• Algebra entails studying arithmetic operation generalizations.
• Geometry is the study of the shapes, sizes, and positions of figures.
• Trigonometry studies the relationships between triangle angles and sides.
• Calculus involves studying continuous change.
• Statistics is the study of data collection, analysis, interpretation, presentation, and organization.
• Logic involves studying valid reasoning and argumentation.
• Number theory studies the properties and relationships of numbers, especially integers.
• Topology studies the properties of spaces that remain unchanged under continuous deformations.
• Discrete mathematics studies mathematical structures that are fundamentally discrete instead of continuous.

Mathematical Notation

• Mathematical notation refers to a symbolic writing system for representing mathematical objects and concepts.
• Mathematics, science, and engineering all use notation.
• It uses simple symbolic representations, such as 0, 1 and 2; letters in algebra; symbols such as sin, +, −, × , ÷, and =; and more complex diagrams.

Proof

• A mathematical proof constitutes an inferential argument for a mathematical statement.
• Mathematicians use proven statements or axioms when proving a statement.
• Axioms are basic assumptions accepted as true without proof, forming the foundation of a mathematical theory.
• Proofs use logic but usually include some amount of natural language which usually admits some ambiguity.
• Most written mathematics proofs aren't formal proofs.
• Formal proofs are written unambiguously in formal language.
• Formal proofs are considered in proof theory.
• The philosophy of mathematics explores the roles of language and logic in proofs.
• Proofs are a type of epistemic justification.
• Proofs aim to demonstrate a statement's truth beyond any doubt.

History of Mathematics

• The development of mathematical knowledge across cultures and millennia is chronicled in the history of mathematics.
• Mathematical discoveries have been essential to scientific advancement from ancient times to the modern era.
• Mathematics plays a crucial role in fields like natural science, engineering, medicine, finance, and social science.
• Statistics and game theory are mathematical fields that are developed in direct correlation with their applications.
• Number theory is a field that is largely developed independently of any application.
• Mathematical discoveries and practical applications have a continuous interplay which fuels the development of both mathematics and the fields that use it.
• Proto-mathematics can be traced back to 30,000 BCE, with tally sticks discovered in Africa and France.
• Egyptians developed practical geometry for surveying after the annual Nile floods.
• Mesopotamians devised a numeral system and solved algebraic equations.
• Greek mathematics which is attributed to Thales of Miletus and Pythagoras of Samos, emphasized theoretical, abstract approaches.
• Euclid's "Elements" systematized Greek mathematical knowledge.
• Archimedes significantly advanced geometry and calculus.
• Indian mathematicians developed zero and trigonometric functions.
• Islamic scholars preserved and translated Greek texts while advancing algebra.
• Mathematical advancements during the Renaissance were crucial to the Scientific Revolution.
• Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus.
• Leonhard Euler standardized mathematical notation and made key discoveries in calculus, number theory, and graph theory.
• In the 19th and 20th centuries, mathematics became more abstract and specialized.
• Georg Cantor's set theory and Gottlob Frege's mathematical logic provided the basis for modern mathematics.
• Computational mathematics and numerical analysis are new areas of mathematical research enabled by computer technology.

Influences

• Mathematics is used globally as a key tool in natural science, engineering, medicine, finance, and the social sciences.
• Applied mathematics applies mathematical tools to address problems in science, engineering, and other fields.
• Applied mathematics is a mathematical science possessing specialized knowledge.
• The term "applied mathematics" also refers to the professional specialty focused on solving practical problems using mathematics.
• Applied mathematics as a professional specialty emphasizes the mathematical modeling of real-world problems.
• To develop and improve new models, applied mathematics employs mathematical techniques.
• Applied mathematics is used in fields like electrical engineering, mechanical engineering, physics, chemistry, computer science, and operations research.
• Finance and economics can drive applied mathematics research.