Introduction to Mathematics

• Mathematics is the study of topics such as quantity (numbers), structure, space, and change
• There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
• Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social science
• Applied mathematics applies mathematical knowledge to other fields
• Pure mathematics studies mathematics for its intrinsic interest
• The concept of proof is central to mathematics

History

• The history of mathematics stretches back thousands of years
• Mathematical methods evolved from counting, measuring, and describing the shapes of objects
• Ancient Egyptians used math for surveying and construction, including basic arithmetic, algebra and geometry
• Ancient Mesopotamians developed a numeral system and could solve algebraic equations
• Greek mathematics refined the methods and expanded the subject matter
• Key Greek mathematicians: Thales, Euclid, Archimedes, and Pythagoras
• The Elements by Euclid introduced mathematical rigor through axiomatic method and proofs
• Indian mathematics made key contributions to the numeral system and algebra
• With the development of the Hindu–Arabic numeral system using a positional notation and zero, mathematics started to develop at an accelerated pace
• Islamic scholars preserved Greek mathematical knowledge and made advancements, particularly in algebra
• During the Renaissance in Europe, there was rapid development in mathematics and science
• 17th century saw innovations such as analytic geometry and calculus
• Isaac Newton used calculus to formulate laws of motion and gravitation
• 18th century saw work on differential equations and new techniques for solving them
• Leonhard Euler standardized mathematical notations
• 19th century saw the rise of abstract algebra and efforts to formalize all of mathematics
• Key figures in 19th century math: Carl Friedrich Gauss, Niels Henrik Abel, and Évariste Galois
• At the International Congress of Mathematicians in 1900, David Hilbert presented a list of 23 unsolved problems that drove much of 20th-century mathematics
• In the 20th century, mathematics became increasingly specialized

Subdisciplines

• Arithmetic: Elementary branch dealing with numbers and basic operations like addition, subtraction, multiplication, and division
• Algebra: Studies mathematical symbols and the rules for manipulating these symbols
• Geometry: Deals with shapes, sizes, relative positions of figures, and the properties of space
• Trigonometry: Studies relationships between angles and sides of triangles
• Calculus: Deals with the study of continuous change, rates of change, and accumulation
• Statistics: Deals with the collection, analysis, interpretation, presentation, and organization of data.
• Probability: Deals with the likelihood of an event occurring
• Set theory: Branch of mathematical logic that studies sets
• Topology: Studies the properties of spaces that are preserved under continuous deformations

Mathematical Notation

• Mathematical notation consists of symbolic representations used to represent mathematical objects and ideas
• Notation can be used for relatively simple operations, such as +, −, ×, ÷
• Variables are symbols that represent a quantity that can change or vary
• Formulas are equations that express a relationship between different quantities
• Examples of mathematical notation and symbols:
  • • for addition
  • • for subtraction
  • = for equality
  • > for greater than
  • < for less than

Proofs

• Proofs are ways to demonstrate that certain statements are necessarily correct
• Mathematical proofs use logic and reasoning to establish the truth of a statement
• Axioms are statements that are assumed to be true without proof
• Theorems are statements that have been proven to be true using axioms and previously proven theorems
• Common proof techniques:
  • Direct proof: The conclusion is shown to be true directly from the axioms, definitions, and previous theorems
  • Proof by contradiction: It is shown that if the statement were false, a contradiction would result, implying that the statement must be true.
  • Proof by induction: Used to prove statements about all natural numbers. It involves showing that the statement is true for the base case (usually 0 or 1), and then showing that if the statement is true for some natural number n, it must also be true for n+1.

Applications

• Mathematics is used in almost every field
• Physics uses math to describe the laws of the universe
• Engineering uses math to design infrastructures like bridges and buildings
• Computer science uses math to develop algorithms and software
• Economics uses math to model economic systems
• Finance uses math to price assets and manage risk
• Cryptography uses math to encrypt and decrypt data
• Statistics allows us to analyze and interpret data

Mathematics explores quantity, structure, space, and change, serving as a vital tool across various fields. It encompasses both pure mathematics, driven by intrinsic interest, and applied mathematics, which addresses real-world problems. Proofs are central to establishing mathematical truths and advancing the field.