Introduction to Mathematics

• Mathematics is the study of topics such as quantity (numbers), structure, space, and change.
• It has no generally accepted definition.
• Mathematicians seek out patterns and formulate new conjectures.
• They resolve the truth or falsity of conjectures by mathematical proofs.
• Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics concerns itself with the use of mathematical tools to solve problems in natural sciences, engineering, medicine, finance, business, computer science, and social sciences.
• Mathematics is used to model real-world phenomena.
• Statistics is concerned with the collection, analysis, interpretation, and presentation of data.
• Computational mathematics includes numerical analysis, symbolic computation, and computer algebra.

History of Mathematics

• The history of mathematics is an ever-growing series of abstractions.
• The concept of abstract numbers and the practice of determining the area of a field were developed early in mathematics.
• The oldest mathematical texts available are from ancient Egypt and Mesopotamia.
• Rigorous proof first appeared in Greek mathematics.
• Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical development.
• The development of infinitesimal calculus by Newton and Leibniz in the 17th century revolutionized mathematics.
• Leonhard Euler made numerous contributions to notation, terminology and mathematical formulas.

Areas of Mathematics

Quantity

• Quantity begins with numbers, first the familiar natural numbers (1, 2, 3, ...).
• Next are integers which include negative numbers (-1, -2, -3, ...).
• Rational numbers arise as quotients of natural numbers (e.g., 3/4).
• Real numbers are all numbers that can be represented by a decimal, and include both the rational numbers and the irrational numbers.
• Complex numbers allow solutions to equations that have no solutions in real numbers.
• Hypercomplex numbers extend the complex numbers.
• The study of numbers is number theory, and includes Fermat's Last Theorem.

Structure

• Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set.
• Abstract algebra is the study of algebraic structures, such as groups, rings, and fields.
• Vector spaces are used in linear algebra.
• Analysis studies structures that include a notion of closeness.

Space

• Space is studied in geometry.
• Trigonometry studies the relationships between the sides and angles of triangles.
• Differential geometry studies curves and surfaces.
• Topology studies properties that are invariant under continuous transformations.

Change

• Understanding and describing change is a common theme in the natural sciences.
• Calculus was developed to investigate change.
• Functions arise here, as central to describing a changing quantity.
• Differential equations describe the relationship between a function and its derivatives.
• Chaos theory deals with dynamical systems that exhibit significant sensitivity to initial conditions.

Foundations and Philosophy

• In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed.
• Mathematical logic deals with putting mathematics on a solid axiomatic framework, and studies the implications of that framework.
• Set theory is the branch of mathematics that studies sets or collections of objects.
• Category theory deals in an abstract way with mathematical structures and relations between them.

Discrete Mathematics

• Discrete mathematics groups together the fields of mathematics which deal with discrete objects.
• Combinatorics studies ways of enumerating certain objects.
• Graph theory studies graphs and networks.

Mathematical Notation, Language, and Rigor

• Most of the mathematical notation in use today was not invented until the 16th century.
• Before that, mathematics was written out in words.
• Mathematical notation makes mathematics easier for the professional to manipulate, but beginners often find it daunting.
• A mathematical proof is sufficient to convince other mathematicians of the truth of the statement.
• The need for mathematical rigor led to the development of mathematical logic and set theory.
• Mathematical rigor differs somewhat from rigor in other sciences.

Pure and Applied Mathematics

• Pure mathematics is mathematics that is developed primarily for its own sake.
• Applied mathematics is mathematics with substantial applications in other areas.
• Many branches of mathematics that began as pure mathematics have later become applied mathematics.
• Applied mathematics is often used to solve practical problems.
• There is no sharp division between pure and applied mathematics.