Introduction to Mathematics

• Math 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

History

• The history of mathematics can be divided into distinct periods - Mesopotamia, Egypt, Ancient Greece, Hellenistic mathematics, Islamic mathematics, India, Renaissance, and 19th century mathematics
• Mathematical discoveries continue to be made today

Branches

• Quantity relates to numbers, describing their actions, and investigating their generalizations
• Structure relates to mathematical objects and their relationships to each other
• Space originates from the representation of the world around us
• Change is perhaps the most self-evident category and is handled by calculus

Foundations and Philosophy

• To clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed
• Mathematical logic deals with applying formal logic to mathematics

Notation, Terminology, and Style

• Mathematical notation has ramified greatly over time
• Mathematical notation uses symbols to represent mathematical objects and concepts
• A theorem is a statement that has been proven on the basis of previously established statements
• According to a widespread convention, articles appearing in mathematical journals and books are generally based on a style in which all formulas have been rendered in italics

Pure and Applied Mathematics

• Pure mathematics explores mathematical concepts without regard to applications outside of mathematics
• Applied mathematics is concerned with the use of mathematical tools to solve problems in science, engineering, and other fields

Relationship with Science

• Mathematics is essential to many fields of science, both theoretical and applied
• Mathematical models are widely used in physics, chemistry, biology, and other disciplines
• Statistics is directly useful in data analysis, allowing researchers to test hypotheses and draw conclusions from empirical data

Beauty

• Many mathematicians are drawn to their subject for its beauty
• One aspect of beauty is its precision
• There are many philosophical debates about mathematics and its impact on society