Introduction to Mathematics

• Mathematics is the study of topics such as quantity (numbers), structure, space, and change
• It has no generally accepted definition

History

• Mathematical study began first in ancient cultures
• As a formal science, mathematics was developed by the ancient Greeks

Branches of Mathematics

• Quantity: Mathematics began with numbers, familiar to nearly all humans
• Structure: Deals with the properties of objects that do not depend on quantity
• Space: Studies the relationships between objects
• Change: Describes how quantities change over time

Mathematical Notation

• As mathematics developed, abstract symbols became commonly used
• Notation allows mathematics to be more efficient
• A mathematical expression is a sequence of symbols that can be evaluated
• Conventions dictate the meaning of mathematical symbols
• Symbols vary in different dialects, or areas in which symbols are used
• The equals sign represents the equality of two expressions

Numbers

• Important number types are natural, integer, rational, irrational, and complex
• Natural numbers are the familiar counting numbers
• Integers are natural numbers with their negatives and zero
• Rational numbers are ratios of two integers
• Real numbers include all rational numbers, together with irrational numbers
• Complex numbers are numbers that include the imaginary unit i.

Equations

• An equation is a problem that can be solved for one or more unknowns
• Equations consist of two expressions separated by an equals sign
• A general equation is one which uses symbols to represent arbitrary values
• Types of equations include algebraic, differential, and integral

Theorems

• A theorem is a statement that has been proven to be true
• Theorems are derived from axioms, which are statements assumed to be true
• A conjecture is a statement that is proposed to be true, but has not been proven
• Theorems can be proven under certain axioms

Calculus

• Studies continuous change
• Has two main branches:
  • Differential calculus is concerned with the instantaneous rate of change of quantities
  • Integral calculus is concerned with the accumulation of quantities
• Calculus is used extensively in physics, engineering, economics, and computer science

Algebra

• Algebra is the study of mathematical symbols and the rules for manipulating these symbols
• Elementary algebra is an essential part of mathematics
• Abstract algebra studies algebraic structures such as groups, rings, and fields

Geometry

• Geometry is one of the oldest branches of mathematics
• Initially concerned with practical problems such as surveying
• Euclidean geometry studies shapes, lines, angles, surfaces, and solids
• Modern geometry includes many non-Euclidean geometries

Trigonometry

• Trigonometry studies relationships between angles and sides of triangles
• Trigonometric functions such as sine, cosine, and tangent are used to describe these relationships

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete
• It includes logic, set theory, combinatorics, graph theory, and cryptography
• Discrete mathematics is used extensively in computer science

Mathematical Proof

• A mathematical proof is an argument that demonstrates that a statement is true
• Proofs use logic and previously-established results to arrive at a conclusion
• Proofs rely on axioms and rules of inference
• Proofs can be direct, indirect, or by contradiction
• Proofs can also use mathematical induction

Statistics and Probability

• Statistics is concerned with collecting, analyzing, and interpreting data
• Probability is the study of chance and uncertainty
• Both are used in decision-making

Topology

• Studies properties that are preserved through deformation
• Has connections to analysis, geometry, and algebra

Applications of Mathematics

• Used in nearly all fields
• Used in the natural sciences, engineering, medicine, finance, and the social sciences
• Essential in computer science and information technology
• Used for creating models, simulations, and predictions

Mathematical Modeling

• Mathematical models are used to study real-world problems
• Models simplify complex systems
• Models can be used to make predictions
• Results can be validated through comparison with observations

Mathematical Finance

• Uses mathematical models for financial markets
• Used for pricing derivatives, managing risk, and portfolio optimization

Mathematical Physics

• Applies mathematics to problems in physics
• Models help to study mechanics, electromagnetism, and thermodynamics

Computational Mathematics

• Deals with using computers to solve mathematical problems
• Includes numerical analysis, symbolic computation, and scientific computing

Set Theory

• Branch of mathematical logic that studies sets
• Sets are collections of objects
• Set theory is foundational for many areas of mathematics

Logic

• Logic is the study of reasoning
• Mathematical logic applies formal logic to mathematics
• Used in computer science for algorithm design and verification

Mathematical Analysis

• Rigorous treatment of calculus
• Includes real analysis and complex analysis
• Used to study limits, continuity, differentiation, and integration

Combinatorics

• Counts arrangements of objects
• Used in computer science and operations research

Graph Theory

• Studies graphs, which are collections of nodes and edges
• Used in computer science, social networks, and operations research

Game Theory

• Studies strategic interaction between individuals
• Used in economics, political science, and biology

Mathematical Optimization

• Finds best solution from a set of available alternatives
• Used in engineering, economics, and operations research