Introduction to Mathematics

• Math encompasses the science and study of quantity, structure, space, and change
• Mathematicians identify patterns, create conjectures, and use axioms and definitions to establish truths through rigorous deduction
• Math provides essential tools applicable worldwide across natural science, engineering, medicine, finance, and social sciences
• Applied mathematics utilizes mathematical tools to address practical problems in natural science, engineering, medicine, finance, and social sciences

Areas of mathematics

• Arithmetic studies numbers, their properties, and basic operations such as addition, subtraction, multiplication, and division
• Algebra involves mathematical symbols and the rules that govern their manipulation
• Geometry examines the properties and relationships of points, lines, surfaces, solids, and their higher-dimensional counterparts
• Trigonometry focuses on the relationships between angles and sides of triangles
• Calculus explores continuous change, including rates of change and the accumulation of quantities
• Statistics includes the collection, analysis, interpretation, presentation, and organization of data
• Logic is the study of valid reasoning, argumentation, consistency, non-contradiction, and truth
• Number Theory studies the properties and relationships of numbers, especially integers
• Topology studies geometric properties that remain unchanged under continuous deformations like stretching, twisting, crumpling, and bending, excluding tearing or gluing
• Discrete Mathematics studies mathematical structures that are fundamentally discrete rather than continuous
• Set Theory studies sets as abstract collections of objects

Basic Mathematical Concepts

• Numbers are used for counting, measuring, and labeling
  • Natural numbers include 1, 2, 3, and so on
  • Integers include ..., -2, -1, 0, 1, 2, and so on
  • Rational numbers can be expressed as fractions
  • Real numbers include both rational and irrational numbers
  • Complex numbers consist of real and imaginary parts
• Variables are symbols representing unknown or changing quantities
• Equations are mathematical statements asserting the equality of two expressions
• Functions represent relationships where each input corresponds to a unique output
• Operations refer to processes like addition, subtraction, multiplication, and division
• Theorems are statements proven true based on axioms and prior theorems
• Proofs demonstrate the truth of a theorem through logical arguments

Fundamental Theorems

• Pythagorean Theorem defines the relationship between the sides of a right triangle
• Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers
• Fundamental Theorem of Calculus connects differentiation and integration

Mathematical Notation

• Mathematical symbols include +, -, ×, ÷, =, <, >, ≤, ≥, etc
• Greek letters represent angles, constants, and variables
• Examples of functions are f(x) and g(x)
• Derivatives are expressed as dy/dx and f'(x)
• Integrals are written as ∫f(x) dx
• Limits are expressed as lim (x→a) f(x)
• Summations are denoted by Σ

Mathematical Reasoning

• Deduction involves reasoning from general principles to specific conclusions
• Induction involves reasoning from specific observations to general principles
• Proof by contradiction involves assuming the opposite of what is to be proven, leading to a contradiction

Applications

• Physics utilizes math to model and understand the physical world
• Engineering uses math to design and construct structures, machines, and systems
• Computer science relies on math for computer algorithms and data structures
• Economics uses math to model economic systems and make predictions
• Finance applies math to manage risk and make investment decisions

History of Math

• Ancient civilizations like the Egyptians and Babylonians developed practical math for surveying, construction, and astronomy
• Greek mathematicians such as Thales, Pythagoras, Euclid, and Archimedes significantly contributed to geometry and number theory
• During the Medieval period, the Hindu-Arabic numeral system was adopted, and algebra was further developed
• The Renaissance saw the development of calculus and analytic geometry
• Modern math includes the development of set theory, topology, and other advanced concepts

Mathematical Skills

• Problem-solving involves analyzing problems, developing strategies, and finding solutions
• Logical reasoning includes the abilities to think critically and construct valid arguments
• Abstract thinking is the ability to understand and manipulate abstract concepts and symbols
• Computational skills include performing mathematical operations accurately and efficiently

Branches of Applied Mathematics

• Numerical Analysis focuses on creating and studying algorithms to solve mathematical problems numerically
• Optimization involves finding the best solution to a problem with specific constraints
• Mathematical Modeling creates mathematical representations of real-world situations
• Financial Mathematics uses math techniques in financial markets and risk management
• Actuarial Science employs mathematical and statistical techniques to assess risks in insurance and finance