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
• Mathematicians seek out patterns and formulate new conjectures
• Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their correctness
• Mathematical concepts can apply to real-world phenomena
• Many areas of mathematics proved to have no application are now essential
• Examples of applied mathematics include mathematical analysis, linear algebra, probability theory, cryptography, game theory
• Pure mathematics drives applied mathematics

History of Mathematics

• Mathematics has been used since ancient times
• Use of math ranges from simple counting to measurement to the more complex calculations
• Rigorous argument first appeared in Greek mathematics
• Greek mathematics greatly refined the methods (especially through the introduction of axiomatic method) and expanded the subject matter of mathematics
• Mathematics has been developing 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 discovery that has continued to the present day

Subdisciplines

• Mathematics is broadly divided into the fields of Geometry, Algebra, Analysis, and Discrete Mathematics

Geometry

• Studies shapes, sizes, and positions of figures
• Includes study of lines, angles, surfaces, and solids
• Has practical applications in surveying, navigation, and engineering
• Includes Euclidean geometry, analytic geometry, differential geometry, and topology

Algebra

• Generalizes arithmetic operations and studies mathematical relations using symbols
• Includes elementary algebra, abstract algebra, linear algebra, and number theory
• Has applications in computer science, physics, and engineering

Analysis

• Deals with continuous change and includes calculus, differential equations, and real analysis
• Includes calculus, real analysis, complex analysis, and functional analysis
• Has applications in physics, engineering, and economics

Discrete Mathematics

• Studies discrete (as opposed to continuous) mathematical structures
• Includes logic, set theory, combinatorics, graph theory, and cryptography
• Has applications in computer science and information technology

Mathematical Proof

• Mathematicians use proof to demonstrate the truth of mathematical statements
• A mathematical proof is an argument using logic and previously established facts to show that a statement is true
• Proofs are essential for ensuring the correctness and reliability of mathematical knowledge
• Different proof techniques are used, including direct proof, proof by contradiction, and proof by induction

Statistics

• Statistics is the science of collecting, analyzing, presenting, and interpreting data
• Statistics deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments
• Involves mathematical models, but requires use of computational and domain expertise

Types of Statistics

• Descriptive Statistics summarize and describe the features of a dataset
• Inferential Statistics uses sample data to make inferences and predictions about a larger population

Descriptive Statistics

• Summarizes and describes the main features of a dataset
• Includes measures of central tendency (mean, median, mode) and measures of variability (range, variance, standard deviation)
• Used to simplify and present data in a meaningful way

Inferential Statistics

• Uses sample data to make inferences and predictions about a larger population
• Includes hypothesis testing, confidence intervals, and regression analysis
• Allows for generalizations and predictions based on incomplete data

Data Collection

• Data collection methods include surveys, experiments, and observational studies
• Proper data collection is essential for ensuring the accuracy and reliability of statistical analysis
• Sampling techniques, such as random sampling and stratified sampling, are used to select representative samples from a population

Data Analysis

• Data analysis involves various techniques to extract meaningful information from data
• Includes data cleaning, data transformation, and statistical modeling
• Statistical software packages, such as R, Python, and SAS, are used to perform data analysis

Probability

• Probability is the branch of mathematics that deals with randomness
• Forms the theoretical foundation for statistics
• Provides a framework for quantifying uncertainty and making predictions about random events

Applications of Statistics

• Statistics is applied in various fields, including science, engineering, medicine, economics, and social sciences
• Used for decision-making, forecasting, and quality control
• Essential for evidence-based research and policy-making