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Pure Mathematics for Beginners

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Based on 46 ratings
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Steve Warner

Pure Mathematics for Beginners consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra. The eight lessons in this course cover basic material from each of these eight topics. In addition, all the proofwriting skills that are essential for advanced study in mathematics are covered and reviewed extensively. Pure Mathematics for Beginners is perfect for

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Pure Mathematics for Beginners consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra. The eight lessons in this course cover basic material from each of these eight topics. In addition, all the proofwriting skills that are essential for advanced study in mathematics are covered and reviewed extensively. Pure Mathematics for Beginners is perfect for

  • an introductory college course in higher mathematics.

  • high school teachers working with advanced math students.

  • high school and college students wishing to see the type of mathematics they would be exposed to as a math major.

The material in this pure math course includes:

  • 8 lessons in 8 subject areas.

  • A friendly but rigorous treatment of all the mathematics covered.

  • Additional analyses before and after proofs to help students gain a deep understanding of the subject matter with the minimum amount of effort.

  • A problem set after each lesson containing problems arranged by difficulty level.

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What's inside

Learning objectives

  • Logic
  • Set theory
  • Abstract algebra
  • Number theory
  • Real analysis
  • Topology
  • Complex analysis
  • Linear algebra

Syllabus

Lesson 1 - Logic
Statements with Words
Statements with Symbols

This is the end of Lesson 1. After completing this part, you should be ready to attempt the problems from the attached problem set.

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This is the end of Lesson 2. After completing this part, you should be ready to attempt the problems from the attached problem set.

This is the end of Lesson 3. After completing this part, you should be ready to attempt the problems from the attached problem set.

Traffic lights

Read about what's good
what should give you pause
and possible dealbreakers
Provides a friendly but rigorous treatment of mathematics, which is helpful for students who want to gain a deep understanding of the subject matter
Covers proofwriting skills, which are essential for advanced study in mathematics and often overlooked in introductory courses
Includes problem sets after each lesson, which allows students to practice and reinforce their understanding of the material
Explores eight different areas of pure mathematics, which provides a broad overview of the field and helps students identify their interests
Requires learners to work through problem sets, which may be time-consuming for some students
Presents additional analyses before and after proofs, which may be challenging for students who are new to abstract mathematical thinking

Save this course

Create your own learning path. Save this course to your list so you can find it easily later.
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Reviews summary

Foundational introduction to pure mathematics

According to students, this course offers a broad introduction to core areas of pure mathematics, providing a solid foundation in proofwriting. Many found it valuable for seeing what higher math entails, appreciating the coverage of multiple topics like Logic, Set Theory, and Abstract Algebra. However, some learners felt the pace can be challenging and that the course might assume some prior familiarity with mathematical thinking beyond calculus. The problem sets require significant effort, with a number of reviewers noting the difficulty level of problems compared to the lecture content. Overall, it's seen as a rewarding first step for those serious about pure mathematics.
Explores many core areas of pure math.
"Liked seeing all the different fields - logic, set theory, algebra, analysis. Gives a great overview."
"Covers a wide range of pure math branches, which was great for exploring interests."
"Introduces you to logic, set theory, abstract algebra, number theory, and more."
Builds essential skills for higher math study.
"This course is a must for anyone wanting to get into pure math. It really hammered home the proofwriting techniques."
"I finally feel like I understand how to construct a proof after taking this course."
"Provides the necessary groundwork in proofs to move onto more advanced studies."
May not be for absolute math beginners.
"Title says 'Beginners' but it felt more like a first course *for math majors*, not necessarily absolute beginners."
"If you haven't done any proof-based math before, this is a steep learning curve."
"Having some prior exposure to abstract concepts is helpful."
Problems are significantly challenging.
"The problem sets are where you really learn, but they are significantly harder than the lectures prepare you for."
"Spent most of my time wrestling with the problem sets. They are tough!"
"Be prepared for problem sets that require a lot of independent thinking."
Some find the speed and difficulty high.
"Moves quite fast, especially if you're completely new to rigorous math."
"Found it challenging and had to pause often to digest the material."
"Assumes a certain level of mathematical maturity that true beginners might lack."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Pure Mathematics for Beginners with these activities:
Review High School Algebra
Solidify your understanding of fundamental algebraic concepts. This will provide a strong foundation for the abstract algebra and real analysis topics covered in the course.
Browse courses on Algebra
Show steps
  • Review key concepts like solving equations and inequalities.
  • Practice factoring polynomials and simplifying expressions.
  • Work through practice problems from a high school algebra textbook.
Read 'How to Prove It' by Velleman
Improve your proof-writing skills. This book will help you understand the logic and structure behind mathematical proofs, which is essential for success in this course.
View How to Prove It: A Structured Approach on Amazon
Show steps
  • Read the book chapter by chapter.
  • Work through the examples and exercises in each chapter.
  • Focus on understanding the different proof techniques.
Practice Set Theory Problems
Reinforce your understanding of set theory concepts. This will help you grasp the more abstract concepts in later lessons.
Show steps
  • Solve problems involving set operations like union, intersection, and complement.
  • Prove set identities using Venn diagrams and algebraic manipulations.
  • Work through problems involving power sets and Cartesian products.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Create a Mind Map of Abstract Algebra Concepts
Organize and visualize the relationships between different abstract algebra concepts. This will help you retain the information and see the bigger picture.
Show steps
  • Identify the key concepts from the abstract algebra lesson.
  • Create a mind map with the central concept at the center.
  • Connect related concepts with branches and sub-branches.
  • Use colors and images to make the mind map visually appealing.
Read 'Naive Set Theory' by Paul Halmos
Deepen your understanding of set theory. This book provides a more advanced treatment of the subject and will help you appreciate its role in the foundations of mathematics.
View Naive Set Theory on Amazon
Show steps
  • Read the book chapter by chapter.
  • Work through the examples and exercises in each chapter.
  • Focus on understanding the more advanced concepts.
Write a Summary of a Research Paper in Topology
Apply your knowledge of topology to understand and summarize a real-world research paper. This will help you see how the concepts you learned are used in practice.
Show steps
  • Find a research paper in topology that interests you.
  • Read the paper carefully and take notes.
  • Write a summary of the paper, highlighting the key concepts and results.
  • Explain how the paper relates to the concepts you learned in the course.
Help Other Students with Linear Algebra Problems
Solidify your understanding of linear algebra by helping other students with their problems. Teaching is a great way to learn!
Show steps
  • Participate in online forums or study groups.
  • Answer questions about linear algebra concepts and problems.
  • Explain your reasoning clearly and concisely.

Career center

Learners who complete Pure Mathematics for Beginners will develop knowledge and skills that may be useful to these careers:
Mathematician
A mathematician uses advanced mathematics and logical reasoning to solve complex problems in a variety of fields. Daily work may involve developing new theories, proving theorems, and creating mathematical models. This course in Pure Mathematics for Beginners, with its lessons in logic, set theory, abstract algebra, and other areas, helps build a foundation for understanding the abstract concepts and proofwriting skills essential to excelling as a mathematician. The course's rigorous treatment of mathematics encourages deep understanding, especially with its lessons in real analysis, topology, and complex analysis, all core subjects for advanced mathematical study. Those who wish to become a mathematician should see if they find this course useful.
See salaries and explore the career path for Mathematician
Mathematical Modeler
Mathematical modelers create and analyze mathematical representations of real world systems to understand their behavior and make predictions. They apply various mathematical techniques to develop and solve models across diverse fields. This Pure Mathematics for Beginners course, with its lessons in Logic, Set Theory, Abstract Algebra, Real Analysis, and Linear Algebra, helps build a foundation for understanding various modeling techniques. The course's emphasis on proofwriting prepares one to think in abstract detail, ensuring a model accurately reflects the system it is trying to model. Those who wish to become mathematical modelers should see if they find this course useful.
See salaries and explore the career path for Mathematical Modeler
Theoretical Physicist
A theoretical physicist conducts research to understand the fundamental laws of the universe, often using mathematical models and abstract theories to explain phenomena. A graduate degree is typically required. This Pure Mathematics for Beginners course, including lessons in Real Analysis, Complex Analysis, Topology, and Linear Algebra, helps build a foundation for the advanced mathematical concepts used in theoretical physics. The course's emphasis on Logic and rigorous proofwriting can sharpen analytical skills, essential for success as a theoretical physicist. With its focus on pure mathematical theory, this is an ideal starting point.
See salaries and explore the career path for Theoretical Physicist
Cryptographer
Cryptographers design and analyze algorithms to encrypt and decrypt information, ensuring secure communication and data storage. They use mathematical principles to develop secure systems and protocols. This Pure Mathematics for Beginners course, which includes lessons in Number Theory, Abstract Algebra, and Logic, helps build a foundation for understanding the mathematical concepts underlying cryptography. The course's emphasis on proofwriting and rigorous thinking can sharpen analytical skills valuable for a future cryptographer. The sections on Set Theory may also be useful.
See salaries and explore the career path for Cryptographer
Operations Research Analyst
Operations Research Analysts use mathematical and analytical methods to help organizations optimize their operations and decision making. They develop models to analyze complex systems and identify areas for improvement. This Pure Mathematics for Beginners course, with its lessons in Logic, Set Theory, and Abstract Algebra, helps build a foundation for understanding mathematical modeling and optimization techniques. The course's sections on Linear Algebra and Real Analysis may be useful for those who wish to become Operations Research Analysts as it further strengthens analytical and problem solving skills.
See salaries and explore the career path for Operations Research Analyst
Research Scientist
Research scientists design and conduct experiments, analyze data, and publish findings in their field of expertise. They may work in academia, industry, or government laboratories, often requiring an advanced degree. This Pure Mathematics for Beginners course, with its focus on Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra helps build essential analytical and problem solving skills. The course's emphasis on proofwriting and rigorous thinking can prepare one for the demands of scientific research. Those who wish to become a research scientist should consider this course.
See salaries and explore the career path for Research Scientist
Statistician
Statisticians apply statistical theories and methods to collect, interpret, and summarize quantitative data to help inform decision making. A statistician often uses statistical modeling, hypothesis testing, and regression analysis. This Pure Mathematics for Beginners course, with its introduction to Logic, Set Theory, and Number Theory, helps build a foundation for statistical reasoning. The course's emphasis on proofwriting and rigorous thinking helps cultivate analytical skills critical for success as a statistician. The lessons in real analysis and linear algebra may be useful to those who wish to become statisticians.
See salaries and explore the career path for Statistician
Quantitative Analyst
Quantitative analysts develop and implement mathematical models to solve complex problems in finance, such as pricing derivatives, managing risk, and optimizing trading strategies. This Pure Mathematics for Beginners course, with its coverage of Real Analysis, Linear Algebra, and Complex Analysis, helps build a foundation for understanding the mathematical concepts used in quantitative finance. The course's emphasis on Logic and proofwriting can sharpen analytical and problem solving skills, valuable in quantitative analysis. The lessons in number theory may also be useful.
See salaries and explore the career path for Quantitative Analyst
Data Scientist
A data scientist analyzes large datasets to extract meaningful insights and develop data driven solutions. They use statistical modeling, machine learning, and data visualization techniques. This Pure Mathematics for Beginners course, which includes set theory, abstract algebra, number theory, and linear algebra lessons, may be useful for those looking to enter the field as it helps build a strong analytical and problem solving skillset. The course's exploration of logic and proofwriting can sharpen critical thinking abilities, valuable in data analysis. The lessons in real analysis can be valuable to those who wish to become a data scientist.
See salaries and explore the career path for Data Scientist
Actuary
Actuaries analyze financial risks and uncertainties to help insurance companies and other organizations manage their liabilities. They apply mathematical and statistical modeling techniques to assess the probability of future events. This Pure Mathematics for Beginners course, which covers Number Theory, Real Analysis, and Set Theory, may be useful to those who want to become an actuary, as it helps build a foundation in quantitative reasoning and mathematical modeling. The course's emphasis on Logic and rigorous proofwriting can sharpen problem solving skills, valuable in actuarial work.
See salaries and explore the career path for Actuary
Financial Analyst
Financial analysts evaluate financial data, provide investment recommendations, and help organizations make informed financial decisions. They analyze financial statements, assess market trends, and develop financial models. This Pure Mathematics for Beginners course, which explores Logic, Set Theory, and Number Theory, helps build a foundation in quantitative reasoning and analytical thinking. The course's coverage of Real Analysis and Linear Algebra may be useful for those who wish to become financial analysts as it further strengthens skills in mathematical modeling and problem solving.
See salaries and explore the career path for Financial Analyst
Econometrician
Econometricians use statistical methods to analyze economic data, test economic theories, and forecast economic trends. This Pure Mathematics for Beginners course, particularly the lessons in Logic, Set Theory, Number Theory and Real Analysis, helps build a foundation for understanding the mathematical underpinnings of econometrics. The course's emphasis on rigorous proofwriting can sharpen analytical skills valuable for econometric research. Those who wish to become econometricians should see if they find this course useful.
See salaries and explore the career path for Econometrician
Software Engineer
Software engineers design, develop, and test software applications and systems. That requires a strong understanding of algorithms, data structures, and programming languages. While seemingly unrelated, this Pure Mathematics for Beginners course helps build a foundation in logical reasoning and problem solving. The course's sections on Logic, Set Theory, and Abstract Algebra may be useful for those who wish to become Software Engineers as it helps foster the ability to think abstractly and design efficient algorithms. The course may be especially useful to software engineers in fields such as cybersecurity, cryptography, or algorithm design.
See salaries and explore the career path for Software Engineer
Bioinformatician
A bioinformatician analyzes biological data using computational tools and mathematical models. The role typically requires strong skills in statistics, programming, and data analysis, often requiring a graduate degree. This Pure Mathematics for Beginners course, with its lessons in Logic, Set Theory, and Linear Algebra, helps build a foundation for these skills. The course's focus on problem solving and rigorous thinking may equip one to tackle challenges in bioinformatics. The lessons in number theory may be useful to those who wish to become bioinformaticians.
See salaries and explore the career path for Bioinformatician
Data Analyst
Data analysts collect, clean, and analyze data to identify trends, patterns, and insights that can inform business decisions. The Pure Mathematics for Beginners course, with its introduction to Logic, Set Theory, and Linear Algebra, helps build a foundation in analytical thinking and problem solving. The course emphasizes logical reasoning and proofwriting, which can be useful for data analysis tasks. Those who want to become data analysts may find this course useful as linear algebra features prominently in data analytics.
See salaries and explore the career path for Data Analyst

Reading list

We've selected two books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Pure Mathematics for Beginners.
Cover image
How to Prove It
Save
Provides a comprehensive guide to mathematical proof techniques. It covers logic, set theory, relations, functions, and mathematical induction. It is particularly useful for students who are new to proof-based mathematics and need a structured approach to learning how to write proofs. This book is commonly used as a textbook in introductory proof courses.
How to Prove It: A Structured Approach
Paperback
$$
How to Prove It: A Structured Approach, 2nd Edition
Paperback
$$$
How to Prove It: A Structured Approach
Paperback
$$$
How to Prove It: A Structured Approach
Kindle Edition
$$
Cover image
Naive Set Theory
Save
Provides a rigorous introduction to set theory. It covers topics such as the axiom of choice, cardinal numbers, and ordinal numbers. While it is not strictly necessary for this course, it provides a deeper understanding of the foundations of mathematics. This book is more valuable as additional reading than it is as a current reference.
Naive Set Theory
Paperback
$

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6.5 total hours
Via
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Instructor
Steve Warner
Language
English
Traffic lights
Read about what's good
what should give you pause
and possible dealbreakers
Provides a friendly but rigorous treatment of mathematics, which is helpful for students who want to gain a deep understanding of the subject matter
Covers proofwriting skills, which are essential for advanced study in mathematics and often overlooked in introductory courses
Includes problem sets after each lesson, which allows students to practice and reinforce their understanding of the material
Explores eight different areas of pure mathematics, which provides a broad overview of the field and helps students identify their interests
Requires learners to work through problem sets, which may be time-consuming for some students
Presents additional analyses before and after proofs, which may be challenging for students who are new to abstract mathematical thinking
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