Pure Mathematics for Pre-Beginners from Udemy

Pure Mathematics for Pre-Beginners consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra. The 8 lessons in this course cover elementary material from each of these 8 topics. A “pre-beginner" is a math student that is ready to start learning some more advanced mathematics, but is not quite ready to dive into proofwriting. Pure Mathematics for Pre-Beginners is perfect for

students wishing to begin learning advanced mathematics, but that are not quite ready to start writing proofs.
high school teachers that want to expose their students to the ideas of advanced mathematics without getting into mathematical rigor.
professors that wish to introduce higher mathematics to non-stem majors.

The material in this math course includes:

8 lessons in 8 subject areas.
Examples and exercises throughout each lesson.
A problem set after each lesson arranged by difficulty level.

There are no prerequisites for this course. The content is completely self-contained. Furthermore, this course will naturally increase a student’s level of “mathematical maturity.” Although there is no single agreed upon definition of mathematical maturity, one reasonable way to define it is as “one’s ability to analyze, understand, and communicate mathematics.” A student with a very high level of mathematical maturity may find this course very easy—this student may want to go through the course quickly and then move on to Pure Mathematics for Beginners. A student with a lower level of mathematical maturity will probably find this book more challenging. However, the reward will certainly be more than worth the effort.

Pure Math Pre-Beginner Book Table Of Contents (Selected) Here's a selection from the table of contents:

Lesson 1 - Logic

Lesson 2 - Set Theory

Lesson 3 - Abstract Algebra

Lesson 4 - Number Theory

Lesson 5 - Real Analysis

Lesson 6 - Topology

Lesson 7 - Complex Analysis

Lesson 8 - Linear Algebra

What's inside

Learning objectives

Logic
Set theory
Abstract algebra
Number theory

Real analysis
Topology
Complex analysis
Linear algebra

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

Syllabus

Lesson 1 - Logic

Statements

Truth Assignments

Logical Connectives

Traffic lights

Read about what's good

what should give you pause

and possible dealbreakers

Provides a gentle introduction to advanced mathematics, making it suitable for students who are not yet comfortable with rigorous proof-writing techniques

Covers a wide range of fundamental topics, including logic, set theory, algebra, number theory, analysis, topology, and linear algebra, which are essential for further study

Offers a self-contained curriculum with examples and exercises, making it easy for instructors to integrate into their existing lesson plans and adapt to different learning styles

Presents complex topics in an accessible manner, which can help students develop their analytical and problem-solving skills, regardless of their academic background

Includes problem sets after each lesson, which allows learners to test their knowledge and understanding of the material and to build a solid foundation for future studies

Focuses on increasing mathematical maturity, which is essential for success in advanced mathematics courses and for developing a deeper understanding of mathematical concepts

Reviews summary

Accessible first look at pure math

According to students, this course, "Pure Mathematics for Pre-Beginners", offers a gentle and clear introduction to advanced mathematical concepts across a broad range of topics like logic, set theory, and algebra. Learners find it an excellent stepping stone for those new to higher-level math, particularly appreciating that it covers ideas without getting into formal proofs. Many feel it successfully builds mathematical maturity and prepares them for future study. While the vast majority praise its accessibility, a few note it can be very basic for those with some prior knowledge. Overall, it's highly recommended for its stated purpose: providing an accessible first look at pure mathematics for pre-beginners.

Focuses on ideas, avoids formal proofs.

"Perfect if you are not ready to dive into proof writing."

"Focuses on conceptual understanding without the rigor of proofs, as stated."

"Good for getting the intuition before the formal details."

"Avoids getting bogged down in proofs."

Good foundation for future learning.

"Highly recommend as a precursor to university math courses."

"It’s a great stepping stone for those curious about higher mathematics."

"Helped me feel more prepared for proof-based courses."

"Built my mathematical maturity."

Explores many core pure math topics.

"...covers several topics like Logic, Set Theory, Abstract Algebra, etc. This gives a good overall understanding..."

"The range of topics introduced is excellent..."

"Exposes you to 8 different areas of pure math."

"Liked getting a taste of various advanced math fields."

Concepts are easy to understand.

"Instructor made complex ideas easy to grasp."

"Found the way concepts were presented very helpful."

"...explains complex topics simply."

"The explanations were clear and concise."

Excellent starting point for beginners.

"This course is perfect for someone who is interested in pure mathematics but does not have a strong background."

"This course is great for pre-beginners looking to dip their toes..."

"Ideal if you are new to higher level math..."

"Provides a very gentle introduction..."

Some found it too simple.

"It is VERY BASIC though."

"If you have prior exposure, it might feel very simple."

"Just a basic overview, didn't offer much new info."

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 Pre-Beginners with these activities:

Review Basic Algebra Concepts

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Strengthen foundational algebra skills to better understand abstract algebra concepts covered in the course.

Browse courses on Algebra

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Review key concepts like variables, expressions, and equations.
Practice solving linear and quadratic equations.
Work through example problems from an algebra textbook or online resource.

Review 'How to Prove It: A Structured Approach'

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Improve understanding of logical reasoning and mathematical arguments, which are essential for grasping the concepts in the course.

View How to Prove It: A Structured Approach on Amazon

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Read the chapters on propositional logic and quantifiers.
Work through the examples and exercises in the book.
Focus on understanding the different proof techniques.

Discuss Logic Problems with Peers

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Reinforce understanding of logic by working through problems and discussing solutions with other students.

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Form a study group with other students from the course.
Choose logic problems from the course materials or other sources.
Work through the problems individually, then compare solutions and discuss any discrepancies.

Four other activities

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Show all seven activities

Practice Set Theory Operations

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Solidify understanding of set theory by practicing set operations and solving related problems.

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Find practice problems involving set operations like union, intersection, and complement.
Work through the problems, paying attention to the definitions of each operation.
Check your answers and review any mistakes.

Create a Concept Map for Abstract Algebra

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Organize and synthesize abstract algebra concepts by creating a visual representation of their relationships.

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Identify the key concepts from the abstract algebra lesson.
Create a concept map showing the relationships between these concepts.
Review and refine the concept map to ensure accuracy and clarity.

Read 'Naive Set Theory' by Paul Halmos

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Expand knowledge of set theory beyond the introductory level covered in the course.

View Naive Set Theory on Amazon

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Read the book, focusing on the chapters that are most relevant to the course.
Take notes on the key concepts and definitions.
Work through the exercises in the book to test your understanding.

Explore Applications of Linear Algebra

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Deepen understanding of linear algebra by researching and presenting real-world applications.

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Research different applications of linear algebra in fields like computer graphics, data science, or engineering.
Choose one application to focus on.
Prepare a presentation or report explaining the application and how linear algebra is used.

Career center

Learners who complete Pure Mathematics for Pre-Beginners will develop knowledge and skills that may be useful to these careers:

Mathematics Teacher

A Mathematics Teacher helps students understand mathematical concepts and principles. To become a successful mathematics teacher, a deep understanding of mathematical fields is essential. This course, with its lessons in logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra, may be useful to build a foundation for teaching mathematics at various levels. The course's content is self-contained, increasing a student's mathematical maturity, which helps with communicating mathematics effectively. The material covered in this course may be particularly helpful for high school teachers exposing their students to advanced mathematics.

See salaries and explore the career path for Mathematics Teacher

Researcher of Mathematics

A Researcher of Mathematics investigates and expands mathematical knowledge through original research. Covering logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra, it provides an initial exposure to the breadth of pure mathematics. This may be useful for identifying potential areas of specialization. A researcher of mathematics typically requires a Ph.D., this course may expose the student to potential fields of study.

See salaries and explore the career path for Researcher of Mathematics

Math Tutor

A Math Tutor assists students in understanding mathematical concepts and improving their problem-solving skills. The comprehensive coverage of logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra makes it easier to tutor across different mathematical subjects. A math tutor helps build a strong foundation in mathematics, enabling them to explain complex topics more clearly and effectively. The self-contained nature of the course makes it suitable for tutors looking to refresh or expand their knowledge.

See salaries and explore the career path for Math Tutor

Professor

A Professor teaches courses and conducts research in a specific academic discipline. This course, covering logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra, provides a broad understanding of pure mathematics. The course may be useful to increase background knowledge outside of the professor's specific field of expertise. The course can expose non-stem majors to higher mathematics. A professor generally holds a Ph.D.

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Curriculum Developer

A Curriculum Developer designs and creates educational materials and programs. This course, covering logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra, is useful for understanding the breadth of mathematical topics. The course material may be useful in order to design effective and engaging math curricula for various levels. Its self-contained nature and focus on mathematical maturity can inform instructional design.

See salaries and explore the career path for Curriculum Developer

Researcher

A Researcher works in academia or industry, conducting investigations to discover new knowledge, new insights, or new products. This course's introduction to a range of mathematical concepts may be helpful in improving one's overall math skills. The lessons include logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra. It may expose students to different areas of mathematics they could study in graduate school. A researcher relies heavily on the ability to understand advanced mathematical concepts.

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Data Scientist

Data Scientists analyze large datasets to extract meaningful insights. This course in pure mathematics provides a strong foundation for the mathematical and logical reasoning required in this field. The lessons on logic, set theory, and linear algebra are especially applicable to data analysis and algorithm development. The course helps with analytical thinking and problem-solving skills which are essential for a data scientist. Since the course includes examples and exercises, it may also help for a junior data scientist to improve their quantitative skills.

See salaries and explore the career path for Data Scientist

Cryptographer

A Cryptographer designs and analyzes encryption algorithms to secure data. This course, covering logic, set theory, abstract algebra, and number theory, helps in developing the mathematical foundations needed for cryptography. The course may be useful to build a foundation in discrete mathematics and abstract structures, which are crucial for cryptographic research and development. The lessons on number theory, in particular, are highly relevant to the principles of cryptography.

See salaries and explore the career path for Cryptographer

Quantitative Analyst

A Quantitative Analyst, often called a Quant, develops and implements mathematical models for pricing and trading securities. This course in pure mathematics helps to build a strong foundation in key mathematical concepts relevant to quantitative finance. The lessons on number theory, real analysis, linear algebra, and complex analysis are particularly applicable. A quantitative analyst relies on advanced mathematical skills to succeed and this course may be useful to build a foundation in mathematics.

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Statistician

A Statistician collects, analyzes, and interprets numerical data to draw conclusions and make informed decisions. The lessons in logic, set theory, number theory, real analysis, and linear algebra may be useful to expand on the mathematical foundations that are essential for statistical analysis. This course helps to develop analytical thinking and problem-solving skills, which are crucial for a statistician. The course may be useful for those preparing for more advanced study in statistics, and its examples and exercises are particularly helpful for building comfort with mathematical concepts.

See salaries and explore the career path for Statistician

Software Engineer

A Software Engineer designs, develops, and tests software applications. This course, covering logic, set theory, and abstract algebra, helps in building a strong foundation in discrete mathematics, which is vital in computer science. The course may be useful in improving problem-solving skills, which are crucial for a software engineer. The course's content is completely self-contained and increases mathematical maturity. The lessons on logic will be especially useful as these are a foundation for computer programming.

See salaries and explore the career path for Software Engineer

Actuary

An Actuary assesses and manages financial risks. This course offers a solid foundation in logic, set theory, number theory, real analysis, and linear algebra, which are relevant to actuarial science. The course may be useful in developing analytical and problem-solving skills, which are crucial for actuarial work. This course focuses on examples and exercises, which will help to reinforce mathematical skills. Moreover, the course helps build a foundation of mathematical knowledge helpful for understanding the financial risk models used by actuaries.

See salaries and explore the career path for Actuary

Operations Research Analyst

Operations Research Analysts use mathematical and analytical methods to improve organizational efficiency. This course, covering logic, set theory, linear algebra, and real analysis, helps with building the mathematical skills required for optimization and modeling. The course helps to develop problem-solving and analytical thinking skills, which are essential for an operations research analyst. The course’s introduction to various fields may be useful as a starting point for more advanced study.

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Financial Analyst

A Financial Analyst analyzes financial data, provides investment recommendations, and helps businesses make informed decisions. This course may be useful to build a foundation in quantitative reasoning, which is essential for financial modeling and analysis. The lessons on number theory, real analysis, and linear algebra are relevant to financial mathematics. The course's examples and exercises reinforce mathematical concepts and increase mathematical maturity. Although a financial analyst might take other courses, this one may be a starting point.

See salaries and explore the career path for Financial Analyst

Economist

Economists study the production, distribution, and consumption of goods and services. This course may be useful to build a foundation in mathematical reasoning, which is essential for economic modeling and analysis. The lessons on real analysis and linear algebra are particularly relevant to economic theory. This course's self-contained content and focus on increasing mathematical maturity provide a useful base for understanding advanced economic concepts. Although economists typically hold advanced degrees, this course might be a good place to start.

See salaries and explore the career path for Economist

Pure Mathematics for Pre-Beginners

What's inside

Learning objectives

Syllabus

Traffic lights

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Reviews summary

Accessible first look at pure math

Activities

Career center

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