Learn Mathematics from START to FINISH (2nd Edition)

The Math Sorcerer
17 Oct 202237:12
EducationalLearning
32 Likes 10 Comments

TLDRThe video script is a comprehensive guide on how to approach learning mathematics from the basics to advanced topics. It introduces three different starting points: algebra, discrete math, and proof writing, and recommends specific books for each. The speaker shares personal experiences and recommendations for books on pre-calculus, trigonometry, calculus, differential equations, probability, statistics, geometry, linear algebra, complex variables, abstract algebra, topology, and more. The emphasis is on the availability of resources for self-study, and the importance of persistence and gradual progression in understanding complex mathematical concepts.

Takeaways
  • 📚 Starting with algebra, especially pre-algebra, is a common and recommended approach for beginners in mathematics.
  • 📈 Discrete mathematics is presented as an unconventional but effective starting point for learning math, accessible even before mastering algebra or calculus.
  • 🔬 Proof writing is emphasized as a crucial skill for advancing in mathematics, with several book recommendations for learning to write mathematical proofs.
  • 🚀 Advanced topics such as calculus, differential equations, and linear algebra are essential for progressing in mathematical studies, with specific textbooks suggested for each topic.
  • 📱 Books on specific topics like discrete math, calculus, and abstract algebra offer various starting points and pathways for learning, highlighting the flexibility in approaching math studies.
  • 📖 Recommendations include both classic texts and modern resources across different areas of mathematics, indicating the depth and breadth of available material for learners.
  • 🖥 Real analysis (Advanced calculus) is identified as a challenging but rewarding area, with books suggested for different levels of complexity and understanding.
  • 💾 Several books are recommended for specialized topics such as cryptography, engineering mathematics, and measure theory, demonstrating the wide range of subjects within the field of mathematics.
  • 🔧 Emphasis on self-study and exploration, encouraging learners to engage with advanced topics even if they don't feel fully prepared, to foster growth and understanding in mathematics.
  • 📝 Collecting mathematics books is highlighted as a personal passion, showing the value and joy in exploring various mathematical subjects through literature.
Q & A
  • What are the three starting points for learning mathematics mentioned in the video?

    -The three starting points for learning mathematics mentioned are starting with basic algebra, starting with discrete math, and jumping into proof writing.

  • Why is the book 'Pre-Algebra Mathematics' by Nickels recommended for beginners in algebra?

    -The book 'Pre-Algebra Mathematics' by Nickels is recommended for beginners because it covers the very basics of algebra, making it a great starting point for those new to mathematics.

  • What makes 'Concrete Mathematics' by Graham, Knuth, and Patashnik a legendary book?

    -It's considered legendary because it's written by very famous individuals, including Knuth, a renowned computer science professor at Stanford who created Tex, which LaTeX is based on, making it instrumental for creating math symbols in modern books.

  • Why is it unconventional to start learning mathematics with discrete math according to the video?

    -It's considered unconventional because, at most colleges and universities in the United States, students are typically required to take calculus 2 before jumping into discrete math, yet discrete math can be learned without any knowledge of calculus.

  • Can you start learning mathematics by learning to write mathematical proofs?

    -Yes, starting with proof writing is highlighted as a fun and viable way to begin learning mathematics, with several books recommended for those interested in this approach.

  • What prerequisites are generally needed before one can start learning calculus according to the video?

    -Before starting calculus, one generally needs to learn more algebra and some trigonometry, often through taking pre-calculus and trigonometry courses.

  • What does the book 'How to Prove It: A Structured Approach' by Daniel Velleman emphasize that's key for learning mathematics?

    -This book emphasizes the importance of understanding logic as a precursor to learning how to write mathematical proofs, which is crucial for advancing in mathematics.

  • Why are older math textbooks like 'Pre-Calculus Mathematics' by Shanks, Fleener, and Eichholtz still valuable?

    -Older textbooks are valued for offering a unique look at how math was taught in the past and for providing different problems and approaches that can be both challenging and educational.

  • How does the video suggest you can proceed after getting started with the basics of mathematics?

    -After getting started with the basics, the video suggests proceeding to more advanced topics such as calculus, differential equations, and beyond, highlighting that having a foundational understanding allows one to explore higher levels of mathematics.

  • Why is it recommended to have more than one book when studying subjects like linear algebra and differential equations?

    -Having more than one book is recommended because it provides a broader range of examples, explanations, and exercises, which can be beneficial for a more comprehensive understanding and for tackling difficult subjects.

Outlines
00:00
📚 Introduction to Learning Mathematics

The video begins with an introduction to learning mathematics from scratch, offering three different starting points. The first is learning algebra, which is the most common approach, starting with a book on pre-algebra mathematics. For those more advanced, 'Elementary Algebra' by Sullivan Struve and Mozzarella is recommended. Another way to start is with discrete math, an unconventional approach that can be beneficial for understanding mathematical proofs. Books like 'Discrete Mathematical Structures' by Coleman Busby and Ross, and 'Concrete Mathematics' by Graham Knuth are suggested. Lastly, jumping into proof writing with books like 'How to Prove It' by Daniel Vellman is also an option. The video emphasizes the importance of logic in learning mathematics and recommends 'How to Prove It' as an affordable and comprehensive choice.

05:01
📈 Progressing in Mathematics: Algebra and Trigonometry

The second part of the video discusses the next steps in mathematics after the basics, focusing on algebra and trigonometry. It's emphasized that to learn calculus, one must have a good understanding of these subjects. The video recommends 'Pre-Calculus' by Stuart Redland and Watson, and 'College Algebra' by Blitzer for more advanced algebra. For trigonometry, 'A Graphical Approach to Algebra and Trigonometry' by Hornsby Lyle and Roxwold is suggested. The video also mentions the importance of having a solid foundation in algebra and trigonometry before moving on to calculus, and provides several book options for each subject.

10:01
📚 Deepening Understanding: Calculus and Differential Equations

The video continues with an exploration of calculus and differential equations. It suggests 'Calculus with Analytic Geometry' by Swachowski for a comprehensive introduction to calculus. For a more modern approach, 'Calculus' by Larson, Edwards is recommended. 'Thomas' Calculus' is highlighted as a legendary book with extensive exercises and explanations. The video also mentions 'A First Course in Differential Equations' by Dennis Zill as a good choice for learning differential equations. The importance of having multiple resources when learning these subjects is emphasized, as books can be challenging and having supplementary materials can be helpful.

15:02
📊 Exploring Further: Probability, Statistics, and Geometry

This section of the video covers probability, statistics, and geometry. For probability and statistics, the Shams outline on probability and statistics is recommended for its affordability and examples. 'Elementary Statistics' by Weiss is suggested for beginners, and 'Mathematical Statistics and Data Analysis' by John Rice for more advanced learners. For geometry, 'Geometry' by Jurgensen is a standard choice, and the video also mentions 'Linear Algebra' by Friedberg as a comprehensive resource for linear algebra topics.

20:04
🌐 Advanced Topics: Complex Variables and Partial Differential Equations

The video delves into advanced topics such as complex variables and partial differential equations (PDEs). For complex variables, the Shams outline is praised as one of the best, and 'Complex Variables and Applications' by Brown and Churchill is recommended for its popularity and comprehensive content. 'Partial Differential Equations' by Zachary Mana glue is suggested for PDEs, despite some errors in the answers. The video also mentions the importance of having a solid foundation in ordinary differential equations before tackling PDEs.

25:05
📖 Diverse Fields: Abstract Algebra and Number Theory

The final part of the video discusses abstract algebra and number theory. For abstract algebra, 'A First Course in Abstract Algebra' by John B. Fraleigh is recommended for beginners, while 'Contemporary Abstract Algebra' by Joseph A. Gallian is praised for its numerous examples. 'Algebra' by Michael Artin is highlighted as a classic text for a more advanced understanding. For number theory, 'Elementary Number Theory' by Underwood Dudley and 'Number Theory' by George Andrews are suggested as excellent starting points.

30:06
📚 Comprehensive Overview: Real Analysis and Topology

The video concludes with an overview of real analysis and topology. 'Principles of Mathematical Analysis' by Rudin is noted for its rigorous approach, while 'Advanced Calculus' by Fitzpatrick is recommended for a more manageable introduction. 'Topology' by monkeys is suggested for those looking to learn about topology, with 'Introduction to Topology' by Bert Mendelson offered as a classic option. The video emphasizes the importance of having a strong foundation in proof writing before attempting these advanced subjects.

Mindmap
Keywords
💡Algebra
Algebra is a branch of mathematics that focuses on mathematical symbols and the rules for manipulating these symbols. It is a fundamental aspect of mathematics and a common starting point for learning more advanced mathematical concepts. In the video, the presenter suggests starting with algebra as a way to begin learning mathematics, recommending books like 'Pre-algebra Mathematics' for beginners and 'Elementary Algebra' for those with some basic algebra knowledge looking to advance.
💡Discrete Math
Discrete mathematics is a branch of mathematics that deals with discrete rather than continuous quantities, such as counting, graph theory, and logic. Unlike calculus, which often deals with continuous functions and rates of change, discrete math focuses on individual, distinct elements and their relationships. The video highlights discrete math as an unconventional yet viable starting point for learning mathematics, with books like 'Discrete Mathematical Structures' by Coleman Busby and Ross, and 'Concrete Mathematics' by Graham Knuth and Patashnik.
💡Proof Writing
Proof writing is the process of formally demonstrating that a statement is true using logical reasoning. It is a critical skill in mathematics, as it allows mathematicians to establish the validity of their claims. The video emphasizes the importance of learning how to write proofs as a way to gain a deeper understanding of mathematical concepts. Books such as 'How to Prove It: A Structured Approach' by Daniel Vellman and 'How to Read and Do Proofs' by Daniel Solo are recommended for learning this skill.
💡Pre-Calculus
Pre-calculus is a mathematics course that typically covers topics which prepare students for the study of calculus. It often includes trigonometry, algebra, and functions, providing a foundation for understanding the concepts necessary for calculus. The video mentions pre-calculus as a step towards learning calculus, with books like 'Pre-Calculus' by Stewart and Redlin providing a solid foundation for these topics.
💡Trigonometry
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles, particularly right triangles. It is closely related to the study of periodic functions and waves. In the context of the video, trigonometry is presented as an essential part of pre-calculus and a prerequisite for calculus. The video suggests resources such as 'College Algebra' by Blitzer for learning algebraic techniques that include trigonometry.
💡Calculus
Calculus is a major branch of mathematics that deals with concepts such as limits, derivatives, and integrals. It is used to study rates of change and accumulation. The video positions calculus as a natural progression from pre-calculus and trigonometry, with several books recommended for self-study, including 'Calculus with Analytic Geometry' by Swachowski and 'Thomas' Calculus', which is noted for its comprehensive coverage and popularity.
💡Differential Equations
Differential equations are equations that involve an unknown function and its derivatives. They are used to model various phenomena in the physical and biological sciences. The video discusses differential equations as a higher-level math topic that comes after learning calculus, with resources like the 'Shams Outline on Differential Equations' and 'A First Course in Differential Equations' by Dennis Zill recommended for study.
💡Probability and Statistics
Probability and statistics are branches of mathematics that deal with the analysis of data and the likelihood of events. Probability focuses on the chance of outcomes, while statistics uses that data to draw conclusions and make predictions. The video suggests learning these topics after a foundation in calculus, with 'Elementary Statistics' by Weiss as a beginner's textbook and 'A First Course in Probability' by Sheldon Ross for more advanced study.
💡Linear Algebra
Linear algebra is the study of linear equations, linear transformations, and vector spaces. It is a crucial tool in many areas of mathematics and science, including geometry, calculus, and computer science. The video presents linear algebra as a fundamental area of study in mathematics, with 'Elementary Linear Algebra' by Howard Anton recommended for beginners and 'Linear Algebra' by Friedberg for those seeking a more advanced treatment.
💡Abstract Algebra
Abstract algebra, also known as modern algebra, is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It abstracts the concepts of number theory and extends them to more general systems. The video highlights abstract algebra as a favorite subject of the speaker and recommends several books for different levels of learning, including 'A First Course in Abstract Algebra' by John B. Fraleigh and 'Contemporary Abstract Algebra' by Joseph A. Gallian.
💡Topology
Topology is a branch of mathematics that deals with the properties of space that are preserved under continuous deformations, such as stretching or bending. It is a form of 'rubber-sheet geometry' and involves concepts like continuity, compactness, and connectedness. The video suggests that topology is typically studied in the senior year of an undergraduate program and recommends books like 'Introduction to Topology' by Bert Mendelson and 'Topology' by猴子 (Monkeys), which is a standard text for advanced undergraduates and beginning graduate students.
Highlights

The video provides a comprehensive guide on how to learn mathematics from the basics to advanced levels.

Three different starting points for learning mathematics are presented: algebra, discrete math, and proof writing.

Pre-Algebra Mathematics by Nickels is recommended for learning the very basics of algebra.

Elementary Algebra by Sullivan, Struve, and Mozzarella is a modern book with plenty of examples for those ready for more advanced algebra.

Discrete Mathematical Structures by Coleman, Busby, and Ross is a beginner-friendly book for starting with discrete math.

Concrete Mathematics by Graham, Knuth, and Patashnik is a famous book written by renowned computer scientists, suitable for more advanced learners.

How to Prove It by Daniel Vellman is highlighted as an affordable and effective book for learning structured proof writing.

The video emphasizes the importance of learning algebra and trigonometry as a foundation for calculus.

Pre-Calculus by Stewart and College Algebra by Blitzer are recommended for those preparing for calculus.

The presenter shares personal experiences and recommendations for various math books, adding credibility to the suggestions.

Calculus with Analytic Geometry by Swachowski is an older book that provides a different perspective on the subject.

Tomas' Calculus is described as a legendary book with extensive exercises and explanations, suitable for learning calculus.

Differential Equations by Nagel, Saf, and Snyder, and Dennis Zill's First Course in Differential Equations are recommended for learning this subject.

Probability and statistics are introduced as topics that can be learned with some calculus background.

Elementary Statistics by Weiss is suggested as a standard modern book for beginners in statistics.

Linear Algebra is discussed, with Elementary Linear Algebra by Howard Anton highlighted as a good choice for beginners.

Complex Variables and Partial Differential Equations are introduced as advanced math topics, with specific book recommendations for each.

Abstract Algebra is covered, with A First Course in Abstract Algebra by John B. Fraleigh and Contemporary Abstract Algebra by Joseph A. Gallian recommended.

Real Analysis and Advanced Calculus are introduced as challenging undergraduate topics, with Principles of Mathematical Analysis by Rudin mentioned as a rigorous book.

Number Theory, Graph Theory, and Topology are also discussed, with specific book recommendations for each area.

The video concludes with a selection of books on various math topics, including cryptography, measure theory, and functional analysis.

Transcripts
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