Learn Mathematics from START to FINISH (2nd Edition)
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
- π Starting with algebra, especially pre-algebra, is a common and recommended approach for beginners in mathematics.
- π Discrete mathematics is presented as an unconventional but effective starting point for learning math, accessible even before mastering algebra or calculus.
- π¬ Proof writing is emphasized as a crucial skill for advancing in mathematics, with several book recommendations for learning to write mathematical proofs.
- π Advanced topics such as calculus, differential equations, and linear algebra are essential for progressing in mathematical studies, with specific textbooks suggested for each topic.
- π± 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.
- π Recommendations include both classic texts and modern resources across different areas of mathematics, indicating the depth and breadth of available material for learners.
- π₯ Real analysis (Advanced calculus) is identified as a challenging but rewarding area, with books suggested for different levels of complexity and understanding.
- πΎ 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.
- π§ 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.
- π 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
π 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.
π 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.
π 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.
π 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.
π 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.
π 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.
π 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
π‘Algebra
π‘Discrete Math
π‘Proof Writing
π‘Pre-Calculus
π‘Trigonometry
π‘Calculus
π‘Differential Equations
π‘Probability and Statistics
π‘Linear Algebra
π‘Abstract Algebra
π‘Topology
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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