Learn Mathematics from START to FINISH

The Math Sorcerer
9 Dec 202018:04
EducationalLearning
32 Likes 10 Comments

TLDRIn this video, the presenter guides viewers on a self-study journey through mathematics, starting from basic logic and proofs to advanced topics like calculus and differential equations. The video recommends a sequence of books for beginners to progressively build their mathematical knowledge, emphasizing that understanding proofs is crucial. It encourages learners to explore their interests and not be discouraged by difficulties, as math is a challenging field that rewards persistence and curiosity.

Takeaways
  • 📚 Start with foundational books on discrete mathematics and proof writing to build a strong base in logic and mathematical reasoning.
  • 🔢 Begin learning mathematics with 'Discrete Mathematics with Applications' by Susannah Epp and a beginner-friendly alternative like Coleman Busby and Roz's book.
  • 📈 Progress to pre-algebra and college algebra to refresh and solidify basic mathematical concepts and skills.
  • 📊 Move on to more advanced topics like precalculus and trigonometry, using resources like the instructor's edition by Hornsby, Lyle, and Roxwoold for guided practice.
  • 🌟 Dive into calculus with popular textbooks like James Stewart's 'Calculus' and Michael Spivak's 'Calculus' for a deeper understanding of the subject.
  • 🔄 After calculus, explore differential equations to understand how to solve them using integration techniques learned earlier.
  • 📐 Continue with linear algebra, a crucial subject for further mathematical studies, using books like Howard Anton's 'Elementary Linear Algebra' for beginners.
  • 🔍 For those interested in statistics, a solid choice of book is available to apply the knowledge of calculus and linear algebra.
  • 🎓 The journey through self-study math can open up to various undergraduate and even graduate-level subjects like complex variables, real analysis, and abstract algebra.
  • 💡 Remember that the learning process is iterative; it's okay to get stuck, move on, and return to difficult concepts later with a refreshed perspective.
  • 🌐 The video script provides a comprehensive roadmap for self-study in mathematics, encouraging exploration of interests and a deep understanding of the subject.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is about teaching oneself mathematics from the basics to advanced topics, focusing on learning logic, proofs, and various branches of mathematics.

  • What are the first two books recommended for beginners in discrete mathematics?

    -The first two books recommended for beginners in discrete mathematics are 'Discrete Mathematics with Applications' by Susannah Epp and 'Discrete Mathematics' by Coleman Busby and Roz.

  • How does the video suggest one can start learning mathematical logic without prior knowledge of algebra?

    -The video suggests that one can start learning mathematical logic without prior knowledge of algebra by using the book 'Discrete Mathematics with Applications' by Susannah Epp, which covers the basics of mathematical logic and does not require algebra.

  • What are the two proof-writing books recommended in the video?

    -The two proof-writing books recommended in the video are 'Mathematical Proofs: A Transition to Advanced Mathematics' by Chartrand, Palomí, and Zang, and 'Mathematical Proofs' by Bonding and Keen.

  • Why does the video recommend starting with discrete mathematics instead of algebra?

    -The video recommends starting with discrete mathematics because it provides a foundation in logic, sets, and proof writing, which are essential skills for understanding more advanced mathematical topics and keeps learners motivated by introducing unique and interesting concepts not typically found in algebra courses.

  • What is the next step after learning basic logic and discrete mathematics according to the video?

    -After learning basic logic and discrete mathematics, the video suggests moving on to pre-algebra books to refresh or learn basic math concepts before progressing to more advanced topics like college algebra and precalculus.

  • Which book is recommended for learning calculus?

    -The video recommends 'Calculus' by James Stewart for learning calculus, as it is a popular and widely-used book in the United States and Canada.

  • What is the significance of learning proof writing in mathematics?

    -Learning proof writing is significant because it is a fundamental skill in mathematics that allows one to rigorously demonstrate and validate mathematical statements, which is essential for advanced study and research in the field.

  • What does the video suggest after completing calculus?

    -After completing calculus, the video suggests learning differential equations, as integration techniques from calculus are helpful in this subject. It also mentions that one could explore other topics like linear algebra, complex variables, or real analysis at this stage.

  • How does the video address the challenge of self-studying mathematics?

    -The video acknowledges that math is hard and it's normal to get stuck or not understand everything. It encourages learners to move on and explore topics of interest, emphasizing the importance of learning for the sake of learning and enriching one's mathematical knowledge.

  • What are some advanced math subjects one can explore after learning calculus and linear algebra according to the video?

    -Some advanced math subjects one can explore after learning calculus and linear algebra include differential equations, complex variables, real analysis, abstract algebra, topology, combinatorics, naive set theory, functional analysis, and graph theory.

Outlines
00:00
📚 Introduction to Self-Teaching Mathematics

The video introduces a method for self-teaching mathematics from the ground up, starting with basic sets, proofs, and discrete mathematics. The speaker recommends a sequence of books for beginners, including 'Discrete Mathematics with Applications' by Susannah Epp and 'Discrete Math' by Coleman Busby and Roz, to build a foundational understanding of logic and mathematical concepts without prior knowledge of algebra. The emphasis is on learning to write proofs and understand basic mathematical logic, which are crucial skills for advanced study in mathematics.

05:01
📈 Progressing from College Algebra to Precalculus

The speaker continues by discussing the transition from college algebra to more advanced topics such as precalculus and trigonometry. Books like 'Precalculus' by Hornsby, Lyle, and Roxwood are recommended, which cover a range of topics including inequalities and matrices. The video also touches on the importance of understanding calculus before moving on to more specialized fields and suggests 'Calculus' by James Stewart for those interested in engineering or a basic understanding of calculus. Additionally, 'Calculus' by Michael Spivak is mentioned for its advanced treatment of the subject, which benefits from prior knowledge of proof writing and logic.

10:02
📚 Exploring Differential Equations and Linear Algebra

The speaker then delves into differential equations, emphasizing the importance of understanding integration before tackling this subject. Books by Zill and Larry Andrews are recommended for beginners, while 'Elementary Linear Algebra' by Howard Anton is suggested for learning linear algebra. The video also mentions the potential to explore various undergraduate-level math subjects once a basic understanding of proofs, calculus, and linear algebra is achieved.

15:02
🌟 Advanced Topics and Self-Directed Learning

The video concludes with a discussion on self-directed learning in advanced mathematical topics. The speaker suggests books for exploring subjects like complex variables, real analysis, abstract algebra, topology, combinatorics, set theory, functional analysis, and graph theory. Notable books include 'Complex Variables' by Safin Snyder and Brown Churchill, 'Analysis One and Two' by Terence Tau, and 'Principles of Mathematical Analysis' by Walter Rudin. The speaker encourages viewers to pursue their interests in mathematics, acknowledging that learning math can be challenging but also rewarding, and that it's okay to move on from topics that prove difficult.

Mindmap
Keywords
💡Discrete Mathematics
Discrete Mathematics is a field of mathematics that deals with discrete rather than continuous quantities, such as integers, graphs, and logic. In the video, the speaker recommends books on discrete math as a starting point for self-study, emphasizing its foundational role in learning mathematical logic, sets, and proof writing, which are essential skills for further mathematical studies.
💡Proof Writing
Proof writing is the process of formally demonstrating that a statement is true through logical reasoning. It is a critical skill in mathematics as it allows mathematicians to rigorously establish the validity of their claims. The video highlights the importance of learning proof writing early in one's mathematical journey and suggests specific books, such as 'Mathematical Proofs: A Transition to Advanced Mathematics' and another by Bonding Keen, to develop this skill.
💡Logic
Logic is a branch of philosophy and mathematics concerned with the principles of reasoning and argument. It involves the study of arguments, their structure, and the rules for constructing and evaluating them. In the context of the video, logic is fundamental to understanding mathematical proofs and discrete mathematics, with concepts like logical implications and truth tables being introduced as part of the beginner's toolkit.
💡Algebra
Algebra is a branch of mathematics that uses symbols and rules of operations to solve equations and problems. It is a unifying thread of almost all of mathematics and includes everything from basic equations to more advanced concepts like matrices and abstract algebra. The video suggests that after learning the basics of logic and proof writing, one should move on to algebra to further build a strong mathematical foundation.
💡Calculus
Calculus is a branch of mathematics that deals with rates of change and accumulation. It is central to understanding many phenomena in the physical world and is used in fields such as physics, engineering, and economics. The video positions calculus as a significant milestone in a self-study journey, with resources like James Stewart's 'Calculus' and Michael Spivak's 'Calculus' being recommended for different levels of learners.
💡Linear Algebra
Linear Algebra is the study of linear equations, linear transformations, and vector spaces. It is a fundamental area of mathematics with applications in fields like computer graphics, quantum physics, and machine learning. The video presents linear algebra as a subject that can be approached after learning calculus and suggests books like 'Elementary Linear Algebra' by Howard Anton and 'Linear Algebra' by Friedberg, Insel, and Spence for self-study.
💡Differential Equations
Differential Equations are equations that involve an unknown function and its derivatives. They are crucial for modeling dynamic systems in physics, engineering, and other sciences. The video suggests that after learning calculus, particularly integration techniques, one can move on to studying differential equations, with books like 'Differential Equations' by Zill and 'Ordinary Differential Equations' by Larry Andrews being recommended for beginners.
💡Abstract Algebra
Abstract Algebra, also known as modern algebra, is the study of algebraic structures such as groups, rings, and fields. It is abstract in the sense that it deals with algebraic operations without giving them a geometric interpretation. The video suggests that after linear algebra, which provides a foundation in proof-based mathematics, one can tackle abstract algebra using resources like 'Abstract Algebra' by Sarah Chino and 'Contemporary Abstract Algebra' by 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 foundational field for many areas of modern mathematics and has applications in physics and computer science. The video suggests that with a solid understanding of calculus and proof writing, one can explore topology through books like 'Topology' by Gamelan and Green, which is noted for having full solutions to all problems.
💡Real Analysis
Real Analysis is the study of the real numbers and their properties using rigorous logical methods. It is considered one of the most challenging courses for math majors due to its high level of abstraction and proof intensity. The video suggests that after learning calculus and linear algebra, one can attempt real analysis with books like 'Analysis I and II' by Terence Tao and 'Principles of Mathematical Analysis' by Walter Rudin, also known as 'Baby Rudin'.
💡Self-Study
Self-study in mathematics refers to the process of independently learning mathematical concepts and techniques without formal classroom instruction. The video encourages self-study as a way to enrich one's mathematical knowledge and to explore various mathematical fields based on personal interest. It emphasizes the importance of persistence and the acceptance of challenges as part of the learning process.
Highlights

The video provides a comprehensive guide on self-studying mathematics from basic concepts to advanced topics.

The recommended starting point is learning basic sets, proofs, and discrete mathematics before moving on to more complex subjects.

Discrete Mathematics with Applications by Susannah Epp is suggested as an excellent beginner book for learning the theory of logic.

The book by Coleman Busby and Roz is another beginner-friendly option for learning discrete mathematics and writing proofs.

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrang, Palomini, and Zang is recommended for learning proof writing.

Proofs and Fundamentals: Calculus by Bonding and Keen is also a good resource for learning proof writing.

Pre-algebra books are suggested for refreshing basic math skills before diving into more advanced topics.

College Algebra is the next step after pre-algebra, focusing on basic algebra skills.

Precalculus and trigonometry are recommended for those who have mastered college algebra, with a book by Hornsby, Lyle, and Roxwoold suggested.

Calculus by James Stewart is a popular choice for learning calculus, with a focus on problems and explanations.

Calculus by Michael Spivak is more advanced and requires prior knowledge of proof writing and logic.

Differential Equations by Zill and Andrews are recommended for learning about integration and other mathematical concepts.

Linear Algebra by Howard Anton and others is suggested for learning basic linear algebra concepts.

Once calculus and linear algebra are understood, one can explore a wide range of undergraduate-level math subjects.

Abstract Algebra, Topology, Combinatorics, and Graph Theory are among the advanced subjects one can study after mastering the basics.

Real Analysis by Royden and Rudin, as well as Complex Analysis, are recommended for those interested in graduate-level mathematics.

The video emphasizes the importance of persistence and exploration in self-studying mathematics, acknowledging that it is normal to get stuck and move on.

Transcripts

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Math Self-StudyTextbook RecommendationsDiscrete MathematicsProof WritingAlgebraCalculusLinear AlgebraDifferential EquationsAbstract AlgebraTopology