Learn Mathematics from START to FINISH

Summary

This video provides a comprehensive guide on how to teach yourself mathematics from start to finish. It starts with learning basic logic and proof writing through discrete math, using recommended books. The video covers essential topics like algebra, calculus, differential equations, and linear algebra, offering book suggestions at each stage. The host emphasizes a flexible, interest-driven learning approach, encouraging viewers to explore various branches of mathematics such as abstract algebra, topology, and real analysis. Overall, the guide aims to equip viewers with a structured, self-directed path to mastering math.

Takeaways

• 📚 The video is about teaching yourself mathematics from the basics up to advanced levels, starting with proofs and discrete math.
• 💡 The first step is to learn how to write proofs and understand mathematical logic, starting with discrete math books such as 'Discrete Mathematics with Applications' by Susanna Epp.
• 🧠 Learning discrete mathematics is foundational for understanding logic, sets, and writing proofs without needing algebra knowledge at first.
• 🔍 Two proof-writing books, 'Mathematical Proofs' by Chartrand et al. and 'Bond and Keene's Proof Writing,' are recommended for developing skills in writing mathematical proofs.
• 📝 After learning proofs and logic, it's suggested to revisit basic algebra through pre-algebra or college algebra books if needed.
• 📊 Once comfortable with algebra, learners can progress to precalculus and calculus, using 'Calculus' by James Stewart and 'Calculus' by Michael Spivak.
• 📘 Differential equations and linear algebra follow calculus, with recommended books like 'Elementary Linear Algebra' by Howard Anton and 'Differential Equations' by Zill.
• 📈 After mastering core subjects like calculus and linear algebra, students can move into more advanced topics such as real analysis, complex analysis, and abstract algebra.
• 🔢 Various advanced math topics like topology, combinatorics, and functional analysis are also recommended for exploration.
• 🚀 The key takeaway is to keep progressing in mathematics at your own pace, focusing on areas of personal interest and not worrying about mastering every topic fully before moving on.

Q & A

What is the starting point for learning mathematics according to the video?

The video suggests starting with basic sets, proofs, and discrete mathematics, emphasizing learning logic and proof writing before moving on to more advanced topics.

Which books are recommended for beginners in discrete mathematics?

Two books are recommended for beginners in discrete mathematics: 'Discrete Mathematics with Applications' by Susanna Epp and a book by Coleman, Busby, and Ross.

Why is learning to write proofs emphasized early in the self-study process?

Learning to write proofs is emphasized because it forms a foundation for understanding more advanced mathematical concepts and allows learners to read more challenging books like those on calculus and real analysis.

What is a suggested next step after learning basic proofs and logic?

After learning basic proofs and logic, the video suggests studying pre-algebra or algebra, depending on the learner's background, before moving on to college algebra, precalculus, and eventually calculus.

Which books are recommended for learning calculus?

The video recommends 'Calculus' by James Stewart for a comprehensive approach and 'Calculus' by Michael Spivak for a more proof-based, rigorous understanding of calculus.

What should learners focus on before starting differential equations?

Learners should focus on mastering integration techniques from calculus, as this knowledge is crucial for understanding differential equations.

What are the two recommended books for learning linear algebra?

The video recommends 'Elementary Linear Algebra' by Howard Anton for beginners and 'Linear Algebra' by Friedberg, Insel, and Spence for a more rigorous, proof-based approach.

What advanced topics can learners explore after mastering calculus and linear algebra?

After mastering calculus and linear algebra, learners can explore advanced topics such as real analysis, abstract algebra, topology, complex analysis, and mathematical statistics.

Why does the video suggest that mastering every concept in a book is not necessary before moving on?

The video suggests that it is normal to get stuck on some topics, and it is okay to move on and explore other areas of interest to keep the learning process enjoyable and engaging.

How can learners make the most of their self-study journey in mathematics?

Learners are encouraged to read what interests them and not feel bound by a strict order. This approach allows them to explore their interests, gain diverse insights, and maintain motivation throughout the self-study process.