Is Calculus the Right Starting Point for an Amateur with a Solid Foundation in Linear Algebra, Functions, Geometry, and Trigonometry?
Exploring the vast universe of mathematics can be both exhilarating and challenging, especially for those who already possess a solid foundation in linear algebra, functions, geometry, and trigonometry. This journey often requires careful selection of the initial stepping stone. For many, calculus represents a natural next step due to its applicability in various scientific and engineering fields. However, starting with calculus might not be the best choice for everyone, especially if one lacks a firm grounding in algebraic and set-theoretic concepts.
Exploring Mathematics Through Courant and Robbins
If you're looking for a comprehensive and broad introduction to the world of mathematics, "What is Mathematics?" by Richard Courant and Herbert Robbins is an excellent choice. Although there might be more specialized and targeted 'scatter gun' introductions to mathematics, this book provides a broad yet deep overview of the discipline. Its engaging writing and broad coverage make it a valuable resource for anyone interested in delving into the vast world of mathematical concepts.
Why Calculus May Not Be the Best Starting Point
Without a strong algebraic foundation, diving into calculus can feel like attempting to swim in a turbulent sea without knowing how to swim. Calculus textbooks often begin with concepts that are closely tied to algebraic theories, such as the set theory and the concept of infinity. These foundational topics can be challenging and overwhelming for those without prior exposure and understanding.
Set Theory and Infinity
The first concepts you encounter in calculus might include set theory, specifically the concept of infinity. Infinity is a complex and fascinating topic that extends far beyond the scope of calculus. Understanding the concept of different sizes of infinity, as described in set theory, is crucial not only in calculus but in many other areas of mathematics. Without this understanding, the foundational ideas of calculus can seem like gibberish, making it difficult to grasp more advanced concepts.
Exploring Calculus After Establishing a Foundational Understanding
If set theory and infinity have sparked your interest, I encourage you to dive deeper into these topics before venturing into calculus. A solid understanding of concepts such as functions, infinity, infinitesimals, limits, and continuity will greatly enhance your ability to appreciate the beauty and depth of calculus. Key texts like "The Joy of Sets" (Kirkham, 1989) can provide a thorough and engaging exploration of these fundamental ideas.
Alternative Approaches to Mathematics
For a different approach, you might consider starting with the panoramic view offered in "What is Mathematics?" by Courant and Robbins. This book provides an extensive introduction to the broader field of mathematics before delving into formal study. It serves as an excellent guide for both beginners and those looking to deepen their understanding of the subject in a more holistic manner.
General Mathematics Fundamentals
For those who are new to the mathematical landscape, or even if you are already familiar with linear algebra, functions, geometry, and trigonometry, it is beneficial to establish a strong foundation in basic algebraic and set-theoretic concepts. Here's a structured approach to building this foundation:
Basic Algebraic Expressions: Learn to handle expressions, including simple operations like x / - and the basic binomial forms ax^2 bx c, a - b^2, a^b. Linear Functions: Understand linear functions of the form fx mx b. Quadratic Equations and Functions: Explore quadratic equations and functions, and learn how to solve and manipulate them. Additional Mathematical Operations: Familiarize yourself with exponential operations, logarithms, and how to algebraically handle basic exponents and roots. The Exponential Function: Learn about the e^x function and how all a^x functions can be represented using e. Sequences and Series: Study arithmetic and geometric sequences, and get acquainted with the concepts of limits and convergence, using examples like geometric series.By building a strong foundation in these areas, you will be better prepared to tackle the challenges of calculus and other advanced mathematical concepts.
Conclusion
While calculus is undoubtedly a powerful and important mathematical tool, it is essential to establish a solid foundation in algebra, set theory, and other fundamental concepts before diving in. Whether you choose to start with Courant and Robbins, explore set theory and infinity, or use a structured approach to build a strong background, the key is to approach mathematics with curiosity and persistence.