On April, 30 2022 I published my first book “An Introduction to Real Analysis – Building Visual Intuition” via Amazon KDP.

The Goal

The title of the book gives away what it should be – an introduction to real analysis – but there are already a ton of these, so what is different? First of all, it should be very affordable. Usually, such textbooks can easily succeed the 100$ mark, and as a person that reads such books on a monthly basis, this is simply unaffordable. Secondly, I found that I could provide what many professors cannot anymore. I can remember what it is like to struggle through the topic, as it wasn’t too long ago when I went through the same material that you would go through if you were to purchase and read my book (subtle hint phrased as a kind request :)). Therefore, I carefully explain the things that I struggled with in a way that I wished it would have been written when I read it the first time.

I was always someone that loved visual explanations, so the subtitle of “An Introduction to Real Analysis – Building Visual Intuition” expresses that I focused on giving visual explanation for many of the theorems that are covered, so that you understand the theorem not only through mathematical, but also through visual reasoning. Also, as highschool student you often do not have a good clue to how proofs are written, so instead of throwing them at your face, I slowly introduce more and more rigor throughout the book, so that you get a smooth transition from lower level to higher level mathematics.

The Contents

You may download the following pdf-file which contains the first two chapters to get a glimpse and to help you decide whether you want to buy my book or not.

An Introduction to Real Analysis - Download

If you want to look into one of the later chapters you may write me me under joelgerlach04@gmail.com.

How it came to be

In Germany, where I am from, it is common that 10th grade students write an essay called “Facharbeit” which is supposed to contain at least 15 pages, and explores a particular topic which is often up to the student to choose. However, you require a teacher that that has some form of knowledge about your topic to supervise you. Due to the Covid-pandamic, my school kind of forgot about it. But near the end of the year, our class teacher gave us the option of still writing one about either astronomy or physics, and I chose physics. In particular, I wanted to write about Lagrangian mechanics, but unfortunately, he found that it was to advanced and I should stick to something a typical 10th grade student could understand.

Naturally, I gave up, at least until I realized that I might get my math teacher to supervise me in some form. I shot him an E-Mail (since we were still in home-schooling) and asked him exactly that. However, to this day, he still hasn’t responded to me. After two weeks of waiting for a response, I started to doubt that he would ever answer me. Still, I had energy in me that wanted to express itself through writing a lot of words, so I jokingly told myself to write an entire book about real analysis instead. After dismissing it for a couple of seconds, I started to think about that it would actually be a pretty cool thing to do – what 16 year old could claim to have published a book about higher mathematics?

Plans for the Second Edition

Although I do not plan to publish a second edition within the next two years, there will most likely be a second edition with more examples and exercises, as well as solutions to them.