How to Build Abstract Thinking in Mathematics Using the Epsilon-Delta Definition?

7 min readSep 4, 2022
Source: “Schedules” by Nate Armstrong on INPRNT

When I close my eyes and go back in time, I see a college student sitting in the back row and looking sad while the professor is standing next to the chalkboard, writing mathematical definitions on it with chalk. The click, click, click sound was still obvious every time the professor wrote on the blackboard. Then, the student went into deep thought when the professor said:

“For every epsilon greater than 0 [ε>0], there exists a delta greater than 0. [ δ>0]”*** ε = epsilon*** δ = delta

Upon hearing this, the student asked himself repeatedly: What does epsilon mean? That student actually was me. Although everyone told me I was smart in high school, I was just an average student at college. The thing that made me an average student was the Greek letter “epsilon.” I could not grasp the idea of epsilon, and as a result, I struggled in Calculus for a long time. Epsilon was a nightmare for me in my college years. I felt, “What the heck! Epsilon is everywhere”. Even the arguments I read about the epsilon simply weren’t convincing enough. I felt obscure every time I saw epsilon because it seemed bizarre to me. I was gauging my understanding of epsilon based on my scores in my Calculus exams. I was told mathematics is fun for everyone in this world, but in my case, I could not enjoy it because of the epsilon. Then I realized that I wasn’t the only one who had an issue understanding the epsilon. The generations before me also had a problem understanding it.

Then something extraordinary happened one autumn evening when I read a book that explained what it meant. After reading the book, I fully understood what epsilon means. The book was The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth.” The book was about Paul Erdös, the great mathematician who called a child “epsilon.” It may seem awkward, but it was at this moment I grasped the idea of epsilon. Epsilon represents a small quantity of a number, and it is not a constant like children. Children are little, and they always change. Then I stopped cursing at the concept of epsilon-delta and started loving it. I also passed my Calculus examination that semester, and I feel happy about it even today.


Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better.