Personal Growth Limit in a Euler Number

Growth is a natural phenomenon. Where we place our attention and focus determines its direction. I remember the exponential growth curve from my physics class. Nature tends to follow patterns of gradual development, often accelerating over time. Just like the curve, which starts slowly and then rapidly increases, many natural and human processes follow similar patterns. This concept not only applies to physical phenomena but also to personal and societal growth. As we invest time and resources, the initial progress might seem slow, but with persistence and dedication, the rate of growth can increase significantly, leading to substantial change or improvement over time.

In the start of 2023, I aimed to expand my knowledge and skills in my career by immersing myself in the latest technological discussions on platforms like X (Twitter), YouTube, and Reddit. I discovered many people talking about NextJS, Vercel, Supabase, Pinecone, OpenAI’s API, RAG, FastAPI, Streamlit, Gradio, Stable Diffusion, RunPod, and more. Within a year, starting from zero experience with these technologies, I am now able to play around to make some project with it. Honestly, there are many things that still needs to learn but from what I feel AI can really leverage our learning of almost everything from fundamental until technical perspective.

In the end of 2023, I was contemplating about the Euler number that equivalent to 2.718. This curiosity led me to the work of Jacob Bernoulli, who first encountered this number while exploring compound interest. Jacob Bernoulli discovered something interesting when he was studying how money grows with interest in a bank. Usually, banks add a bit of extra money (interest) to what you’ve saved. Bernoulli looked at what would happen if the bank did this not just once a year, but more and more often, even every second.

To illustrate, imagine you deposit Rp 1M to a bank account. The bank offers an annual interest rate of 100%. Normally, this interest is added just once at the end of the year. So, if you wait a year, your Rp 1M would double to Rp 2M. This is straightforward — your money just doubles over the year.

But Jacob Bernoulli thought about a different scenario. He wondered what would happen if the bank added interest more frequently. Let’s say the bank decides to add half of the annual interest (which is 50%) every six months instead of 100% at the end of the year. Here’s how it would work:

• First Six Months: You start with Rp 1M. After six months, the bank adds 50% of your initial amount as interest. So, 50% of Rp 1M is Rp 0.5M When this is added to your initial Rp 1M, your total becomes Rp 1.5M.
• Next Six Months: Now, your account has Rp 1.5M. The bank again adds 50% interest, but this time the interest is on the new total amount (Rp 1.5M), not just the initial Rp 1M. So, 50% of Rp 1.5M is Rp 0.75M. Adding this Rp 0.75M to your Rp 1.5M makes your total Rp 2.25M by the end of the year.

This example shows compound interest — where you earn interest not just on your original amount, but also on the interest that has been added previously. Bernoulli’s exploration of compound interest revealed that as you compound more frequently (every six months in this case), the total amount grows more than it would with simple annual compounding. But there’s a limit to how much it can grow, for example from once a year, every month, even every second, this has a limit from 2, 2.25, then the limit number known as e, approximately 2.718.

“Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn’t, pays it.” — Albert Einstein

Then I am thinking of something called growth limit. Maybe we are as a human really have a limit. In a year, no matter how hard we try, we can only achieved 2.718 growth in a year. No matter how hard we try in a day to learn and work as many as we can, it always has a limit. Our cognitive loads has limit. Our energy has limit. We need to sleep in a day. We need to eat. Because we are a human. This understanding has an implication then, we really need to use the time wisely. That’s the challenge here.