I have met ‘e’ right from school. The quintessential natural logarithm or ln as it was called, was supposed to have a base ‘e’. Then I met it again in calculus when
Thank God, one less thing to remember as we were trying to pile up on ‘studying’ for board exams and entrances. Curiously I met it again in my M.Tech in probability but I was not curious enough even then – trying to finish my assignments and prepare for my exams. Later, when the mayhem had settled and I had the patience to be reflective, I went back to learn some maths and boy I found a diamond.
Now, learning has come a long way from great teachers to wonderful books, YouTube, Audible and so on.
But learning with an LLM can be fun and more importantly it is liberating. You are not constrained by the teacher’s knowledge or the content’s scope in a YouTube video. You are not going to stop asking the most basic and ridiculous questions which you would never have done in school. Also you could take several branches to explore other related stuff to come back and continue what you were learning.
That is how I met ‘e’ again. I was learning basic calculus and hit the same derivative of e^x is e^x (rate of change is same as itself), but instead of stopping at that like a trained Maths teacher, the LLM nudged me into giving an extra detail. It said “It is the only function in the universe that is its own derivative and so becomes a native language for higher math”. That piqued my interest and I went probing deep into ‘e’. I realised that it combined Compound Interest (Economics) with growth of Bacteria (Biology) and Radioactivity (Physics) along with funnily, Probability (Maths) as well, other than calculus of course. I will distill some of that here for other curious souls.
Simply put ‘e’ is just a number: 2.7182 (goes on like pi but let’s stop here). What is special about this number between 2 and 3.
Let’s take money for our example because well I like it and can explain better, but this applies to anything for which the growth depends on the amount of existing stuff.
Imagine you have 100 Rs. Now after 5 years with a 100% interest it becomes 200.
1: One option for this growth is at exactly on the midnight of the last day of the 5th year, we put 100 more and make it 200. So 100% in one shot.
2: Another option is to give yearly interest and allow the interest to earn more income. So 100% interest for 5 years is 20% per year. So at the end of year 1 you get 120 Rs and the entire 120 Rs goes to work for you at 20% interest for next year giving you 120 (principal) + 24 = 224 Rs (note that the 4 Rs are from the extra 20 Rs interest which are also working now). This gives you 249 Rs at the end instead of the 200 you got earlier.
3: Naturally the next option is to Split the interest monthly giving 20% / 12 = 1.666% per month. So at the end of first month we have 100 + 1.66 = 101.66 and it goes on giving you 270 Rs at the end.
4: Last option is daily interest which gives you 271.179.
Now you might think that going even finer like hourly, secondly etc can lead you to higher numbers and may be we can even cross 1000 Rs by the end of 5th year going extremely fine. But you will find that how much even finer you go you hit a mathematical ceiling at 271.82 Rs and you will not be able to cross it. Curiouser and curiouser!
(Since we started with 100 Rs we get 271.82 Rs. If we had started with unit rupee (that is just one rupee) we would have reached 2.7182 which is our ‘e’)
See the graph below to see how things grow and how it hits ‘e’ (Note: Daily is hidden for us since it is plotted below blue of ‘e’ both being very close)
‘e’ is nothing but the natural speed limit of a 100% growth.
In fact the same analogy applies to say a culture of bacteria growing in a test tube or population growth or virus spreading etc.
It can even be used to explain the decay on the inverse side like cooling down of things (the cooler they are the faster they cool and become cooler increasing the cooling and so on) or decay of radioactive stuff.
Isn’t this beautiful but does it explain why d/dx of e^x is e^x. Not quite. And that could be part 2 or you hopefully I have kindled some curiosity fire for you to go find it.