If you have a deep understanding of e, this piece is certainly not for you. However, if you want to go ahead and read it, please do contribute a few of your harshest and most painful words to the poverty-stricken comment box.
To establish an understanding of something, we need to define it and we also need to know what should not define it. However I shall employ a different strategy.
Reach out for your phone and look for an online definition of e. This is what you will find:
“The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828.”
I used the copy-paste functionality because typing that out would be quite a shameful exercise I would rather avoid indulging myself in.
Wikipedia is not wrong. The definition above is true but this is why a lot of math haters exist. There must be a better way of learning what e really stands for.
We could start by drawing a few circles.
Steps to follow:
Note: If below 14 years of age you may skip steps 2 and 3
- Take out a sharp pencil and compass.
- Go out and buy a pencil, sharpener and a compass.
- Stop sulking and carry out step 2.
- Draw circles of radius 1cm, 2cm, 3cm and so on.
- You may draw 1mm, 2mm, 3mm or 1 micron, 2 microns and so on, if you have invented superfine rulers and possess amazing microscopic vision.
- Baptise the first circle as the mother of all circles.
- If you mentally did step 6 without hesitating, ask yourself why you did it and then continue reading.
The smallest circle is the mother who gives birth to all the other circles in your drawing. It is like calling number 1 the mother of all positive integers. You can scale down to 1 from 10, 15 or even 347.
The constant, pi, is common to all circles and it would be utter stupidity if we have to give it a puny and meaningless but true definition of 3.14. Instead think of pie as something that represents an idea that all circles, as well as all other shapes derived from circles, share. If you use Mahabharatha to explain this concept, Draupadi would be pi and the Pandavas would stand for all possible circles.
(For readers unfamiliar with Mahabharatha, Draupadi is the wife of five brothers and has been used as a symbol to represent someone common to all five. One of the readers pointed out that Kunti, their mother, might have been a better symbol to use because the Pandavas could have existed without Draupadi but not without Kunti. It might seem like a reasonable argument. However, on closer examination of the epic, you realize that Draupadi was born to marry the five Pandavas. They existed because she did. I would be glad to hear out a differing opinion).
Now for the less dramatic definition of pi;
Pi is the ratio between the circumference and the diameter of any god forsaken circle in this universe. Try to find the truth in the above statement with your own circles.
You could alternatively adopt an eccentric route of taking a stick and pulling it to the ends of the earth from a central point. Make as many circles as you can but pi will stubbornly sit in your equation without flinching.
A million dollars from my pocket to anyone who can draw a circle that is not married to pi.
Likewise, e is a Draupadi constant.
It can be thought of as a common base rate to describe ALL continually growing/decaying processes that we encounter in our daily lives.
One example lies in banking.
Banks sanction loans with a fixed interest rate that increases continually.
Steps involved in loaning 1 rupee to buy coffee toffee:
Interest rate: 50%
There figures represent the loan along with accumulated interest rounded off to two decimal places because beyond that, the figures are “insignificant”.
- At 0 years: 1 rupee .
- After 1 year: 1.50 rupees
- After 2 years: 2.25 rupees
- After 3 years: 3.38 rupees
- After 4 years: 5.06 rupees
- After 5 years: 6.83 rupees
An elaborate explanation of the first three years:
- 50% of 1 rupee is 50 paise interest, therefore 1 rupee + 50 paise= 1.50 rupees.
- In the second year the bank charges you for your loan AND your interest which involves 50% of original 1 rupee loan and 50% of 50 paise interest. This is added to the original 1.50 rupees of the first year (1.50 + 50% of 1 rupee loan + 50% of 50 paise interest i.e. 1.50 + 0.50 + 0.25 = 2.25.
- After the third year, it becomes 50% of each of the components comprising the second year added to 2.25.
The bank wants you to pay for not only the loan you sanction but also for the interest.
To sum it up the bank never forgets, hence the phrase “paisa vasuli”. Your loan and even the interests of your interests that keep piling up, are charged with a fixed interest rate every single term.
Another instance of exponential growth is observed in case of bacterial cell division.
Non biological steps involved in bacterial growth:
- 1 cell divides or multiplies (I am not sure) into two.
- 2 cells become 4 cells (the original 2 cells + 100% of 2 cells)
- 4 cells become 8 cells (the original 4 cells + 100% of 4 cells) and so on.
I do not think (I might be wrong) that 8 cells of E. coli sit and wait for 20 minutes before deciding they need to finally become 16. Probably if you could zoom in and observe the cells in each second comprising those 20 minutes you could possibly observe a slow increase from 8 to 16 involving fractions as well. Since we are talking about living cells we cannot even begin to imagine fractions of it, hence for convenience, we adopt to understanding it in a series of 2 raised to x.
Radioactive decay is another similar process. I leave it to the readers to ponder over it.
Now the question circles back to e which ties up all these processes in a sacred knot of marriage.
Can you connect the dots yourself and define e in your own way?
This time try not to reach out for your phone.
(Edit: One of the readers pointed out a flaw in the analogy which I graciously accepted to an extent after carefully reading it. I pinned the existence of the Pandavas on Draupadi. The same treatment with circles would imply they existed because of pi. Circles were there even before the concept of pi was known. I wanted to convey the idea that you cannot draw a circle leaving out pi. It is an inherent property that has an impact on all calculations of circumference and areas for circles and shapes arising from circles. The analogy, if you read carefully, could go either way. Thank you, Parth Raval, for your feedback).