What is e (2.718…) and why does it literally appear everywhere?
In: 756
**PART 1 –SEQUENCES—**
Life is filled with sequences. Right down to the number of hairs on your head each day or the amount of money in your bank account each month and you’ll get a sequence of numbers.
It’s handy to look at how the terms in a sequence change at each step. The differences between the terms. That gives us some idea of the patterns.
For example, the sequence 1,3,6,10,15
Looks bizarre and uninteresting until you look at the differences (3-1), (6-3), (10-6),…
Hmmm… 2,3,4,5… Ah now it all makes sense.
**PART 2 –MULTIPLICATIVE GROWTH–**
Doubling. It’s a thing that happens a lot.
Here’s a sequence of doubling:
1,2,4,8,16,32…
We call this 2^n, because we are multiplying by 2 “n times” to get to each term.
But maybe we don’t care that we are multiplying by 2 each time. Maybe we want to be able to say what we are adding each time. Well, for that you can write out a new sequence. The differences between each number and the next
1,2,4,8,16,32…
Oh blimey, who saw that coming? Take all the differences of this sequence and you get the same sequence back. This (as it turns out) is pretty special. You don’t get this when you do a sequence of multiplying by 3 (ie the sequence 3^n):
The sequence: 1,3,9,27,81…
The differences: 2, 6, 18, 52…
completely different! Oh wait… Hold on… It’s two (3^n)’s in a trench coat: (1+1), (3+3), (9+9),…
Turns out the difference of any “multiplicative growth” sequence will get you that same sequence back (albeit) with a scale factor.
But 2^n is still special, because it has no scale factor. You don’t need to work out what the scale factor is or really work out anything at all. The answer just appears.
**PART 3 –Continuous sequences–**
Sequences are great, but thier old news. Why track something every day, when you can track it every ALWAYS. Just because something is doubling every day, doesn’t mean it duplicates all at once at the strike of midnight.
It’s probably changing a tiny bit all the time.
So out with sequences and in with functions. Now we’re cooking with gas.
Differences become a bit meaningless now though. I mean how do I find the difference between now and an instant after now. What is an instant? And isn’t that change going to be imperceptibly small?
**Part 4 –DERIVATIVES–**
Derivatives are the answer to continuous change. Don’t ask “how much has this changed this instant”. Instead think “Okay, it’s changing some amount this instant. If it kept changing that amount every instant from now till tomorrow, then how much would it have changed in total.?”
It side steps the question of “how many moments are in a day”, because who cares? We just want to get an idea for how things are changing right now in real terms.
It’s easy to do this if you have a graph. Draw a line to show all of the values at all of the times. You’ll get a lovely smooth curve of some sort. Then, put your ruler up against the point you are interested in and draw a line with the same slope as that part of the curve. The steeper the line, the more change that is “happening”
**PART 5 –THE NEW KID–**
Now, with all this knowledge, you draw yourself a beautiful 2^x curve.
You find the derivative (the slope) at various points and then you excited await the moment that you had with sequences. The moment where all the numbers are the same.
But sadly, you don’t get that lovely moment.
Turns out if the thing you’re looking at doubles at the strike of midnight, then it always instantaneously changing by the current amount.
But if It’s changing continuously, so that every 24hr period sees a doubling no matter what time you start looking at it. Well, then you find that the derivative (slope) at each moment and the amount of stuff at that moment aren’t the same. You end up with something similar, but with a scale factor. It’s numbers in a trench coat all over again.
You cry at the madness of it all. Your once perfect number 2 has been reduced to just another number. It’s not special when you consider continuous change.
But wait! Is there a number where that scale factor is 1? A number where we can recreate the beauty that we once saw in two?
The answer is yes. The number e just so happens to fit this criteria. If you have e^x stuff at any given moment, then your stuff is changing by e^x.
Ah, all is right with the world. You can rest easy knowing that the numbers match.
Lots of stuff in science involves change, so naturally a function that changes in a simple way is very handy.
If physics was interested in discrete instantaneous changes (like one might see in the turns of a board game) then you would see more interest in the number 2.
Latest Answers