There are several different, equivalent definitions of the constant e (it is also called the “Euler Number”). For example, one of the deftnitions is that it is the number whose exponential function is equal to the derivative of it’s exponential function.

It’s just a number that happens to be the solution to a bunch of (mathematical) problems humans have come up with. It doesn’t appear everywhere: the things it appears in are all somehow related, or are purposefully defined to contain it.

[Edit: Pleased disregard anything below this point, I was, or course, thinking of the golden ratio]

I suspect you are thinking about spirals in nature. People really overstate both the commonness and accuracy of those patterns. It’s a pretty normal thing for humans to do: humans like patterns, and they like something cool to talk about. Remember how everyone was talking about the world ending in 2012, or the bermuda triangle? Yeah, those are just cool, mysterious things. Memes basically, like when people put the euler spiral on random pictures. Those pictures don’t actually fit the euler spiral, usually not even remotely. But it’s a fun joke.

Yes, it’s true that some plants or shells seem to resemble it, or that some galaxies are shaped in that way. But even more of them aren’t shaped like that, and those that are, are only approximately so.

*e* (Napier’s Constant) was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.^1

To put it in five year old’s terms, this number is a ratio that you get when you try taking small parts of something, like bank interest, and add them many many times. When you do something like that, taking fractions and doing them a lot, you can see there is some similarity of numbers behaviour. That similarity is often expressed with this ratio number. Just like calculations with circles often involve the number π (3,14159…), e is involved with logarithms and other complicated stuff.

These numbers (e, π, φ…) aren’t magical or made up to create complexity, they describe several relations and patterns in mathematics related to some particular fields.

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^1 This statement is true, the first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier as a list of logarithms to that base, Bernoulli discovered it in 1683 as a formula describing the constant, Leonhard Euler started to use the letter e for the constant in 1727 or 1728 in an unpublished paper, then used it in a letter to Christian Goldbach in 1731 and first publicised it in Euler’s Mechanica in 1736.

https://en.wikipedia.org/wiki/E_(mathematical_constant)

EDIT: removed unnecessary talking to the hypothetical five year old.

It’s a number that comes up when how fast something changes depends on how much of it there already is.

That sounds confusing, but it’s really easy to understand with some examples.

A bunch of rabbits live in the meadow. They do the rabbit thing, and soon enough there are more rabbits.

Now that there are more rabbits, that means more pairs of rabbit parents, which means they reproduce even faster! Which makes more rabbits, which leads to faster reproduction, which makes more rabbits… etc.

As you see, how fast the rabbits reproduce depends on how many rabbits there are. And when they reproduce, it creates more rabbits, which continues the loop! (obviously, for simplicity, I’m not including all the rabbits who die or get eaten or the time it takes them to grow up)

The other common example is interest. You gain interest based on how much money you have. And once you gain interest, now you have more money! So you gain more interest, which gives you more money, which gives you more interest… etc.

It appears in nature, e.g. in the patterns of certain plants, because how fast the plant is able to grow depends on how big it is already. Lots of things in nature, as well as in human society, have this kind of relationship.

Basically, when you write out the equations representing this relationship, e is the number that comes out as the most basic number that, when you put it in the equation, causes the equation to grow at the same rate as its current size. Everything else is just multiplying this basic equation by some other numbers to change the exact rate and amount.

(for the pedants, yes, I should be saying “function” and not “equation”)

To explain it precisely would require quite a bit of maths but basically it’s ‘the solution’ you get when modelling a system where the rate of growth is proportional to the size of the population. ie if the population doubles, the rate at which it grows doubles.

Not that surprisingly this comes up a lot in nature – if each individual (or pair of individuals) is reproducing, that’s the same kind of growth we were talking about.

The most abstract concept is this – you have a group of something (animals, bacteria, money in a bank account whatever) and that thing is creating more of itself (having babies, dividing, gaining interest)

What’s important is that the result of this growth adds on to the original group and helps with the next stage of growth.

Suppose you have $1000 and invest it in something that gives you 10% interest once a year.

A year later you have 1000*1.1=$1100

But suppose instead you compound it twice, so you get 5% after 6 months and then another 5% at the end of the year. After six months you have $1000 times 1.05=1050. But at the end of the year you get $1050*1.05=$1102.50, so you got an extra $2.50. This is the same as $1000 times 1.05^2, or 1.05 to the second power.

If you compound it 4 times, once every quarter, you get $1000 times 1.025^4 or $1103.81.

If you compound it every month, then by the end of the year you get $1000*(1.0083^12) = $1104.71.

At this point math dorks started looking at this to figure out the pattern. What if you compound it 100 times? 1000 times? Every time you get a little bit more, but the increase is smaller and smaller so it’s homing in on some sort of hard limit. What if you compound it an infinite number of times so that your investment is always growing at a rate of 10% per year?

And they found e. The answer to continous compounding is after growing for 1 term at 10%, you have $1000*e^0.1 or $1105.71. When you generalize out from there, you get e^(rt) where r is the interest or growth rate and t is the number of time-lengths your investment grows.

**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.

Most things in physics come from differential equations: nature doesnt directly tell you what something IS, it tells you how it changes. Mathematically this means the laws of physics are equations written in terms of the *rates of change* of the quantity you care about. You then solve these equations to get an expression for that quantity.

For example, newtons 2nd law of motion give you an equation that says the 2nd order rate of change of positoon (acceleration) is proportional to force. Solving that gives you an expression for position.

e is special because the function e^x has the property that it’s rate of change with respect to x is e^x . This is very relevant to differential equations because many equations are written such that the rate of change of your quantity is proportional to your quantity, therefore e^x is a solution.

An example of this is a mass on a spring: the 2nd order rate of change of displacement is proportional to displacement.

There are many others such as the heat equation, diffusion equation, shroedinger equation (ok these are all the same equation), radioactive decay, exponential growth, etc. that are written similarly, and therefore have some form of e^x in their solution.

In conclusion, e appears so much because it is the solution of many laws of physics due to its special rate of change property.

A lot of these answers are missing pieces. Either answering what it is or some places it does up but not answering why. So here is my crack at it.

The number e is special because it is a constant that shows up in a lot of places, and it’s important for math in general much like the more familiar pi or lesser known but equally fun and more important i (square root of negative 1). What is it? It is the number such that the derivative of the function e^x is equal to itself. That seems interesting but not that important. So why do we care? Well the obvious application is in growth. The definition is an exponential function and exponential functions deal with growth and anything modeling growth deals with expectation functions… But that is only a tiny part of the power of e.

A lot of things in math are about finding relationships between other things. And then generalizing those relationships to find the underlying patterns. When you do this you find some freaky and eerie coincidences. These are not coincidences. They are a result of how math works. The numbers that we deal with are full of patterns that exist. The number e deals with a special relationship related to change in general. This is why it shows up when we are looking at growth functions. It also shows up in oscillation functions, because those involve change. So functions involving sine can be expressed as a combination of e to a real and e to an imaginary component. e^(πi)=-1 because that is a special case of an expression of rotation where the real part is 0 and the imaginary part is -1. This has to do with the 2D number plane as opposed to the more familiar number line or more realistic number space that we live in (kinda) but it does illustrate why this stuff shows up. We live in a mathematical world where we can describe things with these functions. As a result, the special numbers we build these functions from, the constants, show up in the universe because they build the universe.

So the tldr, e is defined as the number such a that it’s the derivative of itself in an exponential function and it shows up everywhere because it describes how change happens in math and math is all about change so it shows up everywhere.

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