Why should "e" exist?
September 9, 2012 12:16 AM   Subscribe

Why is the number "e" so...

I get that its important and that it comes up everywhere, in interesting integrals, differential calculus, statistics, everything. So I'm not asking about what cool places this number pops up, but rather why it should pop up in so many of these places. Philosophically, I guess, why should a number like e exist?

But why?

For example, Pi has a very intuitive explanation: the circumference of a circle divided by its diameter. And the fact that it comes up in non-geometric problems, I guess, can be explained by the concept that these geometries are fundamental parts of the physical and mathematical universe, and "pi" is a fundamental part of those.

But what about "e"?

What universal concept or physical property of the universe does it represent. What reason is there that it should be so pervasive and important.

BONUS: under different conditions (more dimensions in our universe, or the presence/absence of string theory, or some other sci-fi universe) would math exist as we know it, especially - would the number "e" exist or have the same value?
posted by nondescript to Science & Nature (17 answers total) 29 users marked this as a favorite
You can use Euler's Formula to express e in terms of π and i. In other words, you can express e in units of a property of a unit circle (and something else that follows certain mathematical rules). If there is a universe where a geometry defines π with a different value, or where math is weird and i has a different value, then clearly e would be different, assuming the Euler relation itself still holds.
posted by Blazecock Pileon at 12:32 AM on September 9, 2012 [2 favorites]

Logarithms and exponents expressed in base e (rather than base 10) turn out to be mathematically elegant. The exponential function, e^x, is its own derivative. The derivative of ln(x) (the natural log, i.e. log in base e) is 1/x. If you try to differentiate log(x) in base 10, the answer is 1/(x ln(10)). The derivative of 10^x is ln(10)*10^x. Basically, if you attempt to do exponential and logarithmic math in bases that aren't e, you get extra e terms all over the place just begging to be cancelled out.

Derivatives of logarithmic and exponential functions are important in differential equations, such as those which model physical systems experiencing exponential growth and decay. See the example for exponential decay, and note how the system is described by a very simple differential equation which has an exponential (ie. e^x) as its solution.

BONUS: The definition of e is "limit as n goes to infinity of (1 + 1/n)^n". It's self-contained, not physically derived or anything, rather hard for me to imagine it not existing or having a different value. Another definition is "the number a for which d/dx a^x = a^x". It gets a bit philosophical here but intuitively I understand e as being the number that facilitates mathematical descriptions of exponential growth and decay which are incredibly fundamental parts of physics (see, for example, all of thermodynamics). As long as the universe contains growth and decay there shall be e.
posted by PercussivePaul at 1:04 AM on September 9, 2012 [11 favorites]

Seconding the importance of e in differential calculus. Many things happen at a rate proportional to how much stuff there is, and thanks to e^x being its own derivative when you see that (population growth or decay, cosmological expansion - especially during inflation when we might speak of e-folds, linear air resistance), one knows that e^x is lurking around the corner.

To answer your bonus about extra dimensions or some stringy description at high energies, theoretical physicists understand and explore these scenarios using mathematics with the same value of e as we have; okay the group theory and differential geometry might be a bit more sophisticated but the mathematical principles are unchanged. After all, e is not a physical constant, it's just an irrational number. But while we're at it, here's a recent peer-reviewed open-access discussion of the variation of fundamental constants.

In order to make sense at the energy and length scales that we see every day, all these candidate theories have to explain all the stuff we've already seen (just like any scientific theory should), and so the usual way of doing this is to formulate an effective field theory that (carefully) discards stuff happening at high energies and small length scales. There's no freedom in that process for e to change: the mathematics cannot depend on the physics in that way.
posted by Talkie Toaster at 2:35 AM on September 9, 2012 [1 favorite]

e comes up a lot when things are increasing or decreasing in a regular way (by a fixed additive or multiplicative amount). There is a sweet spot of things increasing or decreasing where a function is its own derivative. Or, a function can be its own derivative multiplied by a constant, or raised to a power, or or or. This is what is happening when e appears in formulas not by itself, but with some other factors.

A lot of things in nature increase and decrease by regular amounts. Cells multiply... rabbits have sex with each other and make new rabbits... stuff (of all sorts) diffuses away from its source in a regularly decreasing way (e.g. Gaussian spreading distributions have e in them)... etc.

Regarding your bonus question: e is probably most relevant in a universe that includes both time and quantity.
posted by kellybird at 2:37 AM on September 9, 2012

> What universal concept or physical property of the universe does it represent. What reason is there that it should be so pervasive and important.

The part about e that makes it seem plausible to me that it could appear in many places, is that the derivative of e^x is e^x.

Now, not every class of functions can sustain this property. For example, in a quadratic equation (that is, equations of the form f(x) = ax^2 + bx + c for any a, b, and c) there's no specific a's, b's, or c's which can let f(x) equal the derivative of itself.

Exponential functions (equations of the form f(x) = a^x for any a) are the only type of function that can manage this. And the specific a that allows this to occur is our friend, e! So f(x) = e^x is the only function who is its own derivative. (This is one way e can be defined.)

This turns out to be a very useful property whenever you want to deal with both a function and its derivatives, which happens all the time in differential equations. For example, the force on a harmonic oscillator is commonly modeled as a function of its position (if you pull a spring further apart, the force will be stronger) and velocity (the faster its going the more force from frictional force there will be). And force determines an object's acceleration.

So right here in this system we see a differential equation involving position, velocity (which is the derivative of position), and acceleration (which is the derivative of velocity). And as you might be able to intuit, a function whose derivative equals itself comes in handy in working with these types of equations.
posted by losvedir at 4:21 AM on September 9, 2012 [1 favorite]

Another place where e comes up is when thinking about compound interest.

If you have some principal P0 invested at r percent for t years, compounded n times a year, you will end up with P0(1+r/n)nt. We want to investigate what happens as the number of compoundings/year goes to infinity. in other words, what if you compound continuously?

We make the following substitution: mr=n, where m therefore goes to infinity as n does (we are just introducing m, it doesn't hae a natural interpretation. This lets us rewrite the expression above as P0(1+1/m)mrt. If we take the limit as m goes to infinity, the expression (1+1/m)m converges to e (which is a consequence of the fact that ex is its own derivative, as mentioned above. Hence our expression becomes P0ert. This is an upper bound to the money you could get via compound interest with that principal at that interest rate.
posted by Elementary Penguin at 4:36 AM on September 9, 2012 [1 favorite]

And the fact that it comes up in non-geometric problems, I guess, can be explained by the concept that these geometries are fundamental parts of the physical and mathematical universe, and "pi" is a fundamental part of those.

You guess wrong.

But that's not your question, which is about e, and which I'd answer this way.

It's not so much that e is so important. It isn't. It's that the logarithm and its inverse, the exponential, are important. Write L for the log and E for the exponential function. Then e is E(1). Is that important? Well, I dunno, do you think sin(1) is important? Not really, but sine and the other trigonometric functions are important. You often write the exponential function as e^x, which makes it seem that it is somehow "defined" in terms of e, but that's an illusion; in reality, e^x is defined to be the function which sends 0 to 1 and is its own derivative, or as a power series

1 + x/1! + x^2/2! + x^3/3! + ....

e is defined in terms of e^x, not the other way around! Slogan: math is about functions, not about numbers.

Also, math doesn't care whether string theory is true.
posted by escabeche at 4:59 AM on September 9, 2012 [8 favorites]

Math doesn't care if anything is true. As Bertrand Russell put it, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." It's just a bunch of assumptions and definitions and rules to connect and extend them without reference to what does or doesn't exist outside its private world.

Here (with a video!) is someone asking and answering your very question.
posted by Obscure Reference at 5:45 AM on September 9, 2012 [2 favorites]

You might be interested in reading e: The story of a number by Eli Maor. This is the book that pushed me over the edge to become a math major. Use with caution...
posted by El_Marto at 6:11 AM on September 9, 2012 [4 favorites]

If a function f(x) is its own derivative f'(x), then that function is some constant to the x power. We've assigned the letter e to that constant. As to its ubiquity in physics, I suppose if you want, you can consider pi ubiquitous because it deals with "circular things" and e ubiquitous because it deals with "recursively derivative things".
posted by babbageboole at 6:59 AM on September 9, 2012 [2 favorites]

Curious to your impression, El_Marto - how accessible is that Maor book? I have a foundation in higher math (calculus, as a chemistry major, but I haven't done any calculus in at least 18 years) and have sort of been thinking it would be nice to try to somewhat reactivate my "math" brain - but I can't quite make it to trying to integrate recreationally...
posted by nanojath at 7:01 AM on September 9, 2012

The simple version: e amounts to the "unit circle" of exponential growth.

The slightly less simple version (but by no means rigorous): Elementary Penguin's explanation comes the closest to how I think of e, since compound interest lets us tweak the compounding period arbitrarily without worrying about things like fractions-of-a-rabbit.

So, let's say you get 100% interest per month on one dollar. After a month you will have $2.00.
If you break that down into 10% every three days, after a month you will have $2.25937.
If you break that down into 0.1% every 43.2 minutes, after a month you will have $2.62877.

And, if you break that down into infinitely small chunks, after a month you will have... $2.71828!

It turns out that when you want to know the growth rate of a continuous process dependent on the quantity you have "now", your answer will always "scale" with e (I use that word loosely, you don't have a literal multiplicative scaling here). Faster growth / higher interest rate / larger litter size, just a bigger scaling factor; Slower growth / lower interest rate / smaller litter size, just a smaller scaling factor, following the standard formula e(rate * time)–1 (in the above example, I hid the "-1" part of that in the dollar we started with, so the rate actually came out to 1.71828)

Hope that helps!

/ False-start Filter - Female rabbits will pop out a litter of 7ish once a (lunar) month, forever. After three months, the female kits can start doing the same. They can even get "double" pregnant, in that they can carry two litters at once with different delivery dates!
posted by pla at 8:55 AM on September 9, 2012 [1 favorite]

Nthing everything that's been said about differential equations. e--or, more accurately, the exponential function, as escabeche has pointed out--has geometry too, particularly when you start working with complex numbers.

The exponential function e^(ix) traces out the unit circle on the complex plane if x is real, and if we allow x to be a complex number, the resulting function is prototypical for periodic functions on the complex plane. The trigonometric functions, which are our prototypes for periodic functions of real numbers, are actually just rational functions of e^(ix).

Geometrically, numbers of the form e^(ix) serve as the complex analogues of 1 and -1 (which themselves can be written in this form), in that they parameterize operations that don't change the magnitudes of numbers; as a consequence, they are at the heart of operations that don't distort the geometry of a space, i.e., reflections, rotations, and the like. So as soon as you do geometry, e is lurking in the background just as much as pi.
posted by Aquinas at 9:01 AM on September 9, 2012

In order to get a satisfying answer to your first question, I think we unfortunately need to engage in a bit of psychology here. The problem is that mathematics doesn't distinguish between "definitions" of e and "cool places" where it pops up. That's entirely a human thing. So we need to ask what it is that makes the "the circumference of a circle divided by its diameter" explanation so much more "intuitive" than some other explanation of π (such as cos-1(-1) or 4 ( 1/1 - 1/3 + 1/5 - 1/7 + 1/9 + ...) or whatever)?

I think a big part of the answer is simply that we live in a physical world that is roughly Euclidean at our scale. Also, it's always the first definition of π that we get, and unlike most of the definitions e, that definition is totally comprehensible with out any knowledge of calculus, limits, or even trigonometry. So that definition sticks. We carry all the baggage behind the definition of Euclidean space with us at an intuitive level, so we don't see it in the circumference/diameter definition of π.

Unfortunately, e doesn't have such a simple geometric definition. If you know calculus well and have absorbed it a deep level, then the comment about the ex function that several others have made here will have some resonance. It's not e that's so important so much as it is that the* function that is its own derivative that's important, and that function happens to be writable in the form ex. But if calculus doesn't feel as intuitive to you as Euclidean geometry (and for most of us, it isn't), then this feels like just another cool place to find e and not a satisfying intuitive definition.

There's also the famous Euler formula eπi = -1, which can also be used to define e, but if you weren't won over by the calculus definition, this won't seem any more intuitive.

The limit and infinite series definitions of e (such as (1 + 1/n)n as n→∞ or 1/0! + 1/1! + 1/2! + 1/3! + ...) are considerably more satisfying than those typically given for π, although there are definitely π-related numbers with pretty satisfying definitions in this sense (most notably π2/6 = 1/12 + 1/22 + 1/32 + ...). But since it's easy to imagine dozens of comparably simple definitions, this doesn't really help matters.

But! All hope is not lost if you want an intuitive definition that doesn't involve too much advanced math. It's still only an exact definition in the limit, but the circumstances described are pretty mundane. A gambler who sees that they've got a one-in-a-hundred chance at winning might think that it'd be pretty unlikely for them to be able to play the game a hundred times and never win. But they'd be wrong. The probability of not winning any of their hundred games is actually about 0.366. And if they play a million one-in-a-million games, that probability is approximately 0.368. What is this converging to?


Nobody ever uses this as a definition, but why the hell not? It's reality, it's understandable, and you certainly could use it to explain the other definitions if you wanted to, especially the non-calculus ones.

*Technically, there are others, but they're all slight variations of the same thing.
posted by ErWenn at 9:05 AM on September 9, 2012 [2 favorites]

And as for your bonus question:

Both π and e have many different definitions, and many of those definitions are completely independent of any sort of physical reality. And so in that sense, neither π nor e would ever change with the laws of physics or the layout of reality. However, I think what you're driving at is that because the most common definition of π is based on Euclidean geometry, it's quite possible to imagine a world where the ratio of the circumference to the diameter is not approximately 3.14159. So the question now becomes: is there a definition of e that depends on some physical facet of the universe that we could imagine as different?

I don't know of any.
posted by ErWenn at 9:24 AM on September 9, 2012 [1 favorite]

What is this converging to? 1/e. Nobody ever uses this as a definition, but why the hell not?

I like your suggestion, but I would quibble that nobody ever uses it — it's very close to one of the standard definitions you mention, that e = limn→∞ (1+1/n)n, so close that you could argue it's "essentially" the same.

(Details: the probability of losing n games when each has an independent probability of 1/n of being won is (1-1/n)n, so this definition gives e = limn→∞ (1-1/n)-n. Writing m=n-1, a little algebra yields (1-1/n)-n = (1+1/m)m(1+1/m), which since 1+1/m→1 implies that if either of the expressions (1-1/n)-n or (1+1/n)n has a limit, then they both do, and then they have the same limit.)

Still, I agree that it's a very nice elementary way of introducing the number.

BONUS: under different conditions [...] would the number "e" exist or have the same value?

In some discrete math contexts, the natural analogue of the function e^x is the function 2^n. For example, in the "finite calculus" for sequences of numbers (instead of functions on the real line), the analogue of the derivative is the first difference operator, and the sequence which is its own first differences is the sequence 2^n. With a little imagination, you might be able to see this as suggesting that if our universe were discrete in the right way, we'd have e=2 (but I doubt that notion would stand up to scrutiny).
posted by stebulus at 11:00 AM on September 9, 2012 [1 favorite]

Math doesn't care if anything is true.

... if you're a formalist. Platonism asserts that math is just descriptions of a perfect universe that we can't actually access. It's not just that math is true relative to itself, but that it's permanently true. So the concept of "e" just sort of exists whether or not we notice. There are some obvious problems with this idea, not least of which is how do we access this Platonic realm?

Formalism basically says that math is a matter of pushing symbols around and doesn't bear any real relationship with reality. This is somewhat unsatisfactory.

There's a rather more satisfying third option called "embodied mathematics." A cognitive scientist and a psychologist published Where Mathematics Comes From which basically says that mathematics is meaningful because it arises from how we are in the world. It's not about pushing about abstract symbols; at its core, it's a huge tangle of metaphors that build off innate concepts like "this thing is bigger than that thing" and "one thing plus one thing is two things" (knowledge of the latter has been demonstrated in very young infants). I think they actually put forward a (rather complicated) metaphorical explanation of e as an example.
posted by BungaDunga at 9:21 PM on September 9, 2012

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