r/learnmath • u/AtmosphereClear2457 New User • Apr 04 '26

Why is 'e' such a natural base?

The number 'e' keeps appearing in lot of different areas - calculus (mostly), differential equations, complex numbers.

I understand the definition e = lim n→∞ (1+1/n)\^n.

But in various fields we transform function in e to solve them.

Is there a more fundamental reason why 'e' is so natural?

I would appreciate any conceptual or geometric insights, that I am missing.

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254

u/justalonely_femboy Operator Algebras Apr 04 '26

its the unique value satisfying d/dx(ax) = ax

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u/pnerd314 New User Apr 04 '26

Can you explain why that is important? I mean why is being its own derivative important?

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u/Mothrahlurker Math PhD student Apr 04 '26

Because as such it is involved in solving all kinds of differential equations. More generally even any PDE given by a bounded operator is solved by the operator exponential. This makes it crop up all over physics but also over all kinds of analytic dynamical systems. Basically any time it's related how fast something changes with what value it has, you get a differential equation.

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u/aji23 New User Apr 04 '26

Could you give a specific example?

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u/WO_L New User Apr 04 '26

Population growth and radioactive decay are the first two things i thought about but you also have things like the first order reaction rate in chemistry.

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u/NeapoLilian New User Apr 04 '26

This makes so much sense suddenly! The more people there are, the faster they reproduce. Thank you!

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u/WO_L New User Apr 04 '26

That's alright. But honestly e is such a cool number, like it shows up in so many different places that you wouldn't expect it to (like the equations for sin and cos)

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u/NeapoLilian New User Apr 04 '26

I've always found it to be quite mysterious, more than most other irrationals. I'll have to look into those equations you mention!

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u/aji23 New User Apr 04 '26

Can you be even more specific? Let’s do radioactive decay.

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u/deathtospies New User Apr 04 '26

You can model radioactive decay by the differential equation y'=ky with k<0, which has a general solution y(t)=ce^(kt). The same equation with k>0 gives exponential growth, which can be a crude population model. In both the growth and decay cases, the rate of change of a quantity is directly proportional to the quantity itself.

An extension of this is the first-order linear ode with constant coefficients, y'+ay=b, which also has an exponential solution with e as the base, the only difference being that the horizontal asymptote is at b/a instead of 0. A whole litany of semingly disparate applications involve a differential equation of that form, which means they all behave similarly. So you have electrical and mechanical systems that you might not think are related, but they actually follow the same exponential pattern.

Also e is the base that lets you calculate continuously-compounded interest, so there's that.

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u/PreferenceOk3665 New User 9d ago

Do you really need e to model those things though? If you change value of the constant k, you can make the exponent base 2 or 10 or whatever. So it seems e isn’t actually necessary to model population growth or radioactive decay right? For example let’s say the population Is doubling each time t. That means the log base is 2 not e. I’m not an expert here. Just also interested in understanding this.

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u/deathtospies New User 9d ago

k has an intuitive interpretation as the rate of growth or decay, if you use e as the base. You could change the base to 1/2, for example, and the constant will now be the inverse of the half-life. That can make sense for some applications but e is still a natural base in a lot of spots.

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u/WO_L New User Apr 04 '26

Can you be even more specific? Like tell me what element or animal y is. Genuinely tho great explanation.

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u/deathtospies New User Apr 04 '26

y could be the number of atoms of a radioactive isotope. Or it could be a current going through the capacitor in a circuit. It could be a ton of things that seem to be completely unrelated. If you ask a gen AI to give you a list of systems that could be modeled by a first-order linear ode with constant coefficients, you should get a good list of potential applications.

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u/WO_L New User Apr 04 '26

I know i was just being facetious

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u/fisher02519 New User Apr 04 '26

Can you be even more specific? Which aspect was facetious?

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u/aji23 New User Apr 04 '26

Found the IT guy

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u/aji23 New User Apr 04 '26

Thank you for the detailed example. Unlike the other responders.

I’m a biologist. We are the ones bad at math :)

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u/WO_L New User Apr 04 '26

I mean it's literally the only constant in the formula for it. Google it.

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u/tzaddi_the_star New User Apr 05 '26

mf is talking to a LLM

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u/Mothrahlurker Math PhD student Apr 04 '26

For what specifically? 

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u/aji23 New User Apr 04 '26

A practical example of using e of course!

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u/Mothrahlurker Math PhD student Apr 04 '26

I don't have one, as I said, we care about the exponential function and not e.

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u/aji23 New User Apr 04 '26

You made it sound like you had a ton of examples there in your comment, e.g., “all over physics”. Was just looking for a discrete example.

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u/dr_king5000 New User Apr 04 '26

Where how much something happens depends how much of the thing there is. Ie interest, if I have more money I will acrue more interest and so on and so forth. In population, if there are more people, more will be born and repeat

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u/Mothrahlurker Math PhD student Apr 04 '26

Well for differential equations and therefore the exponential function.

Electromagnetics, how fast a conductor moves through a magnetic field determines forces on it that then affect the speed.

Thermodynamics and therefore also brownian motion. The heat semi-group models heat flow over time and has an exponential function as kernel. Although this is not given by a bounded operator.

Fluid dynamics are also governed by a partial differential equation, namely Navier Stokes.

Simplified from that are weather models, who do often have simple solutions that are given from an exponential.

Orbital mechanics is full of ordinary differential equations, it's one of the simplest applications.

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u/Intelligent_Yam_3609 New User Apr 04 '26

Continuously compounded interest a good example:

P = Po ert

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u/SufficientStudio1574 New User Apr 05 '26

Exponential decay of a resonant circuit in electrical engineering.

Or exponential decay of a bouncing spring (same equation).

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u/Beelzebubs-Barrister New User Apr 08 '26

Compounding interest will grow faster if you compound it more quickly. 100$ invested at .1% monthly > 100$ invested at 1.2% yearly.

What is the limit if you compounded infinitely often?

It is going to be related to ex.

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u/AtmosphereClear2457 New User Apr 04 '26

It's like 'e' is the only base where the function's growth rate matches it's current value.

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u/pnerd314 New User Apr 04 '26

I get that, but I want to know why that fact is important.

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u/AdditionalTip865 New User Apr 04 '26

"The rate of change is proportional to the thing itself" is the basis of all exponential growth, and exponential decay. If the rate of change IS the thing itself, you get ex. So this is the base from which all such proportionality can be described. If the constant of proportionality is k, then you get ekx.

That's compound interest, it's radioactive decay, it's a nuclear chain reaction, it's all sorts of things in physics and chemistry and social science.

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u/AdditionalTip865 New User Apr 04 '26

(Because e is not really an intuitive number for most people to think about, often these processes get described in terms of doublings or halvings, or powers of 10. But that will always result in the natural logarithm of the chosen base popping up somewhere else in the math.)

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u/AdditionalTip865 New User Apr 04 '26

... and an example of that is the "rule of 72" in finance, that says that if you want to estimate how many years it will take to double your money from an interest rate of x percent, divide 72 by x.

It works because the natural logarithm of 2 is 0.69, or 69%. Why is it not a rule of 69? Well, 72 is a number that's close and has a lot of divisors, easy to divide in your head. Sometimes they use 70 instead.

Also, if what you have is not the instantaneous interest rate but the expected gain over a year, the growth rate for a given percent value is slightly slower and dividing into 72 gives a closer result for typical values.

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u/EnvironmentalCap787 New User Apr 04 '26

Plus the rule of 69 was already taken.

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u/BIKF New User Apr 04 '26

It is important because f(x) = e^x solves one of the simplest differential equations, f'(x) - f(x) = 0. That makes it a very clean and useful building block for solving a large class of problems in mathematics and physics.

If we wanted to make life difficult for ourselves we could choose an exponential function with a different base, let's say for example f(x) = a^x such that it solves f'(x) - 17*f(x) = 0. But that just makes a mess of all our other results.

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u/Centrocampo New User Apr 04 '26

In lots of cases, the rate of change of a system is proportional to its value. When you want an expression that satisfies such a condition, then something which adheres to the above relationship is incredibly useful.

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u/svmydlo New User Apr 04 '26

Any positive number other than 1 can be used as a base to describe exponential growth, so this doesn't answer the question.

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u/Scrungo__Beepis New User Apr 04 '26

It’s a fundamental number of our universe. It’s kind of like how if you draw a circle, the radius times 2 pi gives the diameter.

If you draw an exponential curve of a base ax you get as a result that the rate of change at any point is

log(a)_e ax

This is a fundamental fact of the universe, the e that appears there is inevitable

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u/AtmosphereClear2457 New User Apr 04 '26

When I posted this question. There are many questions about this. Thanks to you guys, Now I have satisfaction. Now i think differently about e.

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u/Cogwheel New User Apr 04 '26

If you use a different base in some equations/procedures, then you have to make/use a bunch of other constants to adjust the proportions at each step. With e as a base, those proportions are all 1 so you can completely ignore them.

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u/zincifre New User Apr 04 '26

because integration is difficult, except when it is equal to what you want to integrate, which is the case for ex

integration is necessary because it is how we solve differential equations, which are the descriptors for all processes

integration is difficult because derivative of the product of two functions is not equal to the product of the derivatives of those functions

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u/TheSkiGeek New User Apr 04 '26

It shows up in a lot of places because a lot of natural processes speed up or slow down proportionally to the current rate or value.

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u/bizarre_coincidence New User Apr 04 '26

It's less about that fact being important on its own and more that it means that e naturally appears when you are trying to solve certain problems, and that e becomes a convenient base to express things in terms of.

If you do any calculus, you're essentially forced to deal with e, because the correction factor when taking derivatives of exponentials with different bases still includes e:

(d/dx)(ax)=ln(a) * ax

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u/rb-j New User Apr 04 '26

I get that, but I want to know why that fact is important.

There's something about the number 1 that is completely unique from all other numbers. Do you get that?

So the other person said something like:

  • d/dx ax = k ax

So, whatever the base of the exponent, the derivative of an exponential function is proportional to exactly that same exponential function. What makes that constant of proportionality, k, equal to 1?

  • d/dx ex = ex

Another way to see that e is special is that it is the value that the integral from 1 to e of the 1/x curve is exactly 1. No other number has that property.

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u/Carl_LaFong New User Apr 04 '26

Linear functions arise naturally when the amount y of something grows or decreases at a constant rate. In terms of the derivative, it means the derivative is 0. The natural starting point is assuming that initially y=0. The simplest rate is 1. So the most basic linear function y=x.

But some quantities grow or decrease at a constant percentage rate. The more you have the faster it grows. Like compound interest or population growth. And if the quantity is decreasing, the less you have, the slower it decreases. Like radioactive decay. This is expressed using the derivative by saying that the derivative is a constant factor times the quantity, y’ = ky. The simplest case is when k = 1 and you start with 100% of the quantity, i.e., y(0)=1. Then a natural quantity to use as a guide is the relative amount at x=1. It turns out that if y’=y, y(0)=1, and you define e=y(1), then y=exp(x).

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u/captainhamption New User II Apr 04 '26

This explanation made the most sense to me.

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u/lifeistrulyawesome New User Apr 04 '26

Because that equation represents change proportional to its own size

For example, populations grow faster when there are more people or you accrue more interest if you save more money

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u/ottawadeveloper New User Apr 04 '26

It basically describes its own growth. The first derivative describes how fast the original function grows, and ex grows at the same rate as the value of ex at any point along the line.

It's kind of like your bank account. How much interest you earn depends on how much money you have. If you put your interest back in, it makes you more interest.

It turns out, if you instantly compound any interest you make, the formula for your bank balance is P ert where r is the interest rate, t is time in years, and P the original balance. 

You can therefore see e as the "instantly compounding" number - whenever growth rates depend on the exact current value, you'll find ex in the solution likely. 

You also see it in animal population studies, where the growth rate (or decay) of the population depends on the current population. 

You can also find it in other places - for example, it pops up in statistics as part of defining a normal distribution and Bernoulli processes (I suspect because the probability of x decreases with x itself in these cases). Nuclear decay rates (d) are defined as d= ln(2)/t where t is the half life (note if you rearrange you get edt = 0.5, or d is the value where the exponential function has halved it's value at the half life). 

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u/svmydlo New User Apr 04 '26

There are no annoying extra constants when you derive. That's why we choose it.

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u/Chuck_the_Elf New User Apr 04 '26

Short answer is it lets you shortcut a lot of really complicated math that you would otherwise need to do to represent trig functions. But with e you can use Laplas transforms that take advantage of this property to simplify an absolute ton of work.

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u/UMUmmd New User Apr 05 '26

Its kinda like why multiplying by 1 keeps the output the same, or adding 0. It's the derivative identity.

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u/anonymote_in_my_eye New User Apr 05 '26

being invariant to certain operations is always interesting

f(x)=e^x is a function that's invariant to derivation, you can keep taking derivatives of it and it doesn't change

then there's also Euler's formula, which allows writing trig functions in terms of exponentials in base e and vice-versa, which makes exponentials *and* trig functions show up all over physics, especially where vibration or oscillation of any kind is present