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Intuitive Understanding of the constant “e”

Let's say you want to explain this to your teenage son. I understand the technical definition of $e$

$$ e=\lim_{n\to\infty}\left( 1 + \frac1n \right)^n $$

But, I don't want to get lost in technical babble. While specific examples are welcome, I also want to understand the big picture. I want to first know the general significance of why and when e is used. Anyone got a link? Is there a typical "pattern" that it models? In what general sorts of situations does "e" arise?

I only know e from the classic continuous compounding example. But, why does it appear in other applications of growth, science, etc? Are all these examples just variations on this same limit that defines e? This increasingly smaller interval when you apply an infinitely smaller percentage growth (1 + 1/n) but infinitely many times (to the power of n)

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marked as duplicate by sdcvvc, Haskell Curry, Ittay Weiss, Alexander Gruber, JavaMan Jan 8 '13 at 3:00

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Try talking for an hour with someone without using any word containing the letter "e." Just about impossible. –  Will Jagy Jan 8 '13 at 0:55
@WillJagy ... One word: Gadsby. –  AvatarOfChronos Jan 8 '13 at 1:59
@AvatarOfChronos, good, good. The subtitle should have been "Fiction book using only abcdfghijklmnopqrstuvwxyz" –  Will Jagy Jan 8 '13 at 2:08

7 Answers 7

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Not an easy task (the teenage part).

Any system where the rate of change of a quantity is proportional to the amount of the quantity has solutions involving exponentials. This is true for continuous as well as discrete systems. Chemical reactions, electronic circuit behavior, physical systems, etc, are often well approximated by such systems.

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There is also a difference between an interest in understanding

  • the mathematical constant "$e$", which is an irrational, transcendental number, but real number, just as is $\pi$, but a number, nonetheless, which happens to be the value of your limit, and happens to be the value of the infinite sum $$e = \displaystyle \sum_{n=0}^\infty \frac{1}{n!}$$And more: $$e^x = \sum_{n=0}^\infty{x^n \over n!} = 1 + x + {x^2\over2!}+{x^3\over3!}+{x^4\over4!}+\cdots$$ vs.

  • exponential functions (base $e$: (e.g., $f(x) = e^x$, or $e^{i\theta} = \cos \theta + i\sin \theta$), functions to which you seem to refer when you speak of its use in representing exponential growth, and other applications which crop up everywhere, so it seems. Certainly, functions involving $e$ tell us something about $e$, but they tell us so much more than that.

That is, the importance of $e$ itself isn't so much for its significance as a particular real number, but its significance as a base for exponential functions, and in terms of the ways that functions involving $e$ as a base appear in surprising ways, are powerful, and have applications in many domains.

So it's hard to know exactly what you are interested in: the number $e$, or the many functions involving $e$.

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I'm interested in both, I suppose –  JackOfAll Jan 8 '13 at 14:14
@amWhy: That's $e$erily another nice answer! :-) +1 –  Amzoti May 8 '13 at 2:36

If you differentiate an exponential function, you get itself times a constant: $$ \frac{d}{dx} 2^x = \left(\text{constant}\cdot 2^x\right). $$ In other words, the function grows at a rate proportional to its present size.

Only if the base of the exponential function is $e$ is the "constant" equal to $1$, so that you get $$ \frac{d}{dx} e^x = 1\cdot e^x. $$ In other words, the function grows at a rate equal to its present size.

It's the same as the reason why radians are used in calculus. You have $$ \frac{d}{dx} \sin x = (\text{constant}\cdot\cos x). $$ If you use degrees, the "constant" is $\pi/180$. If you use radians, the "constant" is $1$, but only if you use radians.

There's more to the story than that. For example, how does the Poisson distribution arise as the limit of binomial distributions? But the above should show you in what sense $e$ is "natural", and in what sense radians are "natural".

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I discern three aspects to your question: Why are exponential functions $b^x$ important? Why are logarithmic functions $\log_b x$ important? And assuming the first two, which base is best (in what sense?) and why is it $e$?

I assume your question is the last and the best reasons have to do with calculus. Any exponential function can be converted to any other base: $b^x = a^{x \log_a(b)}$. And any logarithmic function can be converted to any other base: $\log_b(x) = \log_a(x)/\log_a(b)$. So why pick $b=e$?

@MichaelHardy points out that the derivative of $e^x$ is $e^x$. Similarly the derivative of a logarithmic function is a constant divided by $x$. For $\log_e(x)$ the constant is $1$, the simplest possible. (Thus, to be clear, the derivative of $\log_e(x)$ is $1/x$.) This makes the base $e$ convenient for both exponential and logarithmic functions.

Many other formulas for exponential and logarithmic functions are simplest when the base is $e$. You've already pointed out the limit $$e=\lim_{n\rightarrow\infty}\left(1 + {1\over n}\right)^n\,$$ Compare the series for $2^x$ with the series for $e^x$: $$e^x = \sum_{n=0}^\infty{x^n \over n!} = 1 + x + {x^2\over2!}+{x^3\over3!}+{x^4\over4!}+\cdots$$ Similarly, $$\log_e(x)=\lim_{n\rightarrow\infty}n(x^{1/n}-1)$$ and compare the series for $\log_2 (1+x)$ with $$\log_e(1+x) = \sum_{n=1}^\infty{(-1)^{n-1}x^n \over n}=x-{x^2\over2}+{x^3\over3}-{x^4\over4}+\cdots$$ Basically you could say that the importance of $e$ comes down the constant Michael Hardy refers to -- you could think of it as a "correction factor," I suppose, or better as a "nuisance factor" -- that constant is $1$ for the base $e$, which is very convenient when you are working things out.

The rest of its importance derives from the importance of exponential and logarithmic functions generally.

(If the teenager cannot appreciate calculus yet, well, perhaps he can appreciate the reason that it is easiest in the cases that matter the most. And if not, then perhaps advise patience --"You'll understand when you're older." :) It's not bad advice, really. The thing that bugs me is what was so important about $\log_{10}$? Well, it's the easiest log. to understand at the beginning, but it's not that important later.)

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Considering functions of the form $f(x)=a\,b^x$ with $b>0$, we will always have an initial value of $a$. Only with the base $b=e$ will we also have an initial rate-of-change equal to $a$.

Demonstrate this with pictures: 2^x, e^x, and 3^x plotted near x=0, together with initial tangent lines

(This image is from lecture notes I am coauthoring.) Even better would be a GeoGebra applet modeled after images like these where $a$ and $b$ are manipulable.

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Let $a > 0$ and consider the exponential function $y = a^x$. Now draw the tangent line to the graph at the $y$-axis. If $a < 1$ this line has negative slope; if $a > 1$ the line has positive slope. There is a unique value of $a$ which gives this tangent line slope 1. That value is $a = e$.

You can derive all of the goodies about the exponential function right from this principle.

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$e$ appears in a lot of places in calculus, but before calculus, it has little significance.

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Limits appear in pre-calculus. Compound interest is a valid topic for a pre-calculus course. One of the numbers that appears when studying limits and compound interest is $e$. –  robjohn Jan 8 '13 at 1:05

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