Demystifying Euler’s number
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Euler’s number $e := 2.71828\dots$ has, to me, always been a semi-mysterious number. While I understood many facts about $e$, I never felt I ever truly understood what it really was – it’s core essence so to speak. Thanks to an excellent explanation by Grant Sanderson’s 3Blue1Brown video, I now better understand this constant. In this blog post, I will attempt to describe, in my own words, my understanding of Euler’s number and expound on Sanderson’s explanation.
Introduction
Euler’s number $e := 2.71828\dots$ has, to me, always been a semi-mysterious number. While I understood many facts about $e$, I never felt I ever truly understood what it really was – it’s core essence so to speak. Sadly, it was only very recently that I finally felt like I truly understood this constant. Sure, I knew certain facts about $e$, but I didn’t really understand its essence. For example, I knew that,
\[\frac{de^x}{dx} = e^x\]I also knew that $e$ was used in calculations involving compound interest due to the fact that
\[e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n\]but these facts were not satisfactory to me because they didn’t explain exactly why this constant is so ubiqituous. Why, for example, do we choose to represent nearly every logarithm with base $e$? More puzzingly, why do we call this the “natural” logarithm? Why does it appear in the probability density function for the normal distribution? Or in Euler’s formula?
With the help of Grant Sanderson’s excellent 3Blue1Brown video, I feel now that I have a much better understanding for it’s essence. In this blog post, I will attempt to describe, in my own words, my understanding of Euler’s number and expound on Sanderson’s explanation.
To spoil the punchline, Euler’s constant is, in a sense, a number that describes all exponential functions – that is, functions of the form $f(x) := a^x$. Because of this, all exponential functions of the form $a^x$ can be written in a “canonical” way using $e$ that is more arguably more informative than $a^x$. Let’s dig in.
The essence of exponential functions
To review, exponential functions are functions of the form:
\[f(x) := a^x\]for some constant $a$. Exponential functions do not only grow, but their growth also grows. When plotted, they look like the following:
In fact, the key characteristic of exponential functions – the very characteristic defines exponential functions – is that their growth grows linearly with the function itself. Stated more rigorously, an exponential’s derivative is proportional to the value of the function itself. That is:
\[\frac{da^x}{dx} = Ka^x\]where $K$ is some constant that is determined by $a$.
That is, no matter what value for $x$ for which we are evaluating the derivative, the derivative of $a^x$ is simply $a^x$ itself multiplied by some constant $K$. Intuitively, this makes sense just by looking at the exponential function curve: the bigger is $a^x$ the steeper the rate of change:
This fact can be proven mathematically. Let’s start with the definition of the derivative of $a^x$:
\[\begin{align*}\frac{da^x}{dx} &= \lim_{h \rightarrow 0} \frac{a^{x+h} - a^x}{h} \\ &= \lim_{h \rightarrow 0} \frac{a^xa^h - a^x}{h} \\ &= a^x \underbrace{\lim_{h \rightarrow 0} \frac{a^h - 1}{h}}_{K} \end{align*}\]Note, that the derivative of $a^x$ is simply $a^x$ scaled by a constant:
\[K := \lim_{h \rightarrow 0} \frac{a^{h} - 1}{h}\]Moreover, we see that this constant is determined by the value of $a$ – that is, it is a function of $a$. Thus, we can write this constant as:
\[k(a) := \lim_{h \rightarrow 0} \frac{a^{h} - 1}{h}\]Defining Euler’s number
Now, the natural question that follows from this observation of exponential functions is: what exponential function, $a^x$, yields a constant of 1? That is, for what value of $a$ do we have $k(a) = 1$?. It’s Euler’s number!
That is, Euler’s number is the base of the exponential function for which the derivative of that exponential function is the exponential function itself:
\[\frac{de^x}{dx} = e^x\]That is, $e$ is the value for $a$ that satisfies the following equation:
\[1 = k(a) = \lim_{h \rightarrow 0} \frac{a^{h} - 1}{h}\]It turns out that that $e$ can be expressed as a limit that enables us to compute numerical approximation to this value (See Theorem 1 in the Appendix to this post):
\[e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n\]With this formula, we can calculate ever close approximations to $e$ by simply plugging in larger and larger values for $n$. If we do so, we find that $e \approx 2.71828$.
In a sense, $e$ defines the “base” exponential function; By “base” I mean the exponential function whose rate of change is itself (i.e., whose constant of proportionality is 1).
All exponential functions can be expressed in a more intuitive way with $e$
The next natural question is, why do we care about $e$? And why do we see so many exponential functions and logarithms involving $e$?
To answer this, let’s first say we have some exponential function $a^x$. As we showed above, the derivative of $a^x$ is proportional to $a^x$ with some constant of proportionality given by $k(a)$. Knowing this constant of proportionality would be quite informative: it tells us exactly how quickly the exponential is growing.
Unfortunately, the specific value, $k(a)$, is not easy to compute directly from $a$. Recall it is given by,
\[k(a) := \lim_{h \rightarrow 0} \frac{a^{h} - 1}{h}\]Is there some way to express $a^x$ in a way that involves $k(a)$? Yes! And that is given by (See Theorem 2 in the Appendix to this post):
\[a^x = k(a) e^x\]Because every value for $a$ is associated with a unique constant $k(a)$, we can express all exponential functions using the constant $k(a)$ instead of $a$ via $k(a) e^x$. This value makes the exponential easier to interpret: whenever you come upon an exponential function, $f(x) := K e^x$, the rate of change of $f(x)$ at $x$ is given by $K$.