Sunday, 22 March 2015

Why does e = 2.71828183? Or how e SHOULD have been discovered.

This is an interesting exercise to do with higher level mathematical students - it shows them that the value of $e$ can be derived purely from it's differential properties.

So we start with the fundamental property of $e$, that $D(e^x)_x = e^x$.
In short, we're looking for the identity function of differentiation - what function stays unchanged.
(As noted in previous posts, I'm using a differential format similar to sagemath's default format.)

If we write this property as a limit:
$f(x) = \lim\limits_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}$
and rearrange outside the limit:
$\delta x f(x) = f(x + \delta x) - f(x)$

$(\delta x + 1) f(x) = f(x + \delta x)$
 We get something we can use as a recurrence equation.

$f(x + \delta x) = f(x)(1 + \delta x)$

$f(x + 2\delta x) = f(x + \delta x)(1 + \delta x)$

$f(x + 2\delta x) = f(x)(1 + \delta x)^2$


$f(x + n\delta x) = f(x)(1 + \delta x)^n$
Thus proving that the identity of differentiation is an exponential of some form.

We represent the base of this exponent by the familiar $e$, and rewrite the limit equation.
$e^x = \lim\limits_{\delta x \to 0} \frac{e^{(x + \delta x)} - e^x}{\delta x}$
We expand the exponents
$e^x = \lim\limits_{\delta x \to 0} \frac{e^{x}e^{\delta x} - e^x}{\delta x}$
and eliminate $e^x$ from both sides (As $e^x$ is guaranteed to be nonzero).
$1 = \lim\limits_{\delta x \to 0} \frac{e^{\delta x} - 1}{\delta x}$

Rearranging outside the limit,
$\delta x = e^{\delta x} - 1$

$1 + \delta x = e^{\delta x}$

$(1 + \delta x)^\frac{1}{\delta x} = e$

We have one final trick to make our limit easy to calculate, we replace $\frac{1}{\delta x}$ with $n$, to get the defining limit of $e$. The advantage of this, is that we end up calculating high order powers, rather than high order radicals.
$e = \lim\limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$
When this function was first discovered? invented? limits like this were hand calculated. You can calculate high order radicals by hand, but it's very slow.

Get the students to calculate this for a couple of large values of n, and they get something that looks like this: 
$n$ 1 10 100 1000 $\infty$
$e$ 2 2.59374246 2.70481383 2.71692393 2.71828183

I've found that my top level mathematics students really appreciate this type of work, because they usually have the type of personality that they feel stress and anxiety if you just introduce content without explaining why.

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