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I have recently worked with some students motivating the development of $e^t$ and $e^{t i}$ as summing change over time, basically informally solving differential equations. My motivation for this is that the students are already used to programming physics simulations which is basically numerically solving differential equations.

The definition that students developed was:

When calculating 100% interest continuously compounded for one year, your money is calculated by $m_1 = m_0 \cdot \lim_{\Delta t \to 0} \left(1+ \Delta t\right)^{\frac 1 {\Delta t}}$

This is simply the usual definition of $e$ replacing $n$ with $\frac1{\Delta t}$.

Based on this the definition, the exponential that they developed was:

$m_t = m_0 \cdot \lim_{\Delta t \to 0} \left(1+ \Delta t\right)^{\frac t {\Delta t}}$

which calculates $e^t$. However this is different from the way I usual derive $e^x$ in which the $x$ is inside the exponential rather being the exponent $$\lim_{n \to \infty} \left(1+ \frac x n\right)^n$$ This is really only substituting $n=\frac t {\Delta t}$ and $x=t$, but massively changes what is looks like.

It occurs to me that this could have potential advantages:

  1. It is immediately obvious that $t$ in $\lim_{\Delta t \to 0} (1+ \Delta t)^{\frac t {\Delta t}}$ is an exponent. The only task is to substitute $ u = \lim_{\Delta t \to 0} \left(1+ \Delta t\right)^{\frac 1 {\Delta t}}$ and the above can be rewritten as $u^t$. This is much easier than proving that the $x$ in $\lim_{n \to \infty} \left(1+ \frac x n\right)^n$ acts like an exponent.

  2. The limit looks like a normal calculus-style limit of $\Delta t \to 0$.

potential disadvantages:

  1. The next task was to calculate position on the complex plane where $\frac {\Delta d}{ \Delta t}$ is always perpendicular to the displacement. They already know that a way to create a perpendicular vector is to multiply by $i$ so this quickly led to the formula $\lim_{\Delta t \to 0} \left(1+ i\Delta t\right)^{\frac t {\Delta t}}$. Unfortunately in this formulation as the $t$ is not inside the base of the exponent, it does not immediately follow that the analytic formula is $e^{it}$ as it does in the formulation $\lim_{n \to \infty} \left(1+ \frac {ix} n\right)^n$

It seems like there are 4 equivalent ways of writing this formula by mixing and matching the following options.

  1. $x$ or $t$ inside the base or as an exponent

  2. the limiting variable: ${n \to \infty}$ or ${\Delta t \to 0}$

Questions

What do people think is the best combination of the above (or other) options?

What are additional advantages/disadvantages or various options?

What are other good ways to motivate the definition of $e^x$?

Are there better direct ways of demonstrating $e^{\theta i}$ (not Taylor series!)?

There is some existing discussion at The definition of natural log and e

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    $\begingroup$ This content is not standard, but it is a story I really like telling as well. I think the algebra problem of switching between these representations is the least of the difficulties. Why not explore all of them thoroughly? $\endgroup$ – Steven Gubkin Mar 25 '16 at 7:37
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    $\begingroup$ @StevenGubkin I would love a single story/representation that is both motivated and communicates the key links. But you are right that there is nothing wrong with working with multiple representation which is what I'm planning to do next to create the link between $e^x$ and $e^{i \theta}$. It would just be nice not to have to take extra steps - every bit of algebra makes the links less intuitive. $\endgroup$ – Richard Mar 25 '16 at 9:59
  • $\begingroup$ @MichaelE2 I was planning on using $\Delta t = x/n$. I'm not sure I understand $\Delta t = \Delta s / i$ intuitively - though algebraically it leads to a beautiful result. I might give it a go. My personal preference is to minimise purely symbolic algebraic transformations as they often reduce the intuitive power of an explanation. How would you explain the meaning of $\Delta s$ under that transformation ... is that a displacement vector? If you want to write it up a bit as an answer, I'll definitely give it a vote! $\endgroup$ – Richard Mar 26 '16 at 0:37
  • $\begingroup$ TBH, I'm not understanding the intuition behind the given formula. If we're looking at a physics simulation, to advance time through a duration $\Delta t$, you might advance the simulation through one step of that size. To get better results, you take two steps of size $\frac{\Delta t}{2}$. Or maybe 100 steps of size $\frac{\Delta t}{100}$ each. And so forth. And this leads directly to something like the "standard" formula; the only intuition I have for the $1/\Delta t$ exponent is a change of variable from the standard one. Maybe it would make more sense if I was there. $\endgroup$ – user797 Dec 5 '17 at 7:50
  • $\begingroup$ Physics simulations are often open ended and it is the whole trace that is interesting not just the end point. $\Delta t$ is not time of interest, it is the simulation step which is decreased to give more accurate traces. This was a while ago, so i need to try to remember the steps in the derivation they came up with. $\endgroup$ – Richard Dec 5 '17 at 23:02
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Probably the most important advantage that the definition

$$ \exp(x) = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n $$

has over the other forms you consider is that the only prior notion of exponentiation required is the case of positive integer exponents.

The usual track for rigorous development from foundations is to define $\exp$ and $\log$ before general exponentiation, which is then defined by the identity

$$ x^y = \exp(y \log(x)). $$

To use the other forms, you have to already have general exponentiation as a given.

However, this is all moot. It's about doing exercises in how to give formal definitions and working their properties out from scratch.

But I really get the impression that you are in a setting where you really should be approaching this as an issue of calculating limits — e.g. proving the theorems that all of these related limits are given by exponential functions.


Incidentally, for the purposes of doing calculations with complex numbers, you don't want the limit above anyways; a more convenient theorem to use is

$$ \lim_{y \to \infty} \left( 1 + \frac{x}{y} \right)^y = \exp(x)$$

The point is that the limit variable need not be restricted to integers. (I assume you take the principal value of the exponentiation)

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  • $\begingroup$ The setting was high school AP, and the purpose was gaining intuitions about exponentials. As you suggest, I would stick with the standard derivations if doing rigorous proofs. $\endgroup$ – Richard Dec 5 '17 at 23:07

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