Stop using e for compound interest(blog.danielh.cc) |
Stop using e for compound interest(blog.danielh.cc) |
This seems like kind of a silly question. The reason is that e is usually introduced before you learn about calculus. You might learn about complex numbers before e, but not in any way that would make "e^i*pi = -1" understandable or even interesting. And you'll certainly learn about e before learning what it means for a function to be periodic in the complex plane.
The reason to start with compound interest isn't that these other topics require prior knowledge of e, it's that they require prior knowledge of those topics. The compound interest explanation doesn't require prior knowledge of anything, because you can build it from the ground up in explaining the concept of compound interest and exponential growth. It's also an explanation with obvious practical relevance. There's still a bit of a leap of faith in assuming anyone will find it interesting that that limit converges to this one particular number, but it's a smaller leap than is required to get people to care about anything involving complex numbers.
Because calculus (with real numbers) is usually covered before complex numbers are introduced in a standard course of study, and the default now is to introduce e and trig functions early so we can talk about their derivatives as soon as we cover the concept of a derivative (this is called "early transcendentals"). So students have covered limits and summation fairly early in their post-graduate education, and complex numbers come later (or not at all, depending on their major). So it is easiest to introduce e as the concept of a limit. Sums are much easier to calculate with uniform interval widths. In real life we also do this- we update bank accounts with updates occurring regularly in time (like a batch job running at midnight every night, or at the end of every week, etc.). So the concept of dividing this update inteval into ever-smaller pieces but keeping the intervals uniformly spaced, is directly applicable to how this is applied in this real-world application.
The fact that compounded interest is the most common illustration is because it is one of the few things that almost every college student will come across in real life, so it is a good example. Even if many students don't care about the value of e, it is a great practical lesson that if you are comparing interest rates you need to use a common time base or you might be misled, which is why we require banks to tell you the APY.
Hardly anyone except mathematicians, physicists, and electrical engineers will care about complex exponentials. They are beautiful but not intuitive to many young students. As a side note, there is a style of education that introduces transcendental functions after the fundamental calculus has been covered (and therefore might be appropriate for defining e as the number satisfying d/dx e^x = e^x). This is not standard in the USA anymore, but the "late transcendentals" is a pedagogical approach that is used in some parts of the world.
In fact, it is practically assumed that elementary algebra students have not worked with complex numbers to the extent necessary to understand complex exponentials, due to the fact that complex exponentials are not algebraic.
For example people don't develop intuition into what an exponential with a complex power should be until after have been introduced to it through power series. But understanding why those power series mean what they mean requires understanding the power series for e^x. Which means that they need to already understand e. Therefore you need to introduce e before you introduce exponentials with complex powers - you can't do it the other way.
That said, you will occasionally encounter mathematicians who advocate for teaching about e by first demonstrating that 1/x has an integral, calling its integral ln(x), then demonstrating that the inverse function to ln(x) is an exponential function and calling it e^x. The argument for this order of presentation is that we can present every step of this with mathematical rigor. The argument against this order of presentation is that students find it very confusing. And very few students will ever notice the logical gaps in the usual presentation.
Students are well aware that the real world analogies presented in math books are overly simplified and imperfect long before they are introduced to Euler's number. Adults on the other hand, long out of secondary school, seem to have forgotten.
Of course, the arithmetic laws of exponential growth of money are exactly the same as the arithmetic laws of exponential growth of anything else. And there is no reason that, just because a topic is historically discovered in some fashion, that must be the way it is taught, or even is a good way to teach it.
But it's not some made up conceit by teachers to connect e and continuously compounded interest. That is historically how e came to be investigated.
https://www.3blue1brown.com/lessons/eulers-number
As suggested by the OP, it approaches the problem from the angle of showing that e^x is the only function that is its own derivative.
There is also a follow up explainer giving intuition for e^ix as being about modeling rotations.
https://www.3blue1brown.com/lessons/eulers-formula-dynamical...
Then later we got introduction to e in terms of derivatives and complex numbers. However, compound interest was never used for exploration, and I only got introduced to the it’s connection to e and as an explanation for what e is late in my thirties.
No, the way a subject is trough in school is not the reason banks have to publish all those numbers.
That a look at all the different ways accountant calculate compound interest, and you will see the reason. Anyway, when was the last time banks united to make some rule to make it easier for laypeople to understand what they do?
Regulation is the other reason. APR is required to be rate-per-period * period-per-year while also accounting for fees. But APY is rate-per-period compounded over a year. These have more to do with the grifts and bubbles that gave birth to the regulations. Again, it made sense at the time and now we’re kind of stuck with it. Not the best standard but better than no standard.
- It's elementary to the point that you can introduce it whenever you want.
- It automatically gives a sense of scale: larger than 2, but not by a lot.
- At least to me, it confers some sense of importance. You can get the sense that this number e has some deep connection to infinity and infinitesimal change and deserves further study even if you haven't seen calculus before.
- It directly suggests a way of calculating e, which "the base of the exponential function with derivative equal to itself" doesn't suggest as cleanly.
I don't know of any calculus course that relies on this definition for much: that's not its purpose. The goal is just to give students a fairly natural introduction to the constant before you show that e^x and ln x have their own unique properties that will be more useful for further manipulation.
- "The Classic": There exists a unique function equal to its own derivative up to a constant.
- "I can't bothered with this": Have a series. It's obviously absolutely convergent. kthxbye.
- "My name is Hardy, G.H. Hardy.": A unique function satisfies exp(x+y) = exp(x)exp(y).
This has nothing to do with e and is satified by 2^x or any a^x, so this wouldn't work for introducing e in particular.
- "The Classic": There exists a unique function equal to its own derivative up to a constant.
Same for this, but if you fix the constant to be 1, then e^x is the only one that works.
I will give the series works too.
You need to impose f'(0) = 1. (If you want to be really technical, also at least some regularity condition, I'll be honest, I don't remember what's the minimal one, let's say continuity)
> Same for this
I did say up to a constant.
The clear usefulness appeared in the case of radioactive decay. We first learned about half-life, and clearly you can write the decay function as N(t) = N0 * 0.5^(at) with 1/a being the half-life. But we then learned that the decay values in the data book are often not the half-life but this special "k" that you put in a decay equation that has e as the: N(t) = N0 * e^(-kx). My reaction was: why did they pick e here? Why not just use 0.5 as base and the book would list the half-life? But then we got to the kicker: we learned the formula: Activity = -k*N, meaning that the activity (decays/time-unit) is proportional to the amount of material with k being the proportionality constant. We hadn't learned of differential equations yet and I was confused about what k was, having first seen it in the decay equation and now in the activity equation which were seemingly very different... how on earth could the same number work in both capacities. And I read a bit ahead in the math book and it all made sense. And then I understood it was because of e as the base that the constant k got this property - which it would only get in that particular base. So this showed to me how e was special, allowing the same constant to serve in these two capacities.
I don't entirely know how to square those two things either, but the evidence is pretty strong.
Meanwhile there are less good jobs that can get by without the advance math that needs better teaching all the time.
Math is full of extremely useful concepts that aren’t otherwise obvious. To me, it’s less about “I’m going to need to use this equation” and more about “This is a pattern I will encounter throughout the world.”
But it's still the adopted standard in the majority of the US. https://en.wikipedia.org/wiki/Common_Core_implementation_by_...
ln(x) (+ a constant) was defined as the integral of 1/x.
Then we proved than ln(x) was a logarithm to some base.
Then e was defined as the base of that function.
Nobody ever said anything about compoun interest.
I wish I knew. But all my "answers" are cynical statements about how all these are parts of a (social technological) scheme to enable exploitation such that the exploited don't understand what's happening to them.
or else euler's famous exponential numerical value is somehow directly correlated with gravitational constant (I say this due to my personal understanding of the nature of time and reality; which in academic terms translates into "but I'm an unaccredited crank")
But yeah, there seems not to be a lot of assumption of familiarity with complex numbers beyond the basics of their existence and maybe some simple arithmetic on numbers in the form a + bi which other than i² = −1 is just following the usual rules for polynomial arithmetic. I was surprised at how much basic content on complex numbers was included in the first chapter of my graduate text on complex analysis.
