Stop using e for compound interest
blog.danielh.cc
Stop using e for compound interest
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Re: Stop using e for compound interest
#2> why is compound interest divided linearly even though the growth is exponential?
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")
Re: Stop using e for compound interest
#3Everybody knows the way we teach math is broken but everybody also knows the way we teach math must not be touched. Ever. The curriculum attained perfection in the 20th century and anyone who proposes changing it is at least an idiot and possibly a heretic.
I don't entirely know how to square those two things either, but the evidence is pretty strong.
Re: Stop using e for compound interest
#4> None of these topics (calculus, complex analysis, and trigonometry) require prior knowledge about specifically, so why bring it up using compound interest?
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.
Re: Stop using e for compound interest
#5> why is compound interest divided linearly even though the growth is exponential?
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.
Re: Stop using e for compound interest
#6The answer is because of how much (or little) background is needed to understand each concept.
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.
Re: Stop using e for compound interest
#7As with many "we should explain core math things better," there is a 3Blue1Brown explainer that attempts to give intuition for e:
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...
Re: Stop using e for compound interest
#8Everybody knows the way we teach math is broken but everybody also knows the way we teach math must not be touched. Ever . The curriculum attained perfection in the 20th century and anyone who proposes changing it is at least an idiot and possibly a heretic. I don't entirely know how to square those two things either, but the evidence is pretty strong.
I'm not sure if the common core curriculum that came about in 2010 is proving you right or wrong, given the vehement "that's not mah math!" response that it got from the public.
But it's still the adopted standard in the majority of the US. https://en.wikipedia.org/wiki/Common_Core_implementation_by_...
Re: Stop using e for compound interest
#9> why is compound interest divided linearly even though the growth is exponential? 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 cov…
In the U.S., by the time a student starts calculus they will usually have already worked with complex numbers in the context of elementary algebra. It's true, though, that this won't necessarily have included complex exponentiation.
Re: Stop using e for compound interest
#10Everybody knows the way we teach math is broken but everybody also knows the way we teach math must not be touched. Ever . The curriculum attained perfection in the 20th century and anyone who proposes changing it is at least an idiot and possibly a heretic. I don't entirely know how to square those two things either, but the evidence is pretty strong.
Don't forget that most people don't even have enough math background to understand why those who do want better teaching. Most of those people don't really need better math for their life, so it is hard to explain why their kids would be better off knowing something they don't need.
Meanwhile there are less good jobs that can get by without the advance math that needs better teaching all the time.