Give me a better way to start. Every other option means introducing something else that because of abstraction won't even make sense until you work with it for a few years.
Stop using e for compound interest
11–20 of 28 posts
Re: Stop using e for compound interest
#12As 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://…
FWIW my school introduced e as a base of natural logarithms with “it’s a special number, don’t worry about it. Good thing: your calculator knows them, too” and for a while we all thought that “natural” somehow related to being calculator-friendly.
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.
Re: Stop using e for compound interest
#13Is that really how the exponential function is introduced? The most common ones as far as I know are:
- "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).
Re: Stop using e for compound interest
#14The goal isn't to teach how compound interest works in practice at banks.
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.
Re: Stop using e for compound interest
#15> 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.
I'm a calculus teacher (at the moment, as I also teach probability and statistics some semesters). Students are aware of the existence of complex numbers from algebra, in the sense that teachers mention that there are guaranteed to be a fixed number of roots of a polynomial, but these roots might be repeated or complex. They hardly do anything with complex numbers outside this, and do not have enough treatment to define e in such a way.
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.
Re: Stop using e for compound interest
#16> Because of this, banks have to publish the interest rate, compound interval, and annual percentage yield separately.
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?
Re: Stop using e for compound interest
#17Is that really how the exponential function is introduced? The most common ones as far as I know are: - "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).
> - "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.
Re: Stop using e for compound interest
#18Is that really how the exponential function is introduced? The most common ones as far as I know are: - "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).
> - "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 work…
> 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.
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.
Re: Stop using e for compound interest
#19A lot of the commentary suggests that this motivation for e in terms of continuously compounded interest is introduced for misguidedly contrived pedagogical reasons. But this is historically the way e was first discovered. e was first discovered precisely through Jacob Bernoulli wanting to understand continuously compounded interest.
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.
Re: Stop using e for compound interest
#20Earlier quoted context omitted.
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.
I'm a calculus teacher (at the moment, as I also teach probability and statistics some semesters). Students are aware of the existence of complex numbers from algebra, in the sense that teachers mention that there are guaranteed to be a fixed number of roots of a polynomial, but these roots might be repeated or complex. They hardly do anything with complex numbers outside this, and do not have enough treatment to def…
I’m aware my son is an outlier, but I’m rather proud of his working out the square root(s) of i without even having algebra yet (he’s in fifth grade). Last year I taught him how to solve simple linear equations (ax + b = c) and expanded that to ax + b = cx + d, but he’s been mostly an autodidact with his advanced math (consulting youtube videos and books from the library).
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.