[parsimo2010]

> 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.

[ameliaquining]

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.

[parsimo2010]

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.

[dhosek]

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.