Stop using e for compound interest

In a typical math class, $e$ is introduced like this:

• Imagine a bank account with 100% yearly interest. This means that anything you deposit will be doubled a year later.
• The bank decides to compound interest twice yearly, instead of once per year. Then, every 6 months, your deposit will increase by 50%, letting you leave with $2.25x$ the amount you put in.
• The bank now decides to compound interest daily, meaning that your deposit will increase by $\frac{100}{365}\mathrm{\%}$ 365 times. After a year, you can leave with approximately $2.714x$.
• Finally, the bank compounds interest continuously, letting you leave with exactly $e$ times the amount you put in after a year.

This makes sense mathematically, and in class, $e=\underset{n\to \mathrm{\infty }}{lim}(1+\frac{1}{n}{)}^n$.

But here's the problem: why is compound interest divided linearly even though the growth is exponential? If a bank account has 100% yearly interest that compounds twice yearly, it would make more sense to grow by $\sqrt{2}-1$ every 6 months instead of 50%. This would keep the total yearly return consistent.

Because of this, banks have to publish the interest rate, compound interval, and annual percentage yield separately.

$e$ doesn't need to be introduced this way. It has many other nice properties:

• $\exp z$ is the only function that is its own derivative (a fixed point for the derivative operator) with the condition of $\exp 0=1$.
• $e^{ix}=\cos x+i\sin x$, so $e^{i\pi }=-1$, and $e^{i\tau }=1$.
• In general, $e^x$ is periodic with a period of $i\tau$, showing a delightful connection between $\exp$ and trigonometric functions.

None of these topics (calculus, complex analysis, and trigonometry) require prior knowledge about $e$ specifically, so why bring it up using compound interest?