Dear Steve, the internet is full of videos on the topic of euler's number, e, so you must think: oh god not another one of those. However I find that there is something that I don't like about most explanation videos: most videos begin by explaining exponential functions in general, then introduce "e" in a way that seems unrelated to exponentials. Finally, the amazing properties of "e" are demonstrated while it remains a mystery why this number has these properties to begin with. As you can tell, I really don't like this approach so I would like present a different explanation. If you like, you are free to use it for a video. First consider trying to find the derivative of f(x) = 10^x We do this from scratch, starting with the definition of a derivative f'(x) = 10^(x+h) - 10^(x) / h = 10^x * (10^h - 1)/h

Note here that the first term 10^x is simply f(x), and that the second term (10^h-1)/h is simply f'(0).

f'(x) = f(x) * f(0)

A pretty cool property that all exponential functions have, no matter what base you choose. However, now we are stuck, because we don't have any method to determine the value of f'(0). WOULDN'T IT BE NICE if f'(0) would just nicely equal to 1? Well, that may not be the case for base 10 exponentials, but we can safely claim that there must be some base value "a" where f'(0)=1. You can support this using a graph like this one

Euler's number can then just be defined as the number that has this property. You could then go into methods to find a good approximation. Summarizing, we have "invented" Euler's number because this was our only hope at ever finding the derivative of an exponential function. No need to convince anyone why it is useful. I hope you like this approach, let me know what you think if you see this :)

Steve, if you see this, I want to close by saying that I love your videos, I watch every single one of them. So thank you!

Regards, Kasper Arfman

Nice approach. That's how I introduced e in high school calculus. We looked at y = 2^x and y = 3^x and zeroed in on a base which must lie between 2 and 3 so that your limit is 1.

Are you familiar with Jacob Bernoulli's method using compound interest to obtain e? He is actually the first person to discover it. He found it by using the compound interest formula for investing $1 for 1 year at 100% per year compounded n times per year. Euler was the one who first who introduced this number as the base for a logarithmic function as he found that by taking the derivative of this this function he produced Bernoulli's expression.

Another fun fact. It was Roger Cotes in 1714 who first wrote out:

ix = ln(cox(x) + i sin(x)) thus beating Euler to the punch line. Euler found the infinite series expansion for for e and thus e^x ( plus a whole whack load of other stuff) and that's probably why he gets his name attached to it.