# How does calculus without Euler's number (e) look? [closed]

I'm starting to study calculus and I've become very interested in Euler's number ($e$). I understand that the property of being its own derivative makes it the "natural" base to work on for studying rates of change.

However, I was wondering what would happen if we pretended not to know about the existence of $e$. Would trying to find the derivative of something like $a^x$ lead us into finding the definition of $e$ or is it possible to avoid $e$ altogether?

In this video it says that not using $e$ in calculus leads to some pretty crazy math. What does that math look like?

• It couldn't change anything. It's just a number. The number would still exist, somewhere between $2$ and $3$, and we would just rediscover $e$ anyway. It has too many nice properties to be ignored. Oct 11, 2017 at 22:32
• I don't understand. What makes that "interesting"? Note that there are other ways to arrive at $e$ (and the exponential function, which is really what you are talking about - not the number $e$). For example, the anti-derivative (indefinite integral) of $1/x$ is the natural logarithm (in base $e$) - up to an additive constant - and this has little to do with differential equations. Oct 11, 2017 at 22:33
• Better to call it Euler's number. Euler's constant is something else
– A.Γ.
Oct 11, 2017 at 22:59
• The derivative of a function equal to Euler's number is 0, not itself. You mean that the exponential function with Euler's number as the base is its own derivative. (Yes, this is nitpicking and I knew what you meant. But this is a math site. Math is one of the few fields where we can have life without much ambiguity, so let's do so.) Oct 12, 2017 at 2:15