e (Euler's Number) - Numberphile

What people said

Recommends Guy Fsone · 2017-10 · Mathematics Stack Exchange · on “How does one show that $\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x$ converges?”

In fact it is possible to predict the existence of e but in the way you thing see this video: [this video] Good watching

Recommends CamilB · 2018-03 · Mathematics Stack Exchange · on “How can we come up with the definition of natural logarithm?”

This excellent Numberphile video [this video] discusses the need for Euler's number, which is the base of the natural logarithm (i.e. $\ln x = \log_e x$). It should provide some support in defining $\ln$.

Recommends vitamin d · 2021-02 · Mathematics Stack Exchange · on “Intuition for Strange Exponential Limit”

If you ask a mathematician what the number $e$ is and what makes it so special, he will either say it's properties in analysis, such as $$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x,\quad \int e^x\, \mathrm{d}x = e^x+c $$ or it's beatiful combination with $i,\pi,1$ and $0$ $$e^{i\pi}+1=0,$$ the famous Euler's identity, which is usually proven by the also beatiful power series for $e$ that Euler discovered: $$\sum_{n=0}^{\infty}\frac{1}{n!}.$$ None of these formulas is the one you stated. But still, this formula has it's own history. It is the definition of $e$ , how it all started, how Euler discovered further properties. Below are two interesting YouTube videos that tell you everything about t

Cites Jaood · 2017-01 · Mathematics Stack Exchange · on “How is calculus written in terms of $e$?”

According to this video [this video] , about 8 minutes in, when doing calculus, write "in terms of $e$" to make things easier. What, exactly, is meant by this? What are some examples of things done "in terms of $e$" to make the maths easier, and what were the alternatives to not doing it in terms of $e$?

Cites albertoeda · 2017-10 · Mathematics Stack Exchange · on “How does calculus without Euler's number (e) look?”

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

Cites Guy Fsone · 2017-10 · Mathematics Stack Exchange · on “How does one show that $\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x$ converges?”

Set $h =\frac1x\to 0~~as ~~x\to\infty$ Then, $$\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x =\lim_{x\rightarrow\infty}\exp\left(\frac{\ln\left(1+\frac{1}{x}\right)}{\frac1x}\right) =\lim_{h\rightarrow 0}\exp\left(\frac{\ln\left(1+h\right)}{h}\right) = e$$ Given that $$\lim_{h\rightarrow 0}\frac{\ln\left(1+h\right)}{h} = 1$$ Edit: For OP doubt. I think he to know how can want guess the existence of $e?$ without appealling $\ln$ and $\exp$. In fact Euler Predicted the existence of $e$ before the creation of the functions $\ln$ and $\exp$: it is explained here: [this video]