e (mathematical constant, $e$)

Euler's number is the value $\exp (1)={\sum }_{n=0}^{\infty }1/n!$ of the exponential function, approximately $2.71828$. It is the base of the natural logarithm. Its factorial-series tail gives an elementary proof of the irrationality of e. The compounding limit for e shows how the same constant arises from repeated small multiplications.

Start at one and apply $n$ updates, each multiplying the current value by $1+1/n$. The final value is $(1+1/n{)}^n$: one update gives $2$, two give $2.25$, ten give approximately $2.59374$, and one hundred give approximately $2.70481$. Increasing the number of steps, while reducing each increment proportionally, approaches Euler's number rather than producing unbounded accumulation.

The natural logarithm makes the limit transparent. For $u>0$, its integral definition gives $u/(1+u)\le \ln (1+u)\le u$, by bounding $1/t$ on $[1,1+u]$. With $u=1/n$,The squeeze theorem gives a logarithmic limit of one; continuity of the exponential function gives $(1+1/n{)}^n\to e$. This links a discrete multiplicative rule with continuous exponential growth, without assuming that a finite number of steps equals the limiting value.

If Euler's number were $a/b$, choose $n\ge \max (b,2)$ and multiply its factorial-series tail by $n!$. This produces an integer, but the tail is positive and smaller than ${\sum }_{j\ge 1}(n+1{)}^{-j}=1/n<1$. The contradiction proves that the number is an irrational number. The argument rules out a hypothetical rational remainder by placing it in an impossible integer interval.