1. d d t e t = e t . {\displaystyle {\frac {d}{dt}}e^{t}=e^{t}.}
2. d d t log e t = 1 t . {\displaystyle {\frac {d}{dt}}\log _{e}t={\frac {1}{t}}.}
3. e = lim n → ∞ ( 1 + 1 n ) n {\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}
4. e = ∑ n = 0 ∞ 1 n ! = 1 0 ! + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + ⋯ {\displaystyle e=\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots } 其中 n! 是 n 的階乘。
5. ∫ 1 e 1 t d t = 1 {\displaystyle \int _{1}^{e}{\frac {1}{t}}\,dt={1}} .
