Irrational number　e

Consider the binominal expansion of (1 + )ⁿ where n is a positive interger.

When n tends to infinity,

tend to zero. 　Also, 　n! =

The coefficient of the general term(the (r+1)th term) is :

We take

When　x = 1,　we have

　　or　

　　　　where

Discovery of Natural Logarithms　自然對數的發現

Consider a function

　　If we change the base of the logarithm from a to e, we have = 1.

　　The logarithm with base e is called Natural Logarithm.

　　Therefore, when 　

　　Hence, 　 + C

　　　　　　or 　　where C' is a constant