Monday, January 11, 2010

e is irrational

#Problem 8
Prove that e is irrational
Solution scheme and approach
Let e is rational so e=p/q
multiplying equation number (i) by q! we get
q!*e=q!+q!/1!+q!/2!+q!/3!+....+q/q!+ other terms
Now as q!+q!/1!+...+q!/q! is an integer and q!*e is also an integer
So we can say other terms should be an integer. Let I signifies the rest terms whose summation is an integer.
=>I< 1
Hence contradiction.
So, e should be an irrational number. Q.E.D.
[This solution has been proposed by Fourier]

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