Thread: irrational number?

Let Prove that x is an irrational number.

Originally Posted by mathman7

Let Prove that x is an irrational number.

Hopefully someone will give you a more elegant soln than mine!

Assume that x is rational so the integers N and M exist such that:

x=N/M

Therefore, x*(1!2!3!.....M!) must be an integer.

But


Now the first M terms of this are integers. Define S to be the integer that is the sum of the first M integers then:




Now provided M > 1


Since this term is between 0 and 1 it is not an integer and we have a contradiction with our statement that N is an integer and hence x must be irrational.