Is Dividing an Irrational Number by a Rational Number Always Irrational?

[srfriggen]

Homework Statement

"Prove that when an irrational number is divided by a nonzero rational number, the resulting number is irrational"

The Attempt at a Solution

By contradiction: Prove that when an irrational number is divided by a nonzero rational number, the resulting number is rational.

a is an irrational number
b is a rational number
c is a rational number

b=d/e, c=f/g, where d,e,f,g are in Z.

b divides a: equivalent to a=bx, where x is in Z.

a=bx=dx/e, so ae=dx

can't seem to get any further than this... also, more importantly, having a hard time figuring out just what result I want from all of this.

 

[srfriggen]

I think I may have figured this out...

I am going to restate the problem, using different variables, to hopefully make my attempt at the solution a bit clearer.

Prove that when an irrational number is divided by a nonzero rational number, the resulting number is irrational.

Attempt:

Assume, to the contrary, that there exists an irrational number x and a nonzero rational number y, such that ylx is rational.

1. y=a/b, where a,b are in Z.
2. ylx: equivalent to x=yk, for some k in Z.

3. x=yk=ak/b

Conclusion: Since ak and b are integers, it follows that x is rational. But this contradicts our original assumption.