# Can the ratio of two irrationals be rational?

Does there exist a rational number ratio for any two irrational numbers?

hypnagogue
Staff Emeritus
Gold Member
Let x be an irrational number.

x / x = 1.

Also, any nonzero integer multiple of x is irrational.

Proof: If x is irrational, then by definition it cannot be expressed in the form p / q, for all integers p and q (q ~= 0). Now let us consider n*x, where n is a nonzero integer. Let us assume that n*x is rational. If this is the case, then n*x = p/q, for some integers p and q. But re-arranging this equation we get x = p/n*q. Since p/n*q is the ratio of two integers, it is a rational number. But this contradicts our initial assumption that x is irrational. Therefore, n*x must also be irrational.

So for any irrational x, n*x/m*x is a rational ratio, where n and m are integers and m ~= 0.

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HallsofIvy
Homework Helper
Your question is ambiguous. The title "Can the ratio of two irrationals be rational?" seems to ask whether there exist two irrationals whose ratio is rational. That is easy to answer: certainly. Take any irrational x, Let y= 2x. Then then y is also an irrational but the ratio of x to y is 2.

But then you ask "Does there exist a rational number ratio for any two irrational numbers?" which, as well as "do there exist any two rational numbers whose ratio is rational" (the question above), could be interpreted as: "Given any two rational numbers, is there ratio rational?" and the answer to that is, just as obviously, no. [sqrt](2) and [sqrt](3) are irrational and their ratio is not rational.

HallsofIvy,
"Given any two rational numbers, is there ratio rational?" and the answer to that is, just as obviously, no.
I assume a typo here on your part. Yes, I tend to make ambiguous mathematical statements. I had originally questioned my use of the word "any." I see your point in this regard. Mea culpa.

hypnagogue,

Your mathematical imagery was simpler than I thought possible for solving the problem at hand. The answer, I guess, is a standard in the field.

Hypnagogics are one of my favorite pastimes, and "help" me visualize possible physical situations.

hypnagogue
Staff Emeritus
I'm quite intrigued by hypnagogics too, as you might be able to tell. Although for me they are usually spontaneous, ie I don't have any conscious control over them.