Is there a smallest positive rational number?

1/infinity?

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Nice idea but unfortunately that is not a rational number, which is defined as the ration of two integers, x/y where y > 0. Since infinity is not an integer, the suggested ratio is not a rational number.

The correct answer is that there is no such number. If any number laid claim to being the smallest positive rational, then half of that number would have a better claim. And then a half of THAT number would be a positive rational that was smaller still. And so on.