Generally, when we describe something as rational, we either mean that it is based on reason and logic or that it is a mathematical expression describable as a fraction of two integers. I propose that these two definitions go hand in hand.
Rational numbers are those numbers, which through their presentation as a fraction, reveal all of the mathematical derivation necessary to fully conceptualize them. The fraction symbol itself denotes the division function. The number can be lightly manipulated into a long division problem, which after some amount of computation will result in an exact decimal representation.
Rational arguments also should be presented in such a way that their derivation is traceable to agreed upon logic and reason, which can be applied consistently by anyone to arrive at the same conclusion.
We can also consider their opposites, the irrational. Irrational numbers are those that cannot be fully depicted in fractional or decimal form. They rely either on the precursor mathematical equations that lead to them (such as describing pi as the circumference of a circle divided by its diameter) or by truncating or rounding them (such as describing pi as 3.14 or 157/50).
In the former case, one is stepping back from the irrational to determine a more precise formulation. One might call this rational thinking. In the later, one is accepting the irrational as the outcome and constraining it into a rational formulation for convenience of further calculation. One might call this rationalization.
And so irrational numbers can also be mapped to irrational arguments. Those arguments that begin with an outcome, such as the preference for a certain political candidate, are not traceable to a unique set of logical considerations. They can be rationalized by working backwards to find a path of reasoning to support them, but examining them more precisely will reveal impreciseness.[1] Such positions can be improved through rational thinking by recognizing the correct process for arriving at conclusions and stepping away from the initial outcome to generalize until an accurate depiction of the situation is found. In the case of the political candidate, this point would perhaps be stepping back to the checklist Eliezer Yudkowsky describes in "A Rational Argument". Then the actual details that define the best candidate can be filled into this theoretical construct the same way the circumference and diameter can be filled into the formula for pi.
Considering pi in parallel here, one might assume it to be 3.14 but upon pressing the "pi" button on a calculator reveal it to be 3.14159. So treating pi as 157/50 now seems inaccurate. But revising pi to be 314,159/100,000 only addresses the problem for this one truncation and doesn't reveal any new underlying logic.