The Day the Earth Stood Still probably has the most stereotypically "rational" man of all-time in its main character Klaatu. Let's go through the checklist:
1) Rudimentary social skills, replete with the inability to complete a coherent sentence. (Check)
2) Exceptionally proficient at mathematics. (Check)
3) A strange lack of any kind of emotion even when displaying some emotion would be highly beneficial to what he is attempting to accomplish. (Check)
4) Attributing poor collective decision-making to the individual stupidity of all persons involved. This is a common error, and it fails to take into account basic principles of the real world, like game theory. (Check)
5) A poor grasp of the market system. (Check--He traded a handful of perfect diamonds for two dollars!)
I don't know why so-called rational people in fiction are so often boring, lifeless, and stupid. The reality is that you don't have to fit any of these stereotypes to be rational. Indeed, all of them except #2 would probably be harmful to your success in the real world.
I will add the caveat that technically he is not a man but instead an alien who looks exactly like a man. And don't get me started on the odds of this happening through convergent evolution.