Mathematicians are really good at calculating things in their heads.

Not usually, but there have been examples in history of mathematicians who were not only great at research but also great mental calculators. E.g. A. Aitken: http://www-history.mcs.st-and.ac.uk/Mathematicians/Aitken.html

To be successful in math, you have to complete your Ph.D. at an extremely young age.

That is not the message of the film. It is simply the case for the mathematician portrayed in the film. He is supposed to be a genius. It is essential to the plot that he is one of the exceptional mathematicians.

People who are good at math are socially awkward.

Also not what the movie says. Usually they are not socially awkward. But this particular mathematician is. And I personally see nothing wrong with it. If someone is socially awkward and isolated but has mathematics as a talent and escape from his suffering, why not? Who would stand in the way of such a person to fulfill his passion and dream just because he is socially awkward. Luckily, in mathematics (still EVEN TODAY) results count in the end.

Number Theory is synonymous with numerology.

Nope. It’s not and it’s not what the movie says. Yeah he stares at numbers etc.. But he also stares at very complicated code. A number theorist who is interested in this kind of question naturally has to write code. It’s applied research in a sense and mathematicians are just as capable of creating algorithms and write code as they are capable of doing pure mathematical research.

And his doodling on the train is just that: doodling… Mathematicians are allowed to doodle and think about philosophy. And that’s all he does there. I really hated it when reviewers got this wrong. He is not doing “research” in this particular scene.

To be good at math, you must be insane.

Well.. most mathematicians are neither particularly clever nor particularly imaginative (just looked who gets hired these days… HAHA), nor creative.. they still do some decent work. They may need 10^(1 + x) co-authors to get anything started (not that collaboration is bad in itself.. but 3 or more co-authors, come one…), but they get something done. They are not insane either. But there are also special individuals. Who are so great because they’re minds are so different! And it’s beautiful!

The number π provides an active area of research for mathematicians.

At least \pi pops up everywhere in research, often unexpectedly. And there are open questions still about it. Normality for example.

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