In my opinion, this film does represent real depictions of schizophrenic patient. The definition of schizophrenia is characterized by continuous or relapsing episodes of psychosis. We can see that Nash's symptoms develop gradually along the course of this film. In both scene, he's talking to Charles, but other people in the library looked at him weirdly, and only sees that he was talking to himself.
In this scene, he sees patterns in numbers, 'secret codes' he called them, about latitudes and longitudes of areas. It is believed that this is a part of his elaborate hallucinations, where in fact he was never invited to the Pentagon to decipher Soviet's plans.
After being recruited into the Department of Defense, he was tasked to find patterns in newspapers and send them to his supervisor, William Parcher. At the beginning, this setting is believable since United States are at the brink of war, and there may perhaps cases where a civilian is recruited to help the cause in thwarting enemy plans.