A Gentleman is Attentive.
He knows that he thereby declares his valuation of the moment.
While a Gentleman is Observant (see Rule Eight) to benefit his reading of life, he is Attentive that he and others might know he did the assigned reading. Attentiveness signals the observed and the self that a certain grant of importance has been offered to each event or persons. It helps you honestly assess whether or not what you claim to be important truly is. A Gentleman ought to express, through his attentiveness, the valuation of a person or moment that it deserves. Not every moment or person deserves undivided attention but all persons, regardless of deservedness, expect your attention to be undivided. Rarely is a Gentleman's own assessment of some other's importance sufficient datum for not regarding said "some other". He must know that the social price he pays will be according to the other's valuation, not his own. For instance, you sit in a music club and the stage act is playing while your companion responds to a question you asked. Your question may be offhand and you may really enjoy the music. The stage act ought to get less attention than your companion because your companion believes it deserves less. When your companion's conversation draws to a close your highest level of attention may shift to the stage. "Doing unto others as you would be done by" means, in this case, if you would wish to be listened to when you speak, listen. Do you want people merely to look your direction and focus on a spot behind your head by ten or so feet? No? How about responding to your comments with something incoherent? I didn't think so.
This is applicable to remembering people when you are introduced. Should you have trouble with names, you show your recognition of their and their name's importance by apologetically asking for it again. This shows that you find the fault in yourself and grant them the level of attention that seeks the information again.