Bugaev was born in Georgia, Russian Empire into a somewhat unstable family (his father was an army doctor), and at the age of ten young Nikolai was sent to Moscow to find his own means of obtaining an education.

Some of his most influential papers offered proofs of previously unproven assertions of Liouville, but his most original work centered around the development of formal analogies between arithmetic and analytic operations.

He also wrote influential philosophical essays in which he trumpeted the virtues of mathematical analysis and decried the influence of geometry and probability.

Their mathematically, musically, and artistically talented son, Boris Nikolaevich Bugaev (14 October 1880 O.S.-8 January 1934), went on to adopt the pseudonym Andrei Bely, under which name he helped found the Symbolist movement.

In view of his father's prejudices, Boris Bugaev was fascinated by probability and particularly by the notion of entropy, which is mentioned in several of his novels and poems.