Nadsat is a constructed set of Russian-based slang invented by the linguist, novelist, and composer Anthony Burgess. See Nadsat lexicon for a full list, along with definitions. It is a transliteration of the Russian suffix for 'teen'.
Nadsat is in fact not so much a language as a register. Alex is capable of speaking standard English when he wants to, and besides, what he says is intelligible to an English speaker. Nadsat is really a lexicon of 'extra' words which Alex uses to describe the world as he sees it:
Nadsat words are all concrete or semi-abstract: to discuss philosophy Alex would have to shift into normal English. The fact that a teen language has no abstract words is perhaps Burgess' reflection on the shallowness of the juvenile delinquent's thought process.
At least one translation of Burgess' book into Russian ingeniously had the protagonist talk in a slang heavy with transliterated English words, in places where Burgess put Russian ones.
A comprehensive Nadsat lexicon lists the terms used in the book with their origins.
Alex's use of Nadsat mirrors what happens in real life - children and teenagers creating slang and registers for talking amongst themselves or in specific sub-cultural groups. That Alex uses a different, unfamiliar language to talk to others of his age reinforces the social apartheid between the young and the old. It reflects a completely different attitude to life.
From a publisher's point-of-view, Nadsat also has the added effect of desensitizing concepts that would otherwise have been considered unprintable.
Burgess was highly influenced by the two youth movements of Great Britain at the time, the Mods and Rockers. Instead of using the language that these two groups were using, which Burgess knew would have changed by the time the book was published, he decided to invent his own, making the youth culture of the book ageless.
Another function of Nadsat was also that the reader was forced to learn it to understand the book, thus making the reader implicated and a part of the youth culture. This also had the added bonus of forcing the reader to identify with Alex.
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