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applied math has 1 distinct synonym spread across 1 of its 1 sense. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
the branches of mathematics that are involved in the study of the physical or biological or sociological world
Synonyms, antonyms and the broader and narrower relations for applied math come from Princeton University’s WordNet. Full sources and licences.