2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75642This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant $\hbar$ tends to zero taking imaginary values.English and Russian versions. This article, a result of a very considerable expansion of math.GM/0501038, is to appear in Journal of Mathematical Sciences (Notes of Scientific Seminars of the V.A. Steklov Mathematical Institute), see <http://tinyurl.com/a3ahc>. Dedicated to Anatoly Vershik with admiration and gratitude. Russian version to be published in "Zapiski Nauchnykh Seminarov POMI"General Mathematics00A05, 81Q20, 14P99, 51P05, 46S10, 49L99 (Primary) 06A99, 06F07, 14N10, 81S99 (Secondary)text