Explore the fascinating world of tropical mathematics in this comprehensive lecture, the first in a series. Delve into the power of min-plus (or max-plus) algebra and discover how tropical objects serve as limits of classical ones while exhibiting polyhedral geometry. Learn about the application of tropical mathematics in solving diverse problems across domains such as deep neural networks, semigroups theory, auction theory, and extreme value statistics over the past decade. Gain insights into tropical convex geometry, hyperplane arrangements, classical polytopes, and agile geometry. Examine the Legendre differential transform, tropical polynomials, and regular subdivisions. Investigate practical applications in auctions, including maximizing utility and understanding demand sets. Uncover the potential of tropical mathematics to tackle hard problems in various fields through engaging examples and theoretical proofs.