On a log scale, multiplication is sliding and stays exact. Addition is almost "take the larger one", and it gets exactly that as the base grows. Turn up the base and watch arithmetic become max-plus.
Position = log. The rule places the number x at position u = log_b x. Multiplying x·y slides by v, so its position is u + v in every base. That part of the slide rule never changes.
Addition becomes max. The sum x + y sits at log_b(bᵘ + bᵛ) = max(u,v) + log_b(1 + b^−|u−v|). The correction is at most log_b 2, which goes to 0 as b grows. In the limit, "plus" becomes "max" and "times" stays "plus": the tropical semiring (max, +).
Polynomials become corners. A polynomial Σ bᵃⁱ·Xⁱ, read on the rule at X = bˣ, has position log_b Σ b^(aᵢ + i x). As b grows it collapses onto max(aᵢ + i x), a piecewise-linear graph. The corners of that graph are the tropical roots, and their positions predict the sizes of the ordinary roots (Newton polygon).
The name. This limit is sometimes called Maslov dequantization: log base b plays the role of Planck's constant going to zero. "Tropical" honors the Brazilian mathematician Imre Simon.