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Non-Archimedean analysis studies fields equipped with an Non-Archimedean absolute value, satisfying $|x+y|\le \max (|x|,|y|)$. The main examples are the P-adic numbers and fields of Laurent series field.

For example, in the 2-adic metric, $1$ and $1+2^{20}$ are extremely close because their difference is divisible by a high power of two. The inequality makes every triangle isosceles with the two larger sides equal. Over a complete non-Archimedean field, a series converges exactly when its terms tend to zero, a stronger conclusion than in ordinary real analysis. The subject makes congruence information accessible to methods involving limits, derivatives, and power series.