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A ring R is said to be Archimedean if the ring is ordered, has a metric $\text{[math]}$ and for all $\text{[math]}$ in R, x non-zero, there exists $\text{[math]}$ in the natural numbers such that $\text{[math]}$ . Here concatentation with $\text{[math]}$ denotes adding $\text{[math]}$ times. Informally, a ring is Archimedean if it has no infinitely small or infinitely large elements. Examples of Archimedean rings include the real numbers and the rational numbers. Examples of non-Archimedean are less simple.

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