Archimedean ordered field

Definition

An ordered field $F$ is Archimedean when, for all $x$ and $y$ in $F$ with $y > 0$, there is a natural number $n$ with $n y > x$. Here $n y$ means $y + y + \dots + y$, with $n$ terms.

Remarks

With $y = 1$, the definition gives the Archimedean property: for each $x$, some natural number exceeds $x$. Conversely, this property implies the definition. An equivalent condition: for each $\varepsilon > 0$ in $F$, there is a natural number $n \ge 1$ with $1/n < \varepsilon$. The field of rational functions $\mathbb{R}(t)$, with the order in which $t$ is larger than every real number, is not Archimedean.

Explanations

L1 human

You can reach any distance with enough small steps of a fixed size. In an Archimedean field there is no number so large that no multiple of a given positive number can pass it.