Is There a NON-Archimedean Ordered Field? | COUNTEREXAMPLES in Analysis | E3

There are Archimedean ordered fields such as rational and real numbers. But does every ordered field have to be Archimedean? We show that the field of rational functions with real coefficients is an example of a non-Archimedean ordered field.

The field of real numbers can be ordered and the field of complex numbers cannot be ordered. But how many distinct orders can an ordered field have? We show that real numbers and natural numbers have unique orders. We also construct a quadratic field that has two distinct orders. So, some fields have no orders, some fields have unique orders, while some fields have several distinct orders.

The animations for this video are made in Manim Community Edition, which is a fork of the original Manim package that Grant Sanderson uses to make his videos for the 3blue1brown YouTube channel.

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[geoffrygifari]

Hmmm what makes archimedean fields special? what would happen if a field is non-archimedean? I imagine in an archimedean field, b (and therefore n, the coefficient of a) can be as large as we like, going infinitely far to the positive direction. but if a field is non-archimedean there can be a largest element of the field if we choose b>a. Is this correct?

[VideoFusco]

The field shown in the video is an example of a Hyperreal field.

[Fine_Mouche]

yes, ok, but i want a animation to see what look a Non-Archimedean geometry World :c