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?
In an Archimedean field, the subfield of rational numbers is unbounded, but in a non-Archimedean field it is bounded. For example, in the field of rational functions, every rational number is less than the linear polynomial x.
Great question! The field of p-adic numbers cannot be ordered for any prime p. The underlying reason for this is similar to that for the complex numbers: there are square roots of some negative numbers in these fields. This video th-cam.com/video/RBAkofUKGlU/w-d-xo.html explains why the field of complex numbers cannot be ordered. I hope this helps.
Not quite. Hyperreal numbers come from nonstandard analysis. They are an extension of real numbers which includes infinitely small and infinitely large numbers. I was fascinated by nonstandard analysis some 10 years ago, and read several books on the subject.
@@mathflipped if you identify constant functions with real numbers, you get that there are elements of this field (for example x) that are greater than any positive real number, such as infinite hyperreals, and other elements of this field (for example 1/x) that are positive but less than any positive real number, such as infinitesimal hyperreals. It seems to me that identification can be made. I am also very interested in non-standard analysis: I have been teaching analysis with this formulation to my pupils for several years now, rather than with the limit-based formulation.
@@VideoFusco From this point of view, yes, there are elements that are smaller or larger than any positive real number. But would the transfer principle hold? I've never seen rational functions used as a model of hyperreals in place of the standard ultrapower construction.
Very cool!
Thanks!
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?
In an Archimedean field, the subfield of rational numbers is unbounded, but in a non-Archimedean field it is bounded. For example, in the field of rational functions, every rational number is less than the linear polynomial x.
@@mathflipped ah "bounded"! yes that's what i meant
Yes, you got the gist of the idea!
Dear sir, the p-adic field Q_p is ordered field or not?@Mathematics Flipped
Great question! The field of p-adic numbers cannot be ordered for any prime p. The underlying reason for this is similar to that for the complex numbers: there are square roots of some negative numbers in these fields. This video th-cam.com/video/RBAkofUKGlU/w-d-xo.html explains why the field of complex numbers cannot be ordered. I hope this helps.
@@mathflipped Thus basically some of the non-Archimedean fields are ordered fields and some are not!
@@sayantanmaity2494 Yes
The field shown in the video is an example of a Hyperreal field.
Not quite. Hyperreal numbers come from nonstandard analysis. They are an extension of real numbers which includes infinitely small and infinitely large numbers. I was fascinated by nonstandard analysis some 10 years ago, and read several books on the subject.
@@mathflipped if you identify constant functions with real numbers, you get that there are elements of this field (for example x) that are greater than any positive real number, such as infinite hyperreals, and other elements of this field (for example 1/x) that are positive but less than any positive real number, such as infinitesimal hyperreals.
It seems to me that identification can be made.
I am also very interested in non-standard analysis: I have been teaching analysis with this formulation to my pupils for several years now, rather than with the limit-based formulation.
@@VideoFusco From this point of view, yes, there are elements that are smaller or larger than any positive real number. But would the transfer principle hold? I've never seen rational functions used as a model of hyperreals in place of the standard ultrapower construction.
yes, ok, but i want a animation to see what look a Non-Archimedean geometry World :c