Hyperreal in a sentence

Shinkai and his talented team of animators at CoMix Wave Films are unmatched at making a hyperreal Japan seem so inviting.

I like the term "hyperslop" to refer to AI content that systematically confuses reality with things posed as reality (AI content that merely tries to achieve realism is also hyperreal, but interesting mostly for its failure aesthetics).

It wasn't actually possible to watch the film's hyperreal visuals as Lee had intended in the U.S. (in 4K 3D at a high frame rate), so there was all the less incentive for audiences to check it out on the big screen.

A hyperreal r is limited (or finite) if and only if its absolute value is dominated by (less than) a standard integer.

Also every hyperreal that is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal.

An intuitive approach to the ultrapower construction The following is an intuitive way of understanding the hyperreal numbers.

As an example of the transfer principle, the statement that for any nonzero number x, 2x ≠ x, is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers.

For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number.

However, the quantity dx 2 is infinitesimally small compared to dx; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.

Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.

In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic.

It turns out that any finite (that is, such that for some ordinary real ) hyperreal will be of the form where is an ordinary (called standard) real and is an infinitesimal.

Non-standard analysis A hyperreal x and its standard part st(x) are equal up to an infinitesimal difference.

See the article on hyperreal numbers for a discussion of some of the relevant ideas.

The Matrix seemed fresh and cool and visually breathtaking; making wonderful, intelligent use of CGI to augment the on-screen action, striking a perfect balance of the real and the hyperreal.

These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.

The transfer principle main The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra.

The ultrapower construction We are going to construct a hyperreal field via sequences of reals. citation In fact we can add and multiply sequences componentwise; for example: : and analogously for multiplication.

This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.