hyperreal
Speech parts
1. hyperreal - Noun
2. hyperreal - Adjective
Meaning
Of or pertaining to philosophical hyperreality; perceivable as real by consciousness, though potentially unreal.
(mathematics, of a number) Belonging to an extension of the real numbers containing those that cannot be produced by repeatedly adding 1; hence infinite.
hyperreal (plural hyperreals)
A hyperreal number.
Synonyms
Phrases with the word
Examples
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.
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.
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.