Nonstandard Integer

	Nonstandard Integer In mathematics, a nonstandard integer may refer to Hyperinteger, the integer part of a hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, R, are an extension of the real numbers R that contains numbers ... an integer in a non-standard model of arithmetic {{Short pages monitor ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Hyperinteger In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals. Discussion The standard integer part function: :\lfloor x \rfloor is defined for all real ''x'' and equals the greatest integer not exceeding ''x''. By the transfer principle of nonstandard analysis, there exists a natural extension: :^\! \lfloor \,\cdot\, \rfloor defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if x = ^\! \lfloor x \rfloor. Thus the hyperintegers are the image of the integer part function on the hyperreals. Internal sets The set ^\mathbb of all hyperintegers is an internal subset of the hyperreal line ^\mathbb. The set of all finite hyperintegers (i.e. \mathbb itself) is not an internal subset. Elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Hyperreal Number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, R, are an extension of the real numbers R that contains numbers greater than anything of the form :1 + 1 + \cdots + 1 (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in R. For example, the commutative law of addition, , holds for the hyperreals just as it does for the reals; since R is a real closed field, so is R. Since \sin()=0 for all integers ''n'', one also has \sin()=0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]

	Nonstandard Integer In mathematics, a nonstandard integer may refer to Hyperinteger, the integer part of a hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, R, are an extension of the real numbers R that contains numbers ... an integer in a non-standard model of arithmetic {{Short pages monitor ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Hyperinteger In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals. Discussion The standard integer part function: :\lfloor x \rfloor is defined for all real ''x'' and equals the greatest integer not exceeding ''x''. By the transfer principle of nonstandard analysis, there exists a natural extension: :^\! \lfloor \,\cdot\, \rfloor defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if x = ^\! \lfloor x \rfloor. Thus the hyperintegers are the image of the integer part function on the hyperreals. Internal sets The set ^\mathbb of all hyperintegers is an internal subset of the hyperreal line ^\mathbb. The set of all finite hyperintegers (i.e. \mathbb itself) is not an internal subset. Elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Hyperreal Number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, R, are an extension of the real numbers R that contains numbers greater than anything of the form :1 + 1 + \cdots + 1 (for any finite number of terms). Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in R. For example, the commutative law of addition, , holds for the hyperreals just as it does for the reals; since R is a real closed field, so is R. Since \sin()=0 for all integers ''n'', one also has \sin()=0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]