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Hyperreal number that is equal to its own integer part
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
Hyperinteger
Calculus using a logically rigorous notion of infinitesimal numbers
infinitesimal if and only if it is infinitely close to 0. For example, if n is a hyperinteger, i.e. an element of *N − N, then 1/n is an infinitesimal. A hyperreal
Nonstandard_analysis
Swiss mathematician (1707–1783)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Leonhard_Euler
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
also has sin ( π h ) = 0 {\displaystyle \sin({\pi h})=0} for all hyperintegers h {\displaystyle h} . The transfer principle for ultrapowers is a consequence
Hyperreal_number
Branch of mathematics
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Calculus
Number in {..., –2, –1, 0, 1, 2, ...}
portal Canonical factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued
Integer
French mathematician and lawyer (1601–1665)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Pierre_de_Fermat
Modern application of infinitesimals
st(xn)=L (here the extension principle is used to define xn for every hyperinteger n). This definition has no quantifier alternations. The standard (ε,
Nonstandard_calculus
Continuous real function on a closed interval has a maximum and a minimum
Proof In the setting of non-standard calculus, let N be an infinite hyperinteger. The interval [0, 1] has a natural hyperreal extension. Consider its
Extreme_value_theorem
Generalization of the real numbers
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Surreal_number
Extremely small quantity in calculus; thing so small that there is no way to measure it
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Infinitesimal
French mathematician (1789–1857)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Augustin-Louis_Cauchy
Mathematical treatise by Archimedes
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
The Method of Mechanical Theorems
The_Method_of_Mechanical_Theorems
Mathematical notation used for calculus
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Leibniz's_notation
Mathematical notion of infinitesimal difference
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Differential_(mathematics)
Real numbers adjoined with a nil-squaring element
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Dual_number
Geometrical concept relating area and volume
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Cavalieri's_principle
Ordered field that does not satisfy the Archimedean property
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Non-Archimedean_ordered_field
Concept in model theory
hyperreal is larger than 1 / n {\displaystyle 1/n} for some positive hyperinteger n {\displaystyle n} ". In other words, the hyperreals appear to be Archimedean
Transfer_principle
Index of articles associated with the same name
In mathematics, a nonstandard integer may refer to Hyperinteger, the integer part of a hyperreal number an integer in a non-standard model of arithmetic
Nonstandard_integer
German polymath (1646–1716)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Gottfried_Wilhelm_Leibniz
an ultrafilter. Here a hyperrational is by definition a ratio of two hyperintegers. Consider the ring B {\displaystyle B} of all limited (i.e. finite)
Construction of the real numbers
Construction_of_the_real_numbers
Mathematical symbol used to denote integrals and antiderivatives
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Integral_symbol
Principle that whatever succeeds for the finite also succeeds for the infinite
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Law_of_continuity
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Criticism of nonstandard analysis
Criticism_of_nonstandard_analysis
System of mathematical set theory
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Internal_set_theory
Mathematical model for describing material deformation under stress
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Infinitesimal_strain_theory
1734 book by George Berkeley
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
The_Analyst
Reduction of a ring by one of its ideals
{\displaystyle F} of finite hyperrationals (i.e. ratio of a pair of hyperintegers), see construction of the real numbers. The quotients R [ X ] / (
Quotient_ring
Calculus textbook by Guillaume de l'Hôpital (1696)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Analyse des infiniment petits pour l'intelligence des lignes courbes
Analyse_des_infiniment_petits_pour_l'intelligence_des_lignes_courbes
Mathematical term
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Microcontinuity
Textbook by Augustin-Louis Cauchy (1821)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Cours_d'analyse
1976 mathematics textbook by H. Jerome Keisler
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Elementary Calculus: An Infinitesimal Approach
Elementary_Calculus:_An_Infinitesimal_Approach
Type of set in mathematical logic
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Internal_set
Proof technique in nonstandard analysis
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Overspill
Lie group of complex numbers of unit modulus; topologically a circle
{Z} /N{}^{*}\mathbb {Z} ,} where N {\displaystyle N} is an infinite hyperinteger, that is a number that is greater than every standard natural number
Circle_group
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Constructive nonstandard analysis
Constructive_nonstandard_analysis
Institution (computer science) Non-standard analysis Non-standard calculus Hyperinteger Hyperreal number Transfer principle Overspill Elementary Calculus: An
List of mathematical logic topics
List_of_mathematical_logic_topics
Named set of points in nonstandard analysis
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Monad_(nonstandard_analysis)
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Glossary_of_calculus
Heuristic principle enunciated by Gottfried Wilhelm Leibniz
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Transcendental law of homogeneity
Transcendental_law_of_homogeneity
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Increment_theorem
System of numbers with non-finite quantities
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Levi-Civita_field
Type of internal set in nonstandard analysis
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Hyperfinite_set
Formalization in mathematical topos theory
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Synthetic differential geometry
Synthetic_differential_geometry
Function from the limited hyperreal to the real numbers
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Standard_part_function
American mathematician
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Abraham_Robinson
Mathematical procedure equivalent to differential calculus
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Adequality
HYPERINTEGER
HYPERINTEGER
HYPERINTEGER
HYPERINTEGER
Girl/Female
Latin Russian
Born feet first.
Boy/Male
Hindu
Mountain
Boy/Male
Hindu, Indian, Marathi
Lord of Nectar
Girl/Female
Indian
Goddess Lakshmi
Male
Russian
(Шура) Short form of Russian unisex Sashura, SHURA means "defender of mankind." Compare with strictly feminine Shura.
Boy/Male
Tamil
Lord Vishnu
Girl/Female
Greek
Oath.
Boy/Male
Indian
Happy jewel
Boy/Male
Arabic, Australian, Celtic, Dutch, Gaelic, Hebrew, Irish, Latin, Muslim
Great; Surname
Boy/Male
Hindu, Indian, Traditional
A Devotee of Kali
HYPERINTEGER
HYPERINTEGER
HYPERINTEGER
HYPERINTEGER
HYPERINTEGER