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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
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
since sin(πn)=0 for all integers n, one also has sin(πh)=0 for all hyperintegers h. The transfer principle for ultrapowers is a consequence of Łoś's
Hyperreal_number
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
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
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
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
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
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 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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
Lie group of complex numbers of unit modulus; topologically a circle
group is the hyperfinite cyclic group CN=∗ℤ/N∗ℤ, where N is an infinite hyperinteger, that is a number that is greater than every standard natural number
Circle_group
Mathematical term
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Microcontinuity
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
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
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
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
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)
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
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
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
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Increment_theorem
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
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
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
American mathematician
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Abraham_Robinson
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
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
numbers Individual concepts Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set
Glossary_of_calculus
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HYPERINTEGER
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