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An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number. In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels; i.e., in the coarsest level, there are no infinitesimals nor unlimited numbers.
The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson, and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy, and others.
Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number ...
Every real number x is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f at a standard real number x in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets
In the 17th century, with the introduction of the infinity symbol [1] and the infinitesimal calculus, ... can be added to the topological space of the real numbers, ...
These include infinite and infinitesimal numbers which possess certain properties of the real numbers. Surreal numbers : A number system that includes the hyperreal numbers as well as the ordinals. Fuzzy numbers : A generalization of the real numbers, in which each element is a connected set of possible values with weights.
A real-valued function f on the interval [a, b] is continuous if and only if for every hyperreal x in the interval *[a, b], we have: *f(x) ≅ *f(st(x)). Similarly, Theorem. A real-valued function f is differentiable at the real value x if and only if for every infinitesimal hyperreal number h, the value
The "infinitesimal microscope" is used to view an infinitesimal neighborhood of a standard real. Nonstandard analysis deals primarily with the pair R ⊆ ∗ R {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} } , where the hyperreals ∗ R {\displaystyle {}^{*}\mathbb {R} } are an ordered field extension of the reals R {\displaystyle ...