Robinson's approach, called non-standard analysis, uses technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus.

A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

See the article on hyperreal numbers for a discussion of some of the relevant ideas.

As noted in the article on hyperreal numbers, these formulations were widely criticized by George Berkeley and others.

For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.

* hyperreal numbers

* Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.

The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis.

The idea of the hyperreal system is to extend the real numbers R to form a system * R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra.

) If formulated in von Neumann – Bernays – Gödel set theory, the surreal numbers are the largest possible ordered field ; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals ; it has also been shown that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

For example, the transcendental function sin has a natural counterpart * sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers has a natural counterpart, which contains both finite and infinite integers.

In the hyperreal numbers and the surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible.

Some mathematical systems such as surreal numbers and hyperreal numbers generate elaborate systems of infinitesimals with amazing properties.

For example, since the hyperreal numbers form a non-Archimedean ordered field and the reals form an Archimedean ordered field, the property of being Archimedean (" every positive real is larger than 1 / n for some positive integer n ") seems at first sight not to satisfy the transfer principle.

In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system.

Its most common use is in Abraham Robinson's non-standard analysis of the hyperreal numbers, where the transfer principle states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.

By a typical application of the transfer principle, every hyperreal x satisfies the inequality

The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted * R called the hyperreal numbers.

Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f < sup >*</ sup > ( see discussion of Transfer principle in main article at non-standard analysis ).

Later examples of the hyperreal approach pioneered by Galton and Simpson in some of their Hancock scripts include The Royle Family, Early Doors, Gavin & Stacey and The Office, as well as many British dramedies.

Thus, the derivative of f ( x ) becomes for an infinitesimal, where st (·) denotes the standard part function, which associates to every finite hyperreal the unique real infinitely close to it.

Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties.

Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers.

The hyperreal numbers, an ultrapower of the real numbers, are a special case of this.

A real function f is nowhere continuous if its natural hyperreal extension has the property that every x is infinitely close to a y such that the difference f ( x )- f ( y ) is appreciable ( i. e., not infinitesimal ).

* If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal ( or appreciable ) hyperreal.

For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

* The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because is the root of.

It is used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences ( Chinese remainder theorem ) or multiplicative inverses of a finite field.

A set of numbers is said to satisfy Benford's law if the leading digit d ( d ∈

Management wishing to show earnings at a certain level or following a certain pattern seek loopholes in financial reporting standards that allow them to adjust the numbers as far as is practicable to achieve their desired aim or to satisfy projections by financial analysts.

The Fermat numbers satisfy the following recurrence relations:

Note that this usage of the term linear is not the same as the above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity.

The motivation for this definition is the fact that all prime numbers n satisfy the above equation, as explained in the Legendre symbol article.

Analogous to numbers ( elements of a field ), matrices satisfy the following general properties.

Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1 / n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε.

A set X of natural numbers is defined by formula φ in the language of Peano arithmetic if the elements of X are exactly the numbers that satisfy φ.

The Catalan numbers satisfy the recurrence relation

The motivation for this definition is the fact that all prime numbers p satisfy the above equation which can be deduced from Fermat's little theorem.

The cube roots of a number x are the numbers y which satisfy the equation

This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers.

In other words, those useful items that are of insufficient quantity to satisfy demand have a price, and those that exist in numbers superfluous to demand ( or that satisfy no wants ) are free.

However, prior to the introduction of the SU-100 it was the only Soviet armored vehicle capable of tackling the German heavy tanks with any kind of reliability, and its ability to satisfy multiple roles meant it was produced in far greater numbers than the SU-100.

In the case of integrable modules, the complex numbers associated with a weight vector satisfy, where ν is an element of the weight lattice, and are complex numbers such that