Editing Hyperreal number (section)

== Use in analysis ==
Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals, like ''dx'', and as the symbol ∞, used, for example, in limits of integration of [[improper integrals]].

As an example of the transfer principle, the statement that for any nonzero number ''x'', ''2x''&nbsp;≠&nbsp;''x'', is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.

Similarly, the casual use of 1/0&nbsp;=&nbsp;∞ is invalid, since the transfer principle applies to the statement that zero has no multiplicative inverse. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite.

For any finite hyperreal number ''x'', the [[standard part]], st(''x''), is defined as the unique closest real number to ''x''; it necessarily differs from ''x'' only infinitesimally. The standard part function can also be defined for infinite hyperreal numbers as follows: If x is a positive infinite hyperreal number, set st(''x'') to be the [[extended real number]] <math>+\infty</math>, and likewise, if x is a negative infinite hyperreal number, set st(''x'') to be <math>-\infty</math> (the idea is that an infinite hyperreal number should be smaller than the "true" absolute infinity but closer to it than any real number is).

=== Differentiation ===
One of the key uses of the hyperreal number system is to give a precise meaning to the differential operator ''d'' as used by Leibniz to define the derivative and the integral.

For any real-valued function <math>f,</math> the differential <math>df</math> is defined as a map which sends every ordered pair <math>(x,dx)</math> (where <math>x</math> is real and <math>dx</math> is nonzero infinitesimal) to an infinitesimal
: <math>df(x,dx) := \operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right) \ dx.</math>

Note that the very notation "<math>dx</math>" used to denote any infinitesimal is consistent with the above definition of the operator <math>d,</math> for if one interprets <math>x</math> (as is commonly done) to be the function <math>f(x)=x,</math> then for every <math>(x,dx)</math> the differential <math>d(x)</math> will equal the infinitesimal <math>dx</math>.

A real-valued function <math>f</math> is said to be differentiable at a point <math>x</math> if the quotient
: <math>\frac{df(x,dx)}{dx}=\operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right)</math>
is the same for all nonzero infinitesimals <math>dx.</math> If so, this quotient is called the derivative of <math>f</math> at <math>x</math>.

For example, to find the [[derivative]] of the [[function (mathematics)|function]] <math>f(x)=x^2</math>, let <math>dx</math> be a non-zero infinitesimal. Then,
: {|
|-
|<math>\frac{df(x,dx)}{dx}</math>
|<math>=\operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right)</math>
|-
|
|<math>=\operatorname{st}\left(\frac{x^2 + 2x \cdot dx + (dx)^2 -x^2}{dx}\right)</math>
|-
|
|<math>=\operatorname{st}\left(\frac{2x \cdot dx +  (dx)^2}{dx}\right)</math>
|-
|
|<math>=\operatorname{st}\left(\frac{2x \cdot dx}{dx} + \frac{(dx)^2}{dx}\right)</math>
|-
|-
|
|<math>=\operatorname{st}\left(2x + dx\right)</math>
|-
|
|<math>=2x</math>
|}

The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square {{Citation needed|reason=if dx got to zero in limit then dx square go there too and doing this faster, this is rigorous|date=February 2018}} of an infinitesimal quantity. [[Dual number]]s are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the ''dx''<sup>2</sup> term. In the hyperreal system,
''dx''<sup>2</sup>&nbsp;≠&nbsp;0, since ''dx'' is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity ''dx''<sup>2</sup>  is infinitesimally small compared to ''dx''; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.

Using hyperreal numbers for differentiation allows for a more algebraically manipulable approach to derivatives.  In standard differentiation, partial differentials and higher-order differentials are not independently manipulable through algebraic techniques.  However, using the hyperreals, a system can be established for doing so, though resulting in a slightly different notation.<ref>{{cite book |last=Fite |first=Isabelle |title=Operator Theory - Recent Advances, New Perspectives and Applications |chapter=Total and Partial Differentials as Algebraically Manipulable Entities |arxiv=2210.07958 |date=2023|doi=10.5772/intechopen.107285 |isbn=978-1-83880-992-8 }}</ref>

=== Integration ===
Another key use of the hyperreal number system is to give a precise meaning to the integral sign ∫ used by Leibniz to define the definite integral.

For any infinitesimal function<math> \ \varepsilon(x), \ </math>one may define the integral <math>\int(\varepsilon) \ </math>as a map sending any ordered triple <math>(a,b,dx)</math> (where<math> \ a \ </math>and<math> \ b \ </math>are real, and<math> \ dx \ </math>is infinitesimal of the same sign as <math> \, b-a</math>) to the value
: <math>\int_a^b(\varepsilon,dx):=\operatorname{st}\left(\sum_{n=0}^N\varepsilon(a+n \ dx)\right),</math>
where<math> \ N \ </math>is any [[hyperinteger]] number satisfying<math> \ \operatorname{st}(N \ dx) = b-a.</math>

A real-valued function <math>f</math> is then said to be integrable over a closed interval<math> \ [a,b] \ </math>if for any nonzero infinitesimal<math> \ dx, \ </math>the integral
: <math>\int_a^b(f \ dx,dx)</math>
is independent of the choice of<math> \ dx.</math> If so, this integral is called the definite integral (or antiderivative) of <math>f</math> on<math> \ [a,b].</math>

This shows that using hyperreal numbers, Leibniz's notation for the definite integral can actually be interpreted as a meaningful algebraic expression (just as the derivative can be interpreted as a meaningful quotient).<ref>Keisler</ref>