Infinitesimal

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The infinitesimal or infinitesimal refers to a quantity closer to zero than any standard real number but not equal to zero. The term began as an informal notion and is not rigorously originally thought of as an "infinitely small quantity," and originally supported some of the reasoning behind calculus. In the crisis of mathematical foundations of the early 19th century, infinitesimals were abandoned by mathematicians, although they continued to be treated informally in applied sciences, and are usually considered as numbers in practice. Only after the second half of the XX century did a fully rigorous approach to infinitesimal numbers appear.

Non-standard analysis introduced in the 1960s by Abraham Robinson is a rigorous axiomatic approach that allows the introduction of infinitesimals (non-zero hyperreal numbers whose absolute value is smaller than any standard real number). Although the results that can be achieved by non-standard analysis can be achieved by standard real number theory, there are many mathematical proofs and deductions that are simpler and shorter when non-standard analysis is used. The multiplicative inverse of an infinitesimal is an unlimited non-standard real number.

Introduction

Infinitesimal calculus was first proposed by Archimedes. It was then used by Isaac Newton and Gottfried Leibniz, at the dawn of the rise of modern mathematical analysis, but was subsequently discredited by George Berkeley and eventually forgotten. During the 19th century Karl Weierstrass and Cauchy began to use the formal definition of a mathematical limit, so the infinitesimal calculus no longer was necessary. However, during the XX century, the infinitesimals were rescued as a tool that helps to calculate limits in a simple way. The use of infinitesimals in the Russian bibliography is quite popular.

Another way to work with the indefinite is to consider them as numbers, and not as limits, that is to work in a set ${displaystyle Re }$ that contains more numbers than usual. They are called hyperreal numbers, and they are a creation of Non-standard analysis.

Standard analysis

General notion

An infinitesimal or infinitesimal is associated with a very small quantity, a possible attempt to formalize is to consider infinitesimal as a function or magnitude that satisfies:

{displaystyle lim _{xto a}f(x)=0}

it is said that f It's an infinite x=a

Some functions are infinitesimal at certain points, for example:

f(x) = x-1 is an infinite x=1.

g(x) = sen(x) is an infinite

{displaystyle 0+kpi }

with

{displaystyle kin mathbb {Z} }

Therefore, any function as it tends to 0 at a point is called an infinitesimal.

Properties of infinitesimals

The finite sum of infiniteness is an infinite.
The product of two infinitesimos is an infinite.
The product of an infinitesimo by an accompanied function is an infinite.
The product of a constant by an infinite is an infinite.

Comparison of infinitesimals

Dadas ${displaystyle lim _{xto a}f(x)=0}$ and ${displaystyle lim _{xto a}g(x)=0}$

Yeah. ${displaystyle lim _{xto a}{frac {f(x)}{g(x)}}=pm infty }$ f and g are infinite comparable in x=a and f is an infinite of order less than g in x=a.
Yeah. ${displaystyle lim _{xto a}{frac {f(x)}{g(x)}}=0}$ f and g are infinite comparable in x=a and f is an infinite of order superior to g in x=a.
Yeah. ${displaystyle lim _{xto a}{frac {f(x)}{g(x)}}=l}$ with $l$ of ${displaystyle mathbb {R} -left{0right}}$ f and g are infinite of the same order x=a.
In particular, if ${displaystyle {underset {xto a}{mathop {lim } }},{frac {fleft(xright)}{gleft(xright)}}=1}$ f is an infinite amount equivalent to g in x=a

If two indefinites are equivalent then one can approach another. I mean, f(x) and g(x) are infinite equivalents when ${displaystyle xto a}$ Then you can say that ${displaystyle f(x)approx g(x)}$ When ${displaystyle xto a}$ . If presented as a factor or divider one can be replaced by another for the calculation of limits when ${displaystyle xto a}$ :

Theorem: if there is the limit of ${displaystyle f(x)/h(x)}$ When ${displaystyle xto a}$ , being $f$ and $g$ infinite equivalents in ${displaystyle x=a}$ , then the limit ${displaystyle g(x)/h(x)}$ equals the limit of ${displaystyle f(x)/h(x)}$ .

Some Infinitely Equivalents

$f(x),$ It's an infinite when ${displaystyle xto 0}$ :

${displaystyle sin(x)approx x}$
${displaystyle tan(x)approx x}$
${displaystyle 1-cos(x)approx {frac {x^{2}}{2}}}$
${displaystyle arcsin(x)approx x}$
${displaystyle arctan(x)approx x}$
${displaystyle e^{x}-1approx x}$
${displaystyle ln(1+x)approx x}$
${displaystyle a^{x}-1approx xcdot ln(a)}$
$1}" xmlns="http://www.w3.org/1998/Math/MathML">(x+1)n− − 1x≈ ≈ n,n▪1{displaystyle {frac {(x+1)^{n}-1}{x}}}approx n,n literal}1}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37801c6a37870a5cc17691852d0dd4786d788258" style="vertical-align: -1.838ex; width:24.963ex; height:5.676ex;"/>$

$f(x),$ It's an infinite when ${displaystyle xto 1}$ :

${displaystyle ln(f(x))approx f(x)-1}$

Non-standard analysis

Main article: Non-standard analysis

Non-standard parsing is a generalization of actual parsing. Non-standard analysis makes it possible to define, in addition to the objects definable in the ordinary theory of real numbers, new objects called "external" or "non-standard". Any object (number, set, or function) definable in conventional real number theory is a "standard" object; within the non-standard analysis. Along with the "standard" Robinson's non-standard analysis allows you to introduce "non-standard objects" as infinitesimal numbers or unlimited (infinite) numbers and handle them fully coherently within the theory.

Non-standard theory begins to introduce a new preaching ${displaystyle scriptstyle mathrm {st} (cdot)}$ , that predicate allows to build a formal language that includes the ordinary theory of real numbers but allows to define new numbers (concretely the notion of "i-small" and "i-large" numbers allow to build infinitesimal numbers and unlimited numbers larger than any actual standard or ordinary number). The preached "standard" is characterized by three additional axioms that do not possess the ordinary theory of real numbers, and therefore create a formal language that allows formalizing additional numbers. Non-standard analysis makes a crucial use of infinitesimal and unlimited numbers:

A ε number is infinitesimal if for any standard integer n it is fulfilled that LICITLES 1/n. The only real standard number with that property is zero, but there is an infinity r non-standard real numbers such that: r 1/nfor any standard integer. The preached inf(·) forms the notion of infinitesimal, from the primitive relation of standard:

$<math alttext="{displaystyle mathrm {inf} (r)Leftrightarrow left[forall n:mathrm {st} (n)land left(-{frac {1}{n}}<rinf(r)Δ Δ [chuckles]Русский Русский n:st(n)∧ ∧ (− − 1n.r.1n)]{displaystyle mathrm {inf} (r)Leftrightarrow left[forall n:mathrm {st} (n)land left(-{frac {1}{n}}{n}}{frac {1}{n}}{n}}{right)}}}}<img alt="{displaystyle mathrm {inf} (r)Leftrightarrow left[forall n:mathrm {st} (n)land left(-{frac {1}{n}}<r$

Similarly an unlimited number (or infinite) can be defined as any real number r such as r ▪ n for all standard integers. The key to that definition is the standard term, in the ordinary theory of the actual numbers, as there is no standard notion, the concept of infinity cannot be formalized. The preached Inf(·) forms the notion of unlimited number, from the primitive relationship of standard:

$<math alttext="{displaystyle mathrm {Inf} (r)Leftrightarrow left[forall n:mathrm {st} (n)land left(rnright)right]}" xmlns="http://www.w3.org/1998/Math/MathML">Inf(r)Δ Δ [chuckles]Русский Русский n:st(n)∧ ∧ (r.− − n r▪n)]{displaystyle mathrm {Inf} (r)Leftrightarrow left[forall n:mathrm {st} (n)land left(r `nlor r/2005nright)right]}}<img alt="{displaystyle mathrm {Inf} (r)Leftrightarrow left[forall n:mathrm {st} (n)land left(rnright)right]}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a07ab760b28c0d031b1194cbcebc757ac84fe76e" style="vertical-align: -0.838ex; width:40.279ex; height:2.843ex;"/>$

The non-standard analysis therefore allows to build a set of numbers that extends to the actual numbers, this set is of the hyperreal numbers and is represented as ${}^*R$ and in it you can define arithmetic rules for infinitesimal numbers (inf(·)), unlimited (Inf(·)), limited (complete of the previous: ¬Inf(·)) and appreciable (nor infinitesimos, nor unlimited: ¬inf(·)¬Inf(·)), from these four sets you have the following Leibniz rules for operations:

+/-	infinitesimal	limited	appreciable	unlimited
infinitesimal	infinitesimal	limited	appreciable	unlimited
limited	limited	limited	limited	unlimited
appreciable	appreciable	limited	limited	unlimited
unlimited	unlimited	unlimited	unlimited	?

For multiplication, the Leibniz rules are as follows:

x	infinitesimal	limited	appreciable	unlimited
infinitesimal	infinitesimal	infinitesimal	infinitesimal	?
limited	infinitesimal	limited	limited	?
appreciable	infinitesimal	limited	appreciable	unlimited
unlimited	?	?	unlimited	unlimited

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