Infinitesimal

The infinitesimal definition in mathematics is “The quantities that are closer to zero than any standard real number but are not zero are Infinitesimals”. 

Infinitesimals don't exist in the traditional real number system, but they do in a variety of other systems, including unreal and hyperreal numbers, which are real numbers augmented with a system of infinitesimal quantities, and infinite quantities, which are the reciprocals of the infinitesimals.

The below image represents the infinitesimals and infinities on the hyperreal number line.

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Define Infinitesimally Small

Infinitesimally small is a synonym of the word infinitesimal which means “very, very small”, or “extremely small” or “vanishingly small” or “smaller than anything”.

An infinitesimal object is an object less than any measurable size, but not so small or small that the available means cannot distinguish it from zero. Therefore, "infinitesimal" means "infinitesimally small" or less than any standard number of the real number when used as a mathematical adjective. Infinitesimals are often compared with other infinitesimals of similar size as a derivative in order to make them meaningful. Infinite numbers are summed together to produce an integral.

Infinitesimalmente is a Spanish word for Infinitesimally small numbers.

History of Infinitesimal

• The insight with the use of infinitesimals was that, although these entities were infinitely small, they could still retain certain specific properties such as angle or slope.

• The usage of the term infinitesimal in a sentence has been taken from a Latin modern ‘infinitesimus’ coinage dating back to the 17th century that originally spoke of the term "infinite-th." Infinitesimals are a key ingredient in Leibniz's infinitesimal calculus procedures, including the continuity law and transcendental homogeneity law.

• Initially, Nicolaus Mercator or Gottfried Wilhelm Leibniz introduced the concept of Infinitesimals around 1670.

• In his work The method of mechanical theorems, Archimedes used what eventually came to be called the indivisible method to identify areas of regions and solid volumes.

• Archimedes solved the same problem by the method of exhaustion in his formally published treatises. The work of Nicholas von Cusa, further developed by Johannes Kepler in the 17th century, particularly in the calculation of a circle area as an infinite-faced polygon, was seen in the 15th century.

• Simon Stevin has prepared the basis for a real continuum in the work on the decimal representation of all the numbers in the 16th century.

• The indivisible approach of Bonaventura Cavalieri led to the results of classic authors being expanded. The method of indivisible geometric figures as composed of co-dimensional entities 1.

• The infinitesimals of John Wallis differ from the indivisible in that he decomposes geometric figures into infinitely thin building blocks that are of the same dimension as that of the figure and prepares the basis of general methods of integral computations. 

• Leibniz's use of infinitesimals depends on heuristic principles, such as the law of continuity: what succeeds for the finite numbers also succeeds the infinite and vice versa; and the transcendent law of homogeneity, which sets out the methods for replenishing expressions with unexpected quantity, with only assignable expressions.

• Mathematics like Leonhard Euler and Joseph-Louis Lagrange used infinitesimals routinely in the 18th century. Infinitesimals have been used both by Augustin-Louis to define continuity in his course of analysis and to define an early form of the function of a delta of Dirac.

• When Cantor and Dedekind developed more abstract versions of Stevin's continuum, Paul du Bois-Reymond produced a number of papers based on functional growth rates on an infinitesimally enriched continuum.

• The work of Du Bois-Reymond inspired Émile Borel and Thoralf Skolem alike. Borel explicitly associated the work of du Bois-Reymond with the work of Cauchy on infinitesimal rates of growth.

• The first non-standard arithmetic models were developed by Skolem in 1934.

• In 1961 Abraham Robinson, a not standard analysis based upon earlier work by Edwin Hewitt in 1948 and Jerzy Los in 1955, achieved a mathematical implementation of both the law of continuity and the law of infinitesimals.

• In the late 19th as Philip Ehrlich documents, the Mathematical Study of infinitesimal systems continued through Levi-Civita, Giuseppe Veronese, Paul du Bois-Reymond, and so on. 

• The infinitesimal can be used as a foundation for calculation and analysis in the 20th century.

First Order Properties of Infinitesimal

In order to extend the real numbers to infinite and infinitesimal quantities, the usual way is to remain as conservative as possible without changing their basic properties. This ensures that the best possible number of familiar results are still available. Typically fundamental means that no sets but only elements are measured. 

As the aim is to build a non-Archimedean system, the Archimedean principle can be expressed through sets. This extended numbering scheme cannot agree with the actuals on all properties that can be expressed through sets by quantification.

We can differentiate between three levels at which a non-archimedean number system can have first-order compatible properties with the real ones:

• All the usual axioms of the real number system can be specified in the first-order logic in an ordered field.

• A real closed field has all the first-order feature of the actual numbers system for statements involving the basic commands +, , and , regardless of whether they are usually taken as axiomatic. This is stronger than the observance of the axioms of the ordered field. In particular, there are additional first-order characteristics, such as the existence of a root for each unusual polynomial.

• The system could have all of the first-order properties of the real system for declarations involving any relationship, irrespective of whether these relationships can be expressed with +, x, and ≤.

Use of Infinitesimal in Different Number Series

Here let us look at the application of Infinitesimal in different number series.

• Laurent Series

The Laurent series of a complex function f(z) is a power series representation of that function that contains negative degree terms. Where a Taylor series expansion isn't possible, it can be used to express complex functions. The Laurent series with only the constant term 1 is known as the real number 1, and the Laurent series with only the linear term x is known as the simplest infinitesimal, from which the other infinitesimals are built. Since the simple infinitesimal x does not have a square root, these infinitesimals have different first-order properties than the reals.

About point c the Laurent series of a complex function is given as follows:

f(z) = \[\sum_{n=-\infty}^{\infty}\] a\[_{n}\] (z - c)\[^{n}\]

Where a\[_{n}\] and c are constants. 

• The Levi-Civita Field

The Levi-Civita field is a non-Archimedean ordered field that consists of an infinite and infinitesimal number system. The Levi-Civita field is algebraically closed, similar to the Laurent series. Although this area is large enough to allow for extensive research, its elements can still be represented on a computer in the same way as real numbers can be represented in floating-point.

• Surreal Numbers

The surreal number system is a fully ordered proper class that includes real numbers as well as infinite and infinitesimal numbers, which are larger or smaller in absolute value than any positive real number. The surreals have many properties in common with the real numbers, such as the ability to perform standard arithmetic operations as a result they form an ordered field.

• Hyperreals

The hyperreal number system is a method of dealing with infinite and infinitesimal quantities. The reciprocals of hyperreal numbers are infinitesimals, and the hyperreal numbers themselves are infinite. For any finite number of terms, the hyperreals, or nonstandard reals, R, are an extension of the real numbers R that includes numbers greater than anything of form 1+1+1+........+1.

• Superreals

The superreal numbers are a class of extensions of the real numbers that are mainly of interest in non-standard analysis, model theory, and the study of Banach algebras. They are a generalisation of the hyperreal numbers. Surreal numbers are a subfield of the field of superreals.

• Smooth Infinitesimal Analysis

The smooth infinitesimal analysis is a modern reformulation of calculus in infinitesimal terms. It considers all functions to be continuous and incomprehensible in terms of discrete entities. It is a branch of synthetic differential geometry as a theory.

Conclusion

Here, we studied the infinitesimal meaning, history of infinitesimal numbers, first-order properties of infinitesimal, applications of infinitesimal on a different number system. In mathematics, an infinitesimal is a quantity that is less than but not equal to zero. Many early attempts to explain calculus were based on questionable reasoning about infinitesimals, derivatives were defined as ultimate ratios of infinitesimals, and integrals were calculated by summing rectangles of infinitesimal width, despite the fact that no such quantity can exist in the real number system. As a consequence, the infinitesimal calculus was originally applied to differential and integral calculus. As rigorous definitions of limit, continuity, and real numbers were established, this terminology faded away.