Infinitesimal

In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than 0 (number) yet smaller than any positive real number. A number ''x'' ≠ 0 is an infinitesimal iff every sum |''x''| + ... + |''x''| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/''x'' is larger than any positive real number. In standard analysis, infinitesimal is only a notional quantity, and there exists no infinitesimal real number. This can be shown using the supremum axiom of the real numbers: consider whether the least upper bound ''c'' of the set of all infinitesimals is or is not an infinitesimal. If it is, then so is 2''c'', contradicting the fact that ''c'' is an upper bound. If it is not, then neither is ''c''/2, contradicting the fact that among all upper bounds, ''c'' is the least. The first mathematician to make use of infinitesimals was Archimedes, although he did not believe in their existence. See how Archimedes used infinitesimals. The Archimedean property is the property of an ordered algebraic structure of having no infinitesimals. When Isaac Newton and Gottfried Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go: ::To find the derivative ''f'''(''x'') of the function ''f''(''x'') = ''x''², let d''x'' be an infinitesimal. Then, :::$f\text{'}(x)=\backslash frac\{f(x\; +\; dx)\; -\; f(x)\}\{dx\}\backslash ,$ :::$=\backslash frac\{x^2\; +\; 2x\; \backslash cdot\; dx\; +\; dx^2\; -x^2\}\{dx\}\backslash ,$ :::$=2x\; +\; dx\backslash ,$ :::$=2x\backslash ,$ ::since d''x'' is infinitesimally small. This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by George Berkeley in his work ''The analyst: or a discourse addressed to an infidel mathematician''. The fundamental problem is that d''x'' is first treated as non-zero (because we divide by it), but later discarded as if it were zero. It was not until the second half of the 19th century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit (mathematics), which obviates the need to use infinitesimals. Nevertheless, the use of infinitesimals continues to be convenient for simplifying notation and calculation. Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation of the derivative of ''f''(''x'') = ''x''² can be justified with a minor modification: we have to talk about the ''standard part'' of the difference quotient, and the standard part of ''x'' + d''x'' is ''x''. Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., ''NOT'' (''a'' ≠ ''b'') does not have to mean ''a'' = ''b''. A ''nilsquare'' or ''nilpotent'' infinitesimal can then be defined. This is a number ''x'' where ''x'' ² = 0 is true, but ''x'' ≠ 0 can also be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above. ==See also== * Hyperreal number * Infinitesimal calculus * Surreal number Calculus

Infinitesimal

A reason to consider that infinitesimals do "exist" in some sense can be found by considerations on density. An ordered set is dense if between any two of its elements is another element of that set; hence between any two elements there are infinitely many elements of the set (meaning arbitrarily many or infinitely many, depending on one's conception of infinity). In a dense set $\backslash left(\backslash mathbb\{R\},\; \backslash mathbb\{Q\}\backslash right)$ there is no immediate successor: i.e. for a given number $x$ it is not possible to state which is next. Consider a dense set of points on a line, these points cannot, in some sense, "touch" each other. If a point $a$ were touching a point $b$ with User:160.53.250.105 08:09, 21 Jan 2005 (UTC)Richard O'Donovan

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Infinitesimal
Infinitesimal
Infinitesimals
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