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In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero 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.
An infinitesimal is only a notional quantity - there exists no infinitesimal real number. This can be shown using the least upper bound 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 2c, 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 Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go:
This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by Bishop BerkeleyGeorge Berkeley (bark-lee) ( March 12, 1685 January 14, 1753), also known as Bishop Berkeley was an influential Irish philosopher whose primary philosophical achievement is the advancement of what has come to be called subjective idealism, summed up in hi in his work The analyst: or a discourse addressed to an infidel mathematician. The fundamental problem is that dx 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 nineteenth centuryAlternative meaning: Nineteenth Century (periodical ( 18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801- 1900. Events The Little Ice Age ended that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limitIn mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculu, 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 analysisModel theory In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only i of Abraham RobinsonAbraham Robinson ( October 6, 1918 April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. He die. In this theory, the above computation of the derivative of f(x) = x² 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 + dx is x.
Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan. 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 nilpotentRing theory In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that x n 0. Examples This definition can be applied in particular to square matrices. The matrix : is nilpotent because A''3 0. It can be 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.