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. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Graph of absolute value function In mathematics, the absolute value (or modulus) of a number is the difference between that number and 0. ... Zero redirects here. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. In mathematics, a number is called positive if it is bigger than zero. ... The text or formatting below is generated by a template which has been proposed for deletion. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...

In standard analysis, infinitesimal is only a notional quantity, and 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. In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...

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. Archimedes of Syracuse (circa 287 BC - 212 BC), was a Greek mathematician, astronomer, philosopher, physicist and engineer. ... The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ... In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals. ...

When Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go: Sir Isaac Newton in Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who wrote... Gottfried Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1, 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...

To find the derivative f'(x) of the function f(x) = x², let dx be an infinitesimal. Then,

since dx 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 Bishop Berkeley 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. In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Bishop George Berkeley George 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 his dictum, Esse est percipi (To be is...

It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit, which obviates the need to use infinitesimals. Alternative 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 in the sense of the Gregorian calendar. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... In 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 sequences elements, as their index approaches infinity. ...

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) = 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. In the most restricted sense, 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 if its absolute value is smaller than any element of F of the form 1... Abraham 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. ...

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. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...

See also

• Hyperreal number
• Infinitesimal calculus
• Surreal number