Infinitesimal
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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.
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.
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:
- To find the derivative f'(x) of the function f(x) = x², let dx be an infinitesimal. Then,
- <math>f'(x)=\frac{f(x + dx) - f(x)}{dx}\,<math>
- <math>=\frac{x^2 + 2x \cdot dx + dx^2 -x^2}{dx}\,<math>
- <math>=2x + dx\,<math>
- <math>=2x\,<math>
- 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.
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.
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.
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.