Infinitesimal
In calculus, an infinitesimal is a number greater than zero yet smaller than any positive real number. If x is an infinitesimal and x>0 then any finite sum x+ ... +x is less than 1, no matter how large the finite number of terms in the sum. Furthermore, 1/x is larger than any positive real number. Of course, there exists no infinitesimal real number.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 f '(x) = (f(x+dx)-f(x))/dx = (x²+2x*dx+dx²-x²)/dx = 2x+dx = 2x, since dx is infinitesimally small.
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 limiting process, 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.
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