infinitesimals real number infinite infinitesimal derivative work
Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small".  For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols 



d
x


{\displaystyle dx}

 and 



d
y


{\displaystyle dy}

 were taken to be infinitesimal, and the derivative 



d
y

/

d
x


{\displaystyle dy/dx}

 was their ratio.
