Definition:Infinitesimal

Definition

Informal Definition

An infinitesimal is a mathematical object $\delta$ resembling a (real) number such that:

$(1): \quad \delta > 0$

$(2): \quad \forall x \in \R_{>0}: \delta < x$

That is, an infinitesimal is a (strictly) positive real number which is smaller than every other (strictly) positive real number.

Formal Definition

An infinitesimal is a variable whose limit is zero.

Order

Let $x$ and $y$ be infinitesimals.

$x$ and $y$ are the same order if and only if $\dfrac x y$ is finite and does not tend to zero.

Also see

• Results about infinitesimals can be found here.

Historical Note

The concept of an infinitesimal was initially proposed by Gottfried Wilhelm von Leibniz when inventing calculus.

The logical inconsistencies of this approach, famously criticised by Bishop Berkeley, caused it to be abandoned.

However, the concept has been revived in recent years with the invention of non-standard analysis.