# Definition:Infinitesimal

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## 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.