Infinitesimals

In this post we're going to take a look at what infinitesimals are and why they are important.
Infinitesimals are an abstract concept of very small values that are impossible to represent quantitatively in a finite system.

# Definition

We define one infinitesimal as:

`ε = lim_{n to \infty}\frac{1}{n}`

with the inequality: `ε > 0`.

In general, the following inequalities hold true:

`\frac{0}{n} < \frac{1}{n} < \frac{2}{n} < ... < \frac{n}{n}`

as `n -> \infty`.

# Appearance

The infinitesimals appear in some fundamental limits, one of which is the limit for the natural exponentiation function:

`lim_{n to \infty}(1 + \frac{\x}{n})^n = \exp(\x)`

Using our infinitesimal notation, we can rewrite the limit as:

`lim_{n to \infty}(1 + ε*\x)^n = \exp(\x)`

where, for `x=1`, we have:

`lim_{n to \infty}(1 + ε)^n = \e`.

# Debate

There was (and, probably, still is) a debate in mathematics whether the following limit:

`lim_{n to \infty}\frac{1}{n}`

is `0` or greater than `0`.

Considering the concept of  infinitesimals, the limit is greater than `0` by definition, which sometimes makes more sense, logically speaking.

In particular, if we take a closer look at the limit for `e`:

`lim_{n to \infty}(1 + \frac{1}{n})^n = e`

we can see that if we replace `\frac{1}{n}` with `0`, we get:

`lim_{n to \infty}(1 + 0)^n != e`

The result of the above limit is actually `1`, and is equivalent with:

`lim_{n to \infty}(1 + 0*ε)^n = 1`

From this, we can conclude that:

`lim_{n to \infty}1^n = 1`

But this is true only if we consider infinitesimals, because we can rewrite the limit as:

`lim_{n to \infty} \exp(\frac{\log(1)}{\frac{1}{n}}) = lim_{n to \infty} \exp(\frac{0}{\frac{1}{n}}) = ?`

where we can have an undefined value (`\frac{0}{0}`) if we ignore the concept of infinitesimals.

However, when infinitesimals are considered, the illegal division disappears and the result is exactly what we expect it to be:

`\exp(\frac{0}{ε}) = \exp(0) = 1`, where `ε > 0` by definition.

# Infinity from infinitesimals

The concept of infinitesimals can, also, be used to work with infinite values in a natural way:

`lim_{n to \infty}n = \frac{1}{ε}`

In general, we have:

`lim_{n to \infty}(n*x) = \frac{x}{ε}`

This gives us the beautiful identity:

`lim_{n to \infty}(ε*\n) = 1`

The advantage of infinitesimals over a classical definition for infinity, is the fact that we can do arithmetical operations on infinity defined in terms of infinitesimals, such as:

`\frac{7}{ε^(-1)} - \frac{5}{ε^(-1)} = \frac{2}{ε^(-1)}`

as long as `ε = ε` everywhere we use it.

# Conclusion

The concept of infinitesimals is an interesting concept, which can extend or give meanings to mathematical formulas in some special cases.