### 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.