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.
`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`.
`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`.
`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.
`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.
# 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.
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