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=limn→∞1n
with the inequality: ε>0.
In general, the following inequalities hold true:
0n<1n<2n<...<nn
as n→∞.
limn→∞(1 + xn)n=exp(x)
Using our infinitesimal notation, we can rewrite the limit as:
limn→∞(1 +ε⋅x)n=exp(x)
where, for x=1, we have:
limn→∞(1 +ε)n=e.
limn→∞1n
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:
limn→∞(1+1n)n=e
we can see that if we replace 1n with 0, we get:
limn→∞(1 +0)n≠e
The result of the above limit is actually 1, and is equivalent with:
limn→∞(1 +0⋅ε)n=1
From this, we can conclude that:
limn→∞1n=1
But this is true only if we consider infinitesimals, because we can rewrite the limit as:
limn→∞exp(log(1)1n)=limn→∞exp(01n)=?
where we can have an undefined value (00) 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(0ε)=exp(0)=1, where ε>0 by definition.
limn→∞n=1ε
In general, we have:
limn→∞(n⋅x)=xε
This gives us the beautiful identity:
limn→∞(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:
7ε-1-5ε-1=2ε-1
as long as ε=ε everywhere we use it.
# Definition
We define one infinitesimal as:ε=limn→∞1n
with the inequality: ε>0.
In general, the following inequalities hold true:
0n<1n<2n<...<nn
as n→∞.
# Appearance
The infinitesimals appear in some fundamental limits, one of which is the limit for the natural exponentiation function:limn→∞(1 + xn)n=exp(x)
Using our infinitesimal notation, we can rewrite the limit as:
limn→∞(1 +ε⋅x)n=exp(x)
where, for x=1, we have:
limn→∞(1 +ε)n=e.
# Debate
There was (and, probably, still is) a debate in mathematics whether the following limit:limn→∞1n
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:
limn→∞(1+1n)n=e
we can see that if we replace 1n with 0, we get:
limn→∞(1 +0)n≠e
The result of the above limit is actually 1, and is equivalent with:
limn→∞(1 +0⋅ε)n=1
From this, we can conclude that:
limn→∞1n=1
But this is true only if we consider infinitesimals, because we can rewrite the limit as:
limn→∞exp(log(1)1n)=limn→∞exp(01n)=?
where we can have an undefined value (00) 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(0ε)=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:limn→∞n=1ε
In general, we have:
limn→∞(n⋅x)=xε
This gives us the beautiful identity:
limn→∞(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:
7ε-1-5ε-1=2ε-1
as long as ε=ε everywhere we use it.
Comments
Post a Comment