# Infinitesimal Analysis?

If there exists a non-standard analysis which addresses an entity which is larger than any real number, perhaps there are an infinite number of such entities which are each larger than the previous. Could a proper analysis be developed, which would give new insight on the concept of infinity?

One can construct the hyperreal numbers, which have the real numbers as a subset. Below, I present an infinite dimensional vector space. Is this space an abstraction of the hyperreal numbers?

$z = a_{-\infty}\cdot 0 + \dots + a_{-3}\cdot\varepsilon^3 + a_{-2}\cdot\varepsilon^2 + a_{-1}\cdot\varepsilon + a_0\cdot 1 + a_1\cdot\Lambda + a_2\cdot\Lambda^2 + a_3\cdot\Lambda^3 + \dots + a_\infty \cdot\infty$

In an approximated form, consider the 3 dimensional space:

$z = a\cdot\varepsilon + b\cdot 1 + c\cdot\Lambda$

Here, $\varepsilon$ represents an infinitesimal element basis vector, which is nonzero, yet smaller than any real number (which has the basis vector 1). Similarly, $\Lambda$ is larger than any real number, yet not infinite.

Currently, I am trying to develop a self-consistent closed algebraic structure from this infinite dimensional vector space. Coming up with a well defined division operator is a challenge. Is this challenge related to the infinitesimals themselves, or the fact that I have a multidimensional vector space?

Addition and subtraction of two $z_1$ and $z_2$ is trivial. Multiplication can be constructed such that $\varepsilon\times\varepsilon = \varepsilon^2$, where $\varepsilon^2$ is an orthogonal basis vector. Furthermore, one can make the choice that $1/\Lambda = \varepsilon$. However, the division of z’s with multiple nonzero elements can be trickier. Furthermore, the meaning of $\sqrt\varepsilon$ is unclear.

Consider this simple first example of division:

$z_1 = 1+\varepsilon$

$z_2=\frac{1}{z_1} = \frac{1}{1+\varepsilon} = \frac{1-\varepsilon}{1-\varepsilon^2}$

One might expect to get $1-\varepsilon$, but a denominator still remains. To remove the denominator would require an infinite series:

$z_2 = \frac{(1-\varepsilon)(1+\varepsilon^2)}{1-\varepsilon^4} = \frac{(1-\varepsilon)(1+\varepsilon^2)(1+\varepsilon^4)(1+\varepsilon^8)(1+\varepsilon^{16})(1+\varepsilon^{32})}{1-\varepsilon^{64}}$

$= (1-\varepsilon)(1+\varepsilon^2)(1+\varepsilon^4)(1+\varepsilon^8)(1+\varepsilon^{16})(1+\varepsilon^{32})(1+\varepsilon^{64})(1+\dots$

$= \sum_{n=0}^\infty (-1)^n\cdot\varepsilon^n$

One may want to find the inverse of $z_3 = a+b\cdot\varepsilon$:

$z_4 = \frac{1}{z_3} = \frac{1}{a}\sum_{n=0}^\infty \left(\frac{-b}{a}\cdot\varepsilon\right)^{n}$

Next, one may want to find the inverse of z with an arbitrary number of terms. From here, division of two arbitrary z’s would be established.

This example seems to imply that a multiplication and division operator cannot be constructed for the 3D vector space.

Next, I will think about if and how this differs from Georg Cantor’s ordinal numbers. Interestingly enough, he interprets $1 + \Lambda = \Lambda$ and $\Lambda +1$ to be unique, making the algebra non-commutative. This is an intriguing philosophical notion!