I recently started working on a project for school, and I would like to extend the real numbers to allow for infinitesimals to be used in conjunction with real numbers. I am currently working solely on the algebraic extension, and currently, I have taken the route of defining each number as a tuple, where my "infinitesimal unit" is denoted $\varepsilon$. In general, a number in this new system can be written:
$x_0\varepsilon^0+x_1\varepsilon^1+x_2\varepsilon^2+...+x_n\varepsilon^n$
The tuple representation for this number would be $(x_0, x_1, x_2,...,x_n)$. I have defined addition and multiplication as follows:
$\displaystyle (x_0, x_1, x_2...)+(y_0, y_1, y_2,...):=(x_0+y_0, x_1+y_1, x_2+y_2,...)$
$\displaystyle (x_0, x_1, x_2,...)\cdot (y_0, y_1, y_2,...) := (z_0, z_1, z_2,...), \text{where} \ z_{j}=\sum_{k=0}^{j}x_{k}\cdot y_{j-k}$
Now, I would like to expand this definition to allow for my variable to exist in noninteger powers as well, because as it currently stands, expressions like $\sqrt{\varepsilon}$ lack any meaning. I fiddled a little bit with the idea of extending the tuples and introducing a subscript, which would "change" the exponent value of each position in the tuple. So for example:
$\varepsilon^{1}:= (0,0,0,0,0,1)_{1/5}=(0,1)$
Here, an increment of 1 in the position of the tuple is equivalent to an increment of 1/5 in the exponent. But as you might imagine, this becomes quite messy. It also feels like one should be able to utilize the definitions already in place to do this. It may be the case that tuples are not at all appropriate for this endeavor. Whatever the case, I would be delighted to hear what you think!