# Reciprocal with infinitesimal number [closed]

Let's assume $a$ is a number greater than zero and smaller than any positive real number (could be expressed as $\frac1∞$). Now let's assume there is a number written as $x+ya;x∈R;y∈Z$. Is there a way then to express $\frac1{x+ya}$ as another $x+ya$? What would $x$ and $y$ be?

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If the number $a$ existed, then the number $\frac{a}{2}$ would contradict the existence of $a$. –  Jared Jan 13 at 18:08
Presumably not. You will need an infinite entity $a^{-1}$, I think. –  user18921 Jan 13 at 18:11
It depends on the value of $a^2$. For example, if $a^2=0$ i.e. you introduce the infinitesimals of the first order and ignore all others then it would be $\frac{1}{x}-\frac{y}{x^2} a$. Otherwise, you need to introduce some higher order things into your arithmetic. –  user68061 Jan 13 at 18:16
I haven't thought it much into details, I am just a mathematics fan ☺, but I guess I will have to consider fractions of $a$ and $a^2$. But thanks for the formula anyway. –  IllidanS4 Jan 13 at 18:24
@ThomasAndrews and others: What exactly is unclear? I am just interested in infinitesimal hypothesis. –  IllidanS4 Jan 13 at 21:18