This seems like a pretty basic question, but I've been searching around and haven't come across the answer.
Consider two infinitesimal numbers, $\epsilon$ and $\epsilon^2$. On the one hand, it would seem that $\epsilon^2 \ll \epsilon$, because $\epsilon^2 / \epsilon = \epsilon$. And on the other hand, it would seem that $\epsilon^2 \approx \epsilon$, because the difference $\epsilon - \epsilon^2$ is infinitesimal (Keisler's definition of $\approx$).
Are both of the above "it would seem" statements generally accepted as true? My instinct is to affirm $\epsilon^2 \ll \epsilon$ and deny $\epsilon^2 \approx \epsilon$, while affirming $1 - \epsilon^2 \approx 1 - \epsilon$ because $(1 - \epsilon^2) / (1 - \epsilon)$ is approximately 1. I guess what it comes down to is that I'd rather define $\approx$ using division, not subtraction. But is this tenable?
Okay, I guess that's more than one question, and maybe they aren't basic. But anyway, that's what I'm wondering.