calculus - Definition of an Infinitesimal

Covering 1.4 of Keisler's Elementary Calculus, "Slope and Velocity; The Hyperreal Line"

That chapter defines: A number $\epsilon$ is said to be infinitely small, infinitesimal, if: $-a < \epsilon < a$. And goes on to an introduction to the hyperreal line.

However, this definition seems to imply an infinitely small number ($\epsilon$) is one which is between $\pm a$, which seems to be a very large range if you choose, for example, $a = 1000$.

I'm obviously missing something obvious.

Answer

An infinitesimal is an $\epsilon$ which is between $a$ and $-a$ for every standard real number $a$. So $0$ is infinitesimal, but ${1\over 100}$ isn't, because ${1\over 100}\not<{1\over 500}$, and ${1\over 500}$ is a standard real number.

In the standard real numbers, $0$ is the only infinitesimal. In the hyperreals, there are lots of nonzero infinitesimals.

---

The role of infinitesimals is to make naive ideas about limits actually work rigorously; so, e.g., if we interpret "The limit of $f(x)$ as $x\rightarrow c$ is $L$" as meaning "If $d$ is infinitely close to $c$ (that is, $c-d$ is infinitesimal), then $f(d)$ is infinitely close to $f(c)$," the framework of hyperreals will make this meaningful.

In standard calculus, this work is done via the $\epsilon-\delta$ definition of a limit, which avoids the use of non-standard real numbers, but is arguably harder to learn.