The Infinite and the Infinitesimal

The Very Large and the Very Small

Aug 15, 2008 Isaac M. McPhee

To many it is surely surprising just how complicated the idea of "infinity" can truly be, for, oddly enough, it can exist both in various forms and in various "sizes."

Beyond the idea that there are various "cardinalities" of infinity (aleph-null being the "countably" infinite and aleph-one being the "uncountably infinite," among many others), there are also different sizes of infinity: There are the very "big" infinite numbers, and the very "small" infinite numbers. These small numbers, which are often overlooked in favor of the large, are called "infinitesimal."

The Infinitely Tiny

A graphical representation of an infinitesimal number might be a curve which possesses zero as a horizontal asymptote - that is, as it curves, it grows ever closer to zero without actually ever touching that line, thus forcing the numbers to grow smaller and smaller - closer and closer to zero.

Mathematically, such a curve might be represented by way of a function: f(x) = 1/x (where, one can see, as the value of x gets larger and larger, the value of y gets smaller and smaller).

What one must avoid is the often-made mistake of considering the term infinitesimal to refer to those negative numbers which are infinitely large. Those numbers, though they might indeed be seen as being very "small" in some sense, are not infinitesimal - they are infinite, but in a different direction than the positive numbers. Infinitesimal numbers can then exist either in the positive or in the negative, as each of these number lines ebbs ever closer to zero.

Infinitesimals in Nature

While mathematically, one might find infinitesimals cropping up all over the place, do such things ever actually exist in nature? Is there really a point to understanding something so infinitely tiny?

Indeed they do, and indeed there is.

These tiny numbers pop up all the time in the study of physics, where one often finds themselves staring at mathematical representations of curves which have zero as a negative asymptote.

One particular example of this might be the study of singularities - a term which is often used to describe the state of the universe immediately prior to the big bang, or, to slightly lesser extent, the state of existence within a black hole. In many physical theories, a singularity is an infinitesimal point of space-time which is immeasurably small. Mathematically, the collapse of a star which has run out of its nuclear fuel (in the case of a black hole) is practically unstoppable, crushing in on itself until it has achieved a size and density which is infinitesimally small - a singularity, wherein the known laws of physics break down.

This is just one of many examples where these tiny numbers - so close to zero that it almost seems that they couldn't possibly matter at all - have an effect on the entire universe as it is currently known. In addition to this, no class in pre-calculus or calculus would be complete without a good dose of analyzing just these very concepts - numbers which are certainly infinite, but infinitely small.

The copyright of the article The Infinite and the Infinitesimal in Math/Chaos Theory is owned by Isaac M. McPhee. Permission to republish The Infinite and the Infinitesimal in print or online must be granted by the author in writing.

Graphical Representation of a Singularity