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Section 1.4
Infinitesimally small and Infinitely Large Quantities


We have dealt with exceedingly small and exceedingly large quantities. But they were finite nevertheless. We could think of numbers even larger or smaller than the largest or smallest we dealt with and pretend that they measure something.

In answering Exercise 1.1.2 ("How far away is 1?"), you found that the distance from the x-axis to 1 was more than $10^{30}$ times the distance on your screen from the x-axis to $6\cdot 10^{-64}$, or about 1 billion trillion billion trips across the visible universe.

This is an impossibly large physical distance. But we can imagine it, and talk about it. Here is a wonderful YouTube video that addresses this very issue of unimaginably large and small numbers in the human experience.

In another module we will find situations where we need to talk about rates of change over intervals of a quantity’s measure that are "infinitesimally small" (or "infinitesimal"), or about accumulations of change over a number of intervals, where the number becomes "infinitely large".

The phrases "infinitesimally small" and "infinitely large" have been controversial in the history of mathematics. We will not ask you to address that controversy.

Instead, we will ask you to form a personal image that will, in practice, serve the purpose that is intended by appealing to "infinitesimally small" and "infinitely large" quantities.

We will speak about intervals that are small enough so that, were they any smaller, it would make no essential difference relative to the problem being addressed.

If you must have a definite length to consider as "infinitesimally small", then think of the Planck Length. It is approximately $1.6 \cdot 10^{-35}$ meters. Theoretically, a Planck Length is the smallest length that can be measured.

For practical purposes, you can think of an interval of one Planck Length on a number line in place of the phrase "infinitesimally small". You can also think of $6.5 \cdot 10^{34}$ meters, the reciprocal of the Planck Length, in place of "infinitely large".