Today’s soundtrack is Ásmegin: Hin Vordende Sod & Sø, a wonderful folk metal album that features a variety of instruments and vocal styles.
This afternoon, I’m reading the second chapter of Thompson’s Calculus Made Easy, “On Different Degrees of Smallness.”
A point that math students often find confusing is how mathematicians refer to orders of magnitude. They can accurately describe both 100 or 1/100 as being the second order of magnitude of 1, as 1 is 1% of 100, and 1/100 is 1% of 1. So we must remember that orders of magnitude can refer to both greatness and smallness. In many cases, small enough orders of magnitude can be safely ignored: the millionaire does not worry about losing a single penny. But sometimes small values can snowball into importance: a millionaire might worry about paying an extra penny per foot of raw materials.
How can we quantify which values are negligible? Is the second order of magnitude (in smallness) negligible? Not necessarily, says Thompson. “If dx be a small bit of x, and relatively small of itself, it does not follow that such quantities as x ⋅ dx, or x²dx, or aˣdx are negligible. But dx x dx would be negligible, being a small quantity of the second order” (Thompson, 1998, p. 43).
That’s it for today! Next time, we’ll be learning about relative growth.