# Numbers of Order Unity

Over at Unqualified Offerings, Thoreau is bemused by his students’ reaction to unusual numbers:

[I]t is fascinating how we condition people to be used to numbers in a certain range, and as soon as a number is either very big or very small it becomes disconcerting. On one level, I’m glad that they are able to do the conversion and that they at least realize that numbers need to be checked. I’ve had people happily measure the dimensions of an object in millimeters, get their conversion to meters wrong, and cheerfully tell me that their tiny metal cylinder has a volume of 27 cubic meters. At that point in the lab, I say “OK, so, your metal block is 3 meters on a side [I take a few large paces], 3 meters on the other side [a few more large paces], and reaching from the floor to some place above the ceiling. Are you absolutely sure about this?”

I’ve had similar experiences myself. And, in fact, one of the most frustrating things about teaching modern physics is that the scale of the answers is so different from the everyday scale. We’ve conditioned students to use SI units for everything, but when you’re doing basic quantum problems, the lengths are all in the nanometer range and the energies in the 10-19 joule range, and they have absolutely no intuition for those. I get completely ridiculous answers handed in, because they don’t have any feel for what the answer ought to be.

Of course, this phenomenon isn’t limited to undergraduates. Professional physicists are also conditioned to expect numbers of order unity.

The difference is, professional physicists expect numbers of order unity in units that are chosen to give that scale. What units those are depend on the problem– in my branch of atomic physics, descended from laser spectroscopy, we tend to use frequency units. I have trouble remembering the Bohr magneton in J/T, but I can tell you instantly that it’s 1.4 MHz/G.

Particle and nuclear physicists tend to work in large multiples of electron volts (MeV or GeV), condensed matter physicists in eV, and high energy theorists are (in)famous for setting Planck’s constant equal to the speed of light, which is equal to one. In every case, the system of units is chosen so that the scale of the typical answers falls in the range of numbers people are comfortable working with (usually between 1 and 1000).

I sometimes wonder if we’re not doing our students a disservice by insisting on SI units (meters, kilograms, seconds) from the beginning, and not training them to switch into whichever system of units gives human-scale numbers. But then, I have enough trouble finding errors when they’re working in a consistent set of units– I don’t like to think about how much confusion could be generated by adding even more unit conversions.

One final note about weird numbers: I took a class in grad school that was basically “QED for Idiots,” and the professor spent a lot of time talking about the history of the Casimir force. One of the calculations he went through, from the original Casimir-Polder paper, wound up getting a coefficient on one of the terms that was something like 23/7. He said that whichever of the two was the grad student had done the calculation, ended up with that result, and brought it to his advisor, who said “You’ve made some sort of mistake. Nothing is 23/7…” and made the student re-do the calculation.

Of course, 23/7 (or whatever) turns out to be the correct result, from evaluating some named equation or another (the Somebody Polynomials, or the FamousName Series). So our innate bias for numbers that “make sense” can trip up even famous theoretical physicists, not just confused undergraduates.