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.


  1. #1 Brian
    March 6, 2009

    In CS, it’s not the size that’s surprising, it’s the format. If you end up with an answer of n^2, or 2^n, no problem. If you end up with 3^n you probably did something wrong…

    I had a friend in grad school who’s thesis ended up being proving that 24/49 was the cut-off between one thing happening and another. We still don’t know where that number came from, and it’s VERY disconcerting.

  2. #2 moshe
    March 6, 2009

    That’s why one should always work in natural units. If you are doing quantum mechanics, set Planck’s constant to 1 (i.e. measure as actions in units of h-bar). If you are doing relativity, set the speed of light to 1, etc, etc. Then, any result is independent of any human conventions, and tells you directly about the physics. As an added benefit you may avoid confusing yourself with non-questions, such as what would happen if the speed of light was lower (answer: nothing).

  3. #3 Uncle Al
    March 6, 2009

    A diverse answer must be a valid answer for the student to be employable after graduation. Welfare ended poverty, health insurance made healthcare affordable, abstinence ended teenage pregnancy and other STDs, Homeland Severity protects you at every airport… derivative mortgage investments created the largest economic boom in the history of the world!

    Disaster descended in every case when social terrorists ignored the mantissa and went looking for the charateristic. Management is about process not product. 27 m^3 is better than the correct answer – it is an answer fit to appear in a corporate prospectus.

  4. #4 Don in Rochester
    March 6, 2009

    Bring back slide rules! I’m (semi-) serious. When I was first doing physics back in the stone ages, that’s what we used, and so we had to know what the range of the answer was; the slide rule was there only to provide the accuracy. Estimation was a very important skill — one that seems neglected when calculators are used . . . .

  5. #5 Jonathan Vos Post
    March 6, 2009

    My late father was a brilliant and wonderful man, but his Math maxed out around the level where he taught navigation as a flight instructor officer in WW II.

    He told me of an exam he took at Harvard in the 1940s (I think probably not the Intro Astronomy he took from Whipple) that had a metal airplane flying at a specific velocity through the Earth’s magnetic field, and asked what voltage would be induced from wingtip to wingtip.

    My Dad blithely ended up with a megavolt, and didn’t (when he wrote that) see a problem with the answer.

  6. #6 Eric Lund
    March 6, 2009

    @moshe: Your approach works well in a purely theoretical framework, but sooner or later you will have to compare your results with experimental data. If you have been too cavalier about setting c = hbar = 1, this can get tricky.

    At least one theorist I know always does his calculations in SI units precisely because he wants to be able to compare his results with experimental data.

    Also, at least in my field, we do have to worry about what happens with a reduced speed of light. The reason is because some simulators artificially reduce c (this procedure is known as the “Boris correction”) in order to keep their simulations numerically stable (the alternative options of reducing the grid spacing or time step are not always feasible–the issue is making sure any waves in the system do not propagate faster than one grid point per time step). These simulation runs can be useful for getting an intuition of what the system will do, but when it comes time to compare with actual data there is always the issue of what is real and what is due to the reduced value of c.

  7. #7 Art Vandelay
    March 6, 2009

    As a grad student grading homework (in a graduate engineering class that deals with a fair bit of modern physics), I see this all the time. I try to make a point of noting comparisons with physical quantities when I get wildly wrong answers. For example, one student calculated the energy of an alpha particle to be 10^27 eV, so I noted that the world’s biggest particle accelerators only get up to TeV energies (10^12).

    Trying to express rest mass in kg is a common mistake; having extra factors of 10^31 flying around in an equation is never a good thing.

  8. #8 dreikin
    March 6, 2009

    “Trying to express rest mass in kg is a common mistake”
    Eh? How so?

  9. #9 moshe
    March 6, 2009

    Eric: if you perform a calculation of a distance in atomic physics, say, just staring at the result in meters doesn’t tell you anything. That’s because meters are a human construct. It is also not the case that you actually go out and measure those quantities using a meter stick, so some (trivial) translation is required in any event.

    I would think that for all those purposes it is easiest if you measure quantities in terms of physical quantities natural for your field, say the Bohr radius in this example. It’s not just a matter of avoiding large or small numbers, it is useful also for example to quantify what results are generic, and what are surprising, etc.

  10. #10 Michael Lugo
    March 6, 2009

    To extend on Brian’s comment (#1): In my corner of mathematics (discrete probability/analysis of algorithms/combinatorics), often sequences of integers a(1), a(2), … arise and you want to know approximately how large the nth term is. For example, the nth Fibonacci number is approximately φn/51/2, where φ = (1+51/2)/2 is the “golden ratio”. The nth Catalan number (another sequence that arises often) is approximately 4n/(π n3)1/2. In general, “many” sequences turn out to satisfy something like

    a(n) ~ p qn (log n)r ns

    where p, q, r, and s are constants. There are deep reasons for this that can’t fully be explained in a blog comment. But what’s surprising is that while p and q are often irrational, r and s are almost never irrational, at least for sequences that arise in the “real world”. Furthermore, they usually tend to be “simple” rational numbers — 3/2, not 26/17. If you told me some sequence of numbers grows like πn I’d be interested. If you told me some sequence of numbers grows like nπ

  11. #11 Mary
    March 6, 2009

    I dunno, I’m of the opinion that it’s a good thing that computers and calculators exist, and are able to perform computations with whatever stupid kinds of numbers you give up them (up to machine precision.) Nondimensionalizing your equations is nice if you’re trying to teach someone something, but if you’re actually trying to calculate something that you’re going to use, I say put everything in SI units and let the computer chug away at it.

    This may give you ugly numbers (but scientific notation makes them prettier — to me 780 * 10^(-9) m reads exactly the same as 780 nm) but it reduces the opportunities for conversion errors, often makes mistakes easier to find, and makes it easier to compare your results with others — no risk of crashing space ships because of different unit systems. Computers were invented to save me the trouble of converting things into weird units, as far as I’m concerned. The odd 10^(-30) doesn’t usually phase Mathematica at all.

  12. #12 Chad Orzel
    March 6, 2009

    I dunno, I’m of the opinion that it’s a good thing that computers and calculators exist, and are able to perform computations with whatever stupid kinds of numbers you give up them (up to machine precision.) Nondimensionalizing your equations is nice if you’re trying to teach someone something, but if you’re actually trying to calculate something that you’re going to use, I say put everything in SI units and let the computer chug away at it.

    That’s risky for a lot of calculations. A lot of numerical methods work a lot more smoothly with numbers of order unity than the sorts of things you get when you keep everything in SI units. The number one failure mode of one of our labs in intro E&M is having Excel choke on the SI-unit version of the fit. Now, granted, Excel is a piece of shit, but I’ve even seen Mathematica choke on simulations where I tried to do everything in SI.

  13. #13 Stephen
    March 6, 2009

    I’ve been reading this blog for a long time, and I know we’re not supposed to feed the trolls, but wow, Uncle Al is particularly incoherent today. I didn’t think the internet could surprise me anymore, but there you go.

  14. #14 Lowly Engineer
    March 6, 2009

    In electrical engineering, we frequently use the same units but always are talking in terms of micro Farads, or milli Volts. On one hand, you could say we’re using different units, but on the other, which I prefer, is that we’re just using shorthand for the order of magnitude. This is somewhat abused, as sometimes large capacitors are listed as being 50,000 microfarads.

    One other issue with being aware of the scale of your numbers relates to numerical (floating point) representations. That is, even if two quantities are represented to 16 digits, if their order of magnitude also differs by anything approaching that amount, you’ve got a possible loss of precision.

  15. #15 Kaleberg
    March 6, 2009

    Lowly Engineer: Yeah, IEEE had an article on why there aren’t any nanofarads, just microfarards and picofarads. They weren’t sure, but there is definitely a real gap.

  16. #16 Ambitwistor
    March 7, 2009

    I remember one assignment where many of my students didn’t pay much attention to what “m” in a heat capacity equation was, and used the mass of an atom instead of the mass of a mole. Their result was off by 10^23 and they didn’t notice. They had no sense of the difference between 10^-23 J, 1 J, and 10^23 J.

  17. #17 Ambitwistor
    March 7, 2009


    Can you point me towards references to the ubiquity of sequences of the form p q^n (log n)^r n^s?

  18. #18 Eric Lund
    March 7, 2009

    That is, even if two quantities are represented to 16 digits, if their order of magnitude also differs by anything approaching that amount, you’ve got a possible loss of precision.

    It’s worse than that: if you are subtracting two such numbers of the same magnitude, you have a possible loss of precision. The fastest way to destroy any precision in your measurements is to compute the small difference between two large numbers. This is why, for example, you should not compute functions like sin(x) or Jn(x) (where the latter is a Bessel function) from their Taylor series, even though it is formally possible to do so, when x is too large: your result depends upon the cancellation of terms with large magnitudes and alternating signs. Or to take another example, try calculating sin(1038) using double precision variables. You will find that you cannot (at least not reliably), because the round-off error in the argument is larger than 2π.

  19. #19 Jonathan Vos Post
    March 7, 2009

    Beyond this issue of natural units where c = 1 (“Forget Quantum of Solace, we would have directly observed the quantum of time,” says Hogan. “It’s the smallest possible interval of time – the Planck length divided by the speed of light.”)…
    15 January 2009 by Marcus Chown
    Magazine issue 2691

    I’d like to dig deeper, and say that there’s a whole theory of Inönü–Wigner contractions, which makes clear how the Galilean group deforms to the Poincaré group as 1/c moves away from 0. This clarifies some aspects of special relativity — for example, how nonrelativistic quantum mechanics deforms to relativistic quantum mechanics.

  20. #20 CCPhysicist
    March 7, 2009

    I hate to belabor what I always thought was obvious, but SI has contained this little concept called “prefixes” for quite a long time. OK, only 20+ years or so for some of them, but if an old goat like me can use them, so can your students. You can (and should) use fm and Tm in SI when appropriate (nuclear dimensions and the astronomical unit). The only things you need that work off to the side of SI are the L and eV. The former is natural, and the latter only requires that you know what e is. A few others require knowing the value of c. (Duh.)

    For example, h*c takes on a particularly simple value if you use eV*nm, and it has exactly the same numerical value in meV*mum, MeV*fm, etc. That scale transcends boundaries between nuclear physics and atomic physics. And since calculators exist that contain these prefixes as an alternative input/output format rather than the antiquated E of FORTRAN and C, you don’t even have to think real hard if you need the answer without the prefix.

    So, in that lab, why not work in cm^3 … and make sure your students can convert between cm^3, L, and m^3 even if they are in a drunken stupor after spring break? Similarly, why not teach your modern physics students WHY you know the Bohr magneton in those “experimental” units?

    The answer to the question @ #8 is that the kinetic energy has been set by accelerating a charge e through a potential in volts. The mass (there being no distinction between mass and “rest” mass) is quite naturally given in units of eV/c^2 so that lots of nasty (and randomly different) powers of 10 aren’t floating around. Calculations are always easier to manage if all of the input data can be expressed in terms of N digit integers where N is small. That is why engineers use “kip” instead of 10^3 pounds-force: to avoid having lots of powers of 10 floating around, trying mightily to make your bridge design collapse.

    Pet peeve: Astronomers who quote everything in cm rather than using Tm or Ym. The only way to learn those prefixes and relevant scale is to use them as intended.

  21. #21 Jonathan Vos Post
    March 7, 2009

    “Astronomers who quote everything in cm rather than using Tm or Ym” — and really, aren’t there roughly 1 mole of stars in the cosmos?

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