By The Metric Maven

Bulldog Edition

When I was a boy, I read a number of books by Dr. Seuss. One that immediately captured my interest was *On Beyond Zebra*. I don’t recall much about the book at this point in my life other than the fact that it involved additional letters of the alphabet. Each new letter was introduced and illustrated by the author. The idea there might be unknown letters piqued my youthful interest. Here are the new letters that appeared in that book:

The first book I ever read by Isaac Asimov (1920-1992) was: *The Universe, From Flat Earth to Quasar*, which I recently re-read. The two books may seem far apart, but I made a connection between them when I came across a section on how the sun generates its energy. The Sun uses nuclear fusion to convert mass to energy. This process is understood using the famous equation *E* = *mc*^{2} developed by Albert Einstein (1879-1955).

The pressures at the center of the sun cause four hydrogen atoms to fuse into a single helium atom. After this process occurs there is a mass imbalance, the four hydrogen atoms have more combined mass than the resulting single helium atom, and the extra mass is converted into energy.

Dr. Asimov states that about 4.2 Tg (Teragrams) of mass is converted to energy every second inside of the sun. He uses pre-metric terms to describe this value as “4 600 000 tons of mass per second.” Unfortunately so does Wikipedia: “the Sun fuses about 620 million metric tons of hydrogen each second.” As I understand it 1 million is 10^{6} and a “metric ton” is a Megagram or 10^{6} grams for 4.2 x 10^{12} grams per second or 4.2 Tg per second. That’s a lot of grams. Dr. Asimov inquires: “Is it possible for the Sun to support this steady drain of mass at the rate of millions of tons per second? Yes, it certainly is, for the loss is infinitesimally small compared with the total vast mass of the sun.” The currently accepted mass of the sun is, approximately 2 x 10^{30} kg. This means it’s 2 x 10^{33 } grams, and the proper metric prefix would be?—oh, well, there isn’t exactly a metric prefix for this value. The last magnifying metric prefix is Yotta, which allows the mass to be written as 2 000 000 000 Yg (Yottagrams). Which by current convention it appears there are about three extra metric prefixes needed to express the mass of the sun with a 2, and a minimum of two extra prefixes to use 2000 as a magnitude.

So what does “infinitesimally small” mean? Well the mass lost each second, divided by the total mass of the sun, is 4.2 x 10^{12} grams/2 x 10^{33} grams. This value is one divided by 476.19 x 10^{18} or 0.000 000 000 000 000 000 002 which is quite a tiny ratio. I believe this is indeed a small enough ratio to be “infinitesimally small.” Recall we are talking about 4.2 Tg per second of mass loss. Each gram has 90 TJ (Terajoules) of energy contained within it’s mass. If my computation is correct, then 378 x 10^{24} joules are released each second. This would be 378 YJ (Yottajoules) per second. We are approaching the limits of the metric prefix Yotta, and in only 1000 seconds we would have 378 000 YJ and see that a new prefix *might* be useful to describe the power released.

What is notable is that the mass of the sun is not readily expressed with a metric prefix, and it’s not all that massive for a star. It appears that the masses of stars are indeed astronomical. The most massive star is suspected to be R136a1 which is approximately 256 solar masses (a solar mass is the mass of the Sun). This means it has a mass of 512 x 10^{33} grams or 512 000 000 000 Yg. Clearly we are *on beyond Yotta* at this point. While I’ve made it clear in the past that astronomical distances are readily expressed with metric prefixes, this is not the case for stellar masses. One can see why R136a1 is described in terms of an equivalent number of solar masses and the metric system is not employed.

Asimov also makes this surprising statement:

Release of energy is always at the expense of disappearance of mass, but in ordinary chemical reactions, energy is released in such low quantities that the mass-loss is insignificant. As I have just said, 670,000 gallons of gasoline must be burned to bring about the loss of 1 gram (1/27 of an ounce). Nuclear reactions produce energies of much greater quantities, and here the loss of mass becomes large enough to be significant.

What I’ve been able find in my research on this subject is both minimal and contentious. It is mostly stated that the amount of mass lost in chemical reactions is “unmeasurable.” The few who venture to put numbers to paper (including a textbook example) end up with magnitudes on the order of 10^{-33} grams. One example computation has 70 x 10^{-33} grams as the amount of mass lost in the given chemical reaction. This would be 0.000 000 070 yg (yoctograms) and would indicate a possible need for at least two more metric prefixes. It appears that, at least in theoretical discussions, it *might* be useful to have two more metric prefixes on the dividing side of the prefixes.

Currently there are 20 metric prefixes from yocto to Yotta. Adding two more prefixes on the magnification side would be useful for some of this astronomical work. It would probably make sense to add a pair to the reducing prefixes also. This would increase the total number to 24 metric prefixes. This is a lot of prefixes, but is far less than the number of magnitudes scientific notation would allow, which would be 60. What I would propose is to *consider* adding the new prefixes, but at the same time remove the prefix cluster around unity: deca, hecto, deci and centi. They could be separated and relegated into a set of *atavistic prefixes* which are no longer considered proper modern usage. They would be included as an appendix to the modern prefixes for historical reference, but discouraged for modern use. This simplification would reduce the number of prefixes back to 20 and also provide a larger dynamic range for scientific description.

In early grades it makes sense to me that only the prefixes micro, milli, Kilo and Mega would be taught as the *Common Set of Prefixes*. These would be the prefixes that students would generally encounter in everyday life (if the US was metric and fully engaged). In Junior High and High School the new set of prefixes I’ve proposed could be taught as the *Complete Set of Prefixes*. I would argue that all students (and their teachers) should have to memorize and use all these metric prefixes (without the prefix cluster around unity) in their instruction. Textbook authors should not shy away from using Megameters for planetary dimensions, Gigameters for the solar system, and all the other appropriate uses of metric prefixes.

People have objected to my proposal that we teach all students to use all the metric prefixes. They employ the argument that the Common Set of Prefixes is all that is needed for an ordinary person, and the Complete Set of Prefixes is for engineers, scientists and technical people. I reject this view entirely. It produces a scientific apartheid that keeps the public from understanding the important issues of the day, which involve engineering and science more and more everyday. What I have discovered when working with large questions, such as how much the salinity of the ocean would change if we dumped all our fresh water into it, or how much carbon is being belched into our atmosphere over a given period of time, is that these problems are tamed using appropriate metric prefixes. They allow an ordinary citizen to comfortably work with the magnitudes involved. If one talks about hundreds of billions of tons, that is a metaphor, and is not information. If the goal of education in the US is to create the most numerate population on the planet, then a good command of the magnitudes of all the metric prefixes is essential.

I would like to see a song which fixes the order of the metric prefixes in a person’s mind from the smallest to the largest, something similar to Tom Lehrer’s Element Song. Some manner of meaningless acrostic or other method of recalling the order of the 1000 based prefixes should also be developed. With the prefix cluster around unity eliminated, all the magnitudes will be of 1000 and any parsed base unit can be determined. This would allow anyone to look at 1 000 000 000 000 000 grams and immediately relate it to the acrostic or song and “sound out” the size of the number as 1 Pg (Petagram), or conversely be able to take the 1 Pg and work out how many sets of three zeros one would need to express it. This would also be the case for 0.000 000 000 000 001 grams. It could be “sounded out” as 1 fg (femtogram).

When all the metric prefixes no longer apply, that’s when a modern student should viscerally realize they are discussing dimensions that are so large or so small they are mind blowing, and on beyond yocto and Yotta. These values truly exist in an amazing far distant realm.

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You “yotta yotta’d” over the best part!

Sorry for the lame Seinfeld joke!

I attended your astronomical metric talk with the Humanists of Colorado recently, and was stunned by the number of metric prefixes I already didn’t know. But adding more sounds like the right move to take in to account these kinds of measurements. Finally, I do think teaching a common and complete set of metric prefixes makes a lot of sense. Even if not memorized completely, knowledge of them would help people understand the things they should be seriously concerned about. Thanks!

The way i see it is, the level of numeracy in our country is so low, I don’t think these prefixes would really help.

If you must have more prefixes, there should be one for each power of 1000 up to 1000^10 (that is, 10^30), and from there we can stack prefixes.

Personally, I think basic arithmetic is a much higher priority. I would much rather have a populace that can add and subtract without pulling out their smartphone than one that can rattle off every prefix known to man.

Throughout my years of learning, I have experienced subjects that require both simple and complex understanding in order to be truly learned well. The SI is one of those subjects, and I agree with you wholeheartedly, Maven, that BOTH common and complete sets of SI prefixes should be mastered. The SI should be appreciated for its full power, i.e., that it has both plebeian and patrician aspects, that there is a time for a kilogram, and, fer shur by gum, there, good buddy, time to ditch the improper metric ton in favor of the proper SI megagram!!! Now, THERE is a piece of housekeeping in our metric lexicon that is long, long overdue. Get rid of the frikkin’ “ton”nomenclature, and embrace the SI all the way! Calling something a tonne or a metric ton is a throwback to ancient metrology, and we are not ancient now.

“Boil that dust speck!”

Maven, have you considered that there might be an alternative solution to this problem? The SI is already full of disparate prefixes, all of which must be learned separately, by and large bear no phonetic resemblance to their respective degrees of magnification or diminution, and have potentially confusing abbreviations – as someone who works in the medical field, I shudder to think about the number of times that patients prescribed 50 micrograms of a drug get 50 milligrams instead, with disastrous consequences. Even though I use computers a lot, I cannot work out from the prefixes used exactly how many bytes of information can be stored in my new hard drive (is a gigabyte 1 000 000 000 or 1 000 000 000 000 bytes again? I had to hesitate.)

It strikes me that adding new prefixes to the SI because the current range is insufficient misses an opportunity to overhaul the system entirely. Have you ever come across Systematic Dozenal Notation (SDN)? This is a system widely used by people dabbling in base-12, and is the exclusive system used in well-developed base-12 rivals to the SI, such as the TGM system. It is of course equally applicable to the SI’s decimal units. SDN simplifies both magnification and diminution prefixes. Instead of writing “kilo” for 1 000, or “mega” for 1 000 000, you simply prefix the unit with “qua-” to indicate magnification, and then prefix the power to which the base is raised. To avoid confusion, the powers are written with the standard elemental numeral names used by IUPAC. For example, a kilometre, or 1 x 10^3 m, could be written as one triquametre. A megagram/tonne would be a hexquagram. The mass of the Sun is simply 512 tritriquagrams (512 x 10^33 g). For diminution, simply swap in “cia-” for “qua-“. A millimetre would be a triciametre (1 x 10^-3 m), and so on.

To me, this is a simple and elegant solution that can clearly express any power a physicist – no, wait, anyone at all – could possibly need, no matter how large or small. Instead of converting constantly from kilo to giga to tera (as one might today convert between inches, feet, yards, and miles), one simply slides up and down a scale that measures one thing – mathematical power – just as the SI measures every distance, from the subatomic to the intergalactic, using the humble metre.

Food for thought?

Wow! That makes perfect sense and it really is a clean and organized solution to prefixes. I can’t tell you how many times when dealing with large numbers that I count the zeros and wish there was a way of just shortening it out and you did just that. Fascinating.

Certainly more rational than the current SI prefixes, at least from a theoretical point of view…

But how should, for example, the unit km (or Km, à la Maven) be written in real life, e.g. on road signs…? 1 km = 1 3qm? Of course, in this case, the proximity of two numbers is not so optimal for readability. Or 1 km = 1 tqm (or 1 Tqm)? But in this case we return to some form of unique prefixes, each one requiring a letter, as it is today.

The “3” in 3qm could perhaps be written in small type, as a prefixed exponent (thus also implying a hidden “10^”): but, again, not so optimal for readability.

But certainly the SI needs to be overhauled in many ways, sooner or later, in order to make it future-proof…

I don’t see the point of defining more than yotta honestly. Prefixes are useful up to a point. 10^33 or 10^-33 is perfectly fine to use in literature and more importantly when you are actually working with the numbers 2.1342×10^33 m is an easy measurement to understand. Why do we need a prefixs really other than for simple shortenings for quick reading. I get the k,M,G,T prefixes as well as the diminutives but those are more for convenience for quick reading. When I do work I always use 10^3m or 2×10^6 W. Watts are annoying as many things are measured in kW so it requires a final conversion at the end i.e. 2×10^3 kW.