Lies, Damned Lies, and Scientific Notation

By The Metric Maven

Bulldog Edition

My first extended use of scientific notation was in my introductory chemistry class, Chem 147, at a large university. Anyone who has taken a chemistry class will certainly remember 6.022 x 1023 as Avagardro’s number. It appears to be nice, compact and expressive. I was certainly beguiled with scientific notation as I watched my professor manipulate magnitudes large and small with apparent ease. The numbers would cause one to think, “that’s really big”, or “that’s really small.” It all seemed so nice and orderly—and useful. The one question I did not ask was: “what is the actual relation of the magnitudes expressed on the blackboard to the physical world?”

What happened next did not really provide an answer, but introduced another question. I had purchased a Hewlett-Packard RPN calculator, which has a beautifully written manual. It was while perusing the manual that I ran across settings for the display. I could have fixed, scientific and engineering notation displayed. Engineering notation?—I had not heard of it. It was never mentioned in High School. I then saw a small table which showed it. Scientists use the term order of magnitude to describe a power of ten. This is the superscript used in scientific notation. Engineering notation had categorized scientific notation into groups which differ by three orders of magnitude. It automatically eschews the prefix cluster around unity (i.e centi, deci, deca and hecto). In a millisecond I was enamored with engineering notation, but it would be many, many years before I really understood the beauty and utility of it.

The current set of three magnitude multipliers is eight. This group spans a range from 103 to 1024. The current set of magnitude reducers is also eight which covers  10-3 to 10-24. The entire range is 10-24 to 1024. When this range is expressed using scientific notation there are 48 different and separate magnitudes to wrap one’s mind around. With engineering notation it is 16. What is incredibly useful when one uses engineering notation, is the seamless integration of linguistic expression and numerical expression.

Let me back up for a moment, and explain what I mean. When I was taking classes on circuit theory we used a number of components, the most prominent being resistors. Resistors are marked with colors, and the colors are coded for resistance values. I recall looking for a specific resistor I needed in my small stash and slowly working out the color sequence from a chart. My professor looked over my shoulder and said here, and picked one out immediately. I asked how he figured it out, and he said “I don’t, I just see the colors and I know.” It seemed impossible, but within two years—I could “see” the colors as numbers—and seldom needed a chart.

Within a couple of years I had a similar experience with engineering notation. When I would see 11.5 mm it would immediately be punched into my calculator as 11.5 x 10-3. If I saw 1.575 GHz it would go into my calculator as 1.575 x 109. The language designation and the numerical magnitude were indistinguishable. Without thought, my mind knew the equivalence between the prefixes and numbers. The prefixes milli, micro, nano, kilo, mega and giga required no thought, they were 10-3, 10-6, 10-9, 103, 106, 109. The engineering prefixes meld literacy and numeracy. Scientific notation has no such general linguistic equivalents. They are barren in providing an idea of their size using compact language. The use of scientific notation actually obscures numerical comparison. Here is an example from an article about Global Warming from New Scientist; it expresses four possible scenarios with respect to the amount of energy we use:

click to enlarge

Let’s compare the scientific notation values from the article with engineering notation using Pat Naughtin’s Whole Number Rule:

Global Energy Use:

Scientific                     Engineering

1)   8 x 1020 joules            800 Exajoules
2)   1 x 1021 joules          1000 Exajoules
3)   8 x 1020 joules            800 Exajoules
4)   1.75 x 1021 joules     1750 Exajoules

Which column provides you with a better numerical “feeling,” as well as the ability to directly express the size of the number involved, as a number? If we lived in an effectively metric and numerate country, every pupil in grade school would have been taught, and know, that Exa is 1018. Despite living in non-metric America, I’m sure they’ve probably heard of Exabyte drives.

The use of engineering notation allows for a nice continuum of numerical expressions, which are immediately expressible in words alone—yet express an exact numerical magnitude. Scientific notation promotes unit proliferation. For many years, light was expressed in angstroms. One would have to recall that an angstrom is 10-10 meters. There is no metric prefix. There is no clue in the word angstrom as to what  its magnitude might be. In recent years the angstrom has been thankfully abandoned and light is generally expressed with nanometers, which I immediately know is 10-9 meters, just from the prefix nano. There are cases where values which are outside of the range of SI notation appear in engineering and scientific research work. Scientific notation alone must be used for this work.  But in all these cases it should remain without prefix designation as a value in scientific notation.  No googol for 10100 or logoog for 10-100 period. If they’re too big for SI one should leave them unnamed—until they are officially.

I came across the notes for a university course on the environment when I was searching to find out how many joules of energy are in a given quantity of gasoline. I came across this table which has the answer:

Energy Unit  ————————————————————–  Joules Equivalent (S.I.)

This is a clear example of an instructor believing that scientific notation allows for a meaningful comparison of values. Your friendly neighborhood Maven sees exactly the opposite. This person does not use any SI units in the “Energy Unit” column. There are gallons, pounds, tons, a barrel (which is in reality actually statistical), and a cubic foot. Not one metric unit appears on the left and only scientific notation is found on the right, with joules assumed for the column. I have sympathy for the instructor. This is a difficult set of numbers to express because of their large dynamic range, but metric prefixes can help considerably, and should have been employed. I will reorder the table, change it to metric in the left hand column, and compare a one kilogram mass of each substance:

Alternative Table — (click to enlarge)

I believe the use of increasing energy content is a good way to compare these energy sources. The new table shows that the energy density of the majority of the substances we use to supply energy have similar magnitudes. Five of the seven entries are from 20-53 megajoules. The large amount of energy in Uranium-235 is clearly evident when we keep the megajoule prefix, although it’s a very large number. The kilogram of AA batteries is only 0.211 megajoules. When we start with kilograms, the “conversion” of coal and uranium entries to megagram quantities (i.e. “metric tons”) may be done in one’s head. When pounds are used, one needs to multiply by 2000 to obtain a short ton and 2240 to convert to long tons. There is no designation within the table that explains which ton is used. Clearly in a course about environmental concerns, the most efficient and succinct way of presenting energy numbers is desired. The use of a mixture of units and scientific notation obscures the fact that most of the substances have similar energy content.

I see very poor expression of numbers and numerical data in popular science magazines, technical papers, and the worst, mass media publications. When I took English, I demonstrated a complete inability to internalize its grammatical intricacies. What I was told by the instructor was that the worst use of grammar could be found in newspapers and magazines. One could see the irony that these were people making a living using the English language, ostensibly professionals, who used poor grammar. When I first realized how badly numerical data was often presented in technical papers, I had a kinship with my English teacher that I had never experienced before. The lack of the metric system in the US, stunts the ability of professionals to express numbers in a cogent manner. The lack of the metric system in the US prevents teachers from instructing children at the earliest age possible about how to use metric prefixes, as they will not experience them in this society. The lack of the metric system in the US encourages the continued overuse of scientific notation which is an opaque way to express numbers. The lack of the metric system helps to keep us innumerate.

Updated 2014-05-30

Related essay:

The Expanding Universe

NIST: The Metric Cheese Shop

By The Metric Maven

Extra Bulldog Edition

On Sunday, February 7, 1904, a fire began in Baltimore. It would take 1,231 firefighters to bring the fire under control and when it was over 1,500 buildings would be destroyed. One reason the fire burned unchecked for so long was the absence of national standards for fire-fighting equipment. Fire engines from Philadelphia, Washington D.C., Atlantic City, New York City and other metropolitan areas arrived on the scene. Unfortunately, many fire departments were unable to help as their hose couplings would not fit Baltimore’s fire hydrants. Those fire-fighters could only watch as the fire engulfed more and more of the city. It has been claimed that over 600 different sizes and variations of fire hose couplings existed at the time. This was similar to what French Engineer Charles Renard encountered with balloon cables, which caused him to develop preferred numbers.

The National Bureau of Standards was founded in 1901. Two metrication advocates championed its creation, James H. Southard, and John Shafroth. The Great Baltimore Fire directly demonstrated the need for mandatory standardization of fire-fighting equipment. Furthermore, no standards for building construction (building codes) existed, which allowed the fire to rapidly spread.

One would think that with the lessons of the Baltimore fire, and the establishment of a government agency for standards, that soon fire departments across the country would be induced to adopt national standards for fire-fighting equipment.  On March 22, 1975 a fire started at Unit one of the Browns Ferry Nuclear Reactor.  Plant employees attempted to extinguish the fire despite the fact that professional firemen from Athens, Alabama were on the scene. They mistakenly believed there was a problem with a nozzle at the end of a fire hose. This in turn caused the employees at the plant to request a replacement nozzle from the Athens fire department. The threads on the the fire department’s nozzle were not compatible with those of the fire fighting equipment purchased by Browns Ferry.  Because of this, the nozzle would not stay on the end of the hose.

Well, certainly by now, well over a century after the founding of NIST, we would have national standards for fire couplings and this would not be a problem right?  According to Wikipedia:

A national standard for fire hydrant and hose connections was adopted by the National Fire Protection Association. However, inertia remained, and conversion was slow; it still remains incomplete. One hundred years after the Baltimore Fire, only 18 of the 48 most populous American cities were reported to have installed national standard fire hydrants.[18] Hose incompatibility contributed to the Oakland Firestorm of 1991: although the standard hose coupling has a diameter of 2.5 inches (64 mm), Oakland‘s hydrants had 3-inch (76 mm) couplings.[19]

The idea of standardization strangely seems to be at the bottom of the priority list of many engineers. Those who have seen the movie Apollo 13 were reminded that the the carbon dioxide scrubbers for the Command Module and the LEM were not compatible. Fortunately, they were able to engineer their way to compatibility with a duct-tape solution. One should not rely on good fortune instead of planning and standardization, but in the US hoping for good fortune appears to be the standard back-up plan.

The lack of standardization in the US can and has cost lives. The acronym NIST stands for The National Institute of Standards and Technology. A year ago on May 24th 2013 (2013-05-24), on the Friday before Memorial Day weekend, a time when bureaucrats know that news media is generally not paying attention, the Director of NIST, Patrick D. Gallagher, penned a response to a citizens petition requesting that the metric system be adopted as the sole measurement system in the US. His response can be succinctly stated as he supports a “do your own thing” approach to standardization. Standardization is just too confining of a concept for a standards institute to embrace apparently. The title of his response, in case readers have forgotten, is Supporting American Choices on Measurement.  It is well known to metric advocates that 95% of the world’s population uses the metric system. It would appear from just a cursory inspection of this  fact, that one could, with reasonable certainty, state that the metric system is probably the most successful standard in the history of humanity.  The director of the US government body which is tasked with standards, cannot even agree with a petition that the metric system should be the standard of the US?

When one is confronted with Dr. Gallagher’s assertion that the best standard is a lack of standards, and  I remind you he is the director of the standards body of the US, one’s mind can only interpret the strange dark and contradictory humor of this apparently willful cognitive dissonance in but one way—–by resorting to a Monty Python Metaphor. One of the most famous of the Python’s sketches is The Cheese Shop. A patron walks into a cheese shop and requests some cheese. He requests all different manner of cheeses one by one, red Leicester? Tilsit? Caerphilly? Bel Paese? Red Windsor? Stilton? Ementhal? Gruyere? Norweigan Jarlsburg?….. These requests continue ad nausium until finally:

Mousebender Well let’s keep it simple, how about Cheddar?
Wensleydale Well, I’m afraid we don’t get much call for it around these parts.
Mousebender No call for it? It’s the single most popular cheese in the world!
Wensleydale Not round these parts, sir.

The exchange continues as the patron continues to request cheese after cheese until finally he states:

Mousebender It’s not much of a cheese shop really, is it?
Wensleydale Finest in the district, sir.
Mousebender And what leads you to that conclusion?
Wensleydale Well, it’s so clean.
Mousebender Well, it’s certainly uncontaminated by cheese.

I could see a similar exchange with the Director of NIST acting as a standards proprietor where one could request mandatory metric industry standards for fire hose couplers, foot measurement, wire sizes, drill bit sizes, sheet metal thicknesses, medical weights and heights of humans, over the counter medical dosages and on and on. Each time the Director would parrot back “no.”  And when one states “it’s not much of a Standards Institute is it?” this phrase might be met with “finest in the US sir.” Indeed, NIST appears to be quite clean, and uncontaminated with metric standards for the US. As in the sketch, the most popular world measurement standard, which is metric (aka Cheddar) is to be found nowhere as a standard in the standards shop.

It is hard to take NIST’s assertion that it is a standards institute seriously when it promotes the notion that a lack of standards is of exceeding utility to the US, and serves as an illustration of  what makes our nation great. NIST is a Metric Cheese Shop, with no Cheddar, and it is completely uncontaminated by cheese as far as I can tell. It is sad that a scientific standards organization has been turned into a worldwide metric joke. At least the Python players had much better writing, and were actually funny while making important points. Patrick D. Gallagher’s response last year was so feckless, it was almost a killer joke to metric advocates. Now stop me if you’ve heard the one about the 600 choices of hose couplings available to the Baltimore fire department.

Postscript:

In 2012, I wrote an essay entitled Feral Units Endanger Our Health. In it I detailed the well known problem of the confusion between teaspoons and tablespoons. I pointed out that confusion between the two units can lead to a 3:1 or 1:3 dosage mistake. I then cited a column from JAMA, The Journal of The American Medical Association, dated September 20, 1902 (page 712), which is reproduced here in the upper left. The 1902 JAMA column advocates for mandatory implementation of the metric system through the Shafroth Bill. It was brought to my attention (thanks Dr. Sunshine) that just two days ago (2014-05-21) JAMA published a column which yet again addresses the same issue over 111 years later. The new column is entitled Group Urges Going Metric to Head off Dosing Mistakes and is authored by Bridget M. Kuehn (pp. E1-E2). The article opens with modern day examples of the problem:

The article goes on to state that “about 3000 to 4000 children are treated in emergency departments each year as a result of medication errors by a caregiver. Poison control centers in the United States also field approximately 10 000 calls each year about dosing confusion,..”

It has been said that a working definition of insanity is to do the same thing over and over and expecting a different outcome. This apocryphal quotation danced in my mind as I read “The CDC worked with the US Food and Drug Administration and the Consumer Healthcare Products Association to develop voluntary guidelines that were published published in 2011.”  Our current answer to all measurement problems in the US is to adopt voluntary guidelines.

The suggestions are all mostly reasonable, such as including a dosage device with medication which does not use “unusual units” (I guess they mean metric?), adding zeros before decimal points (they could also adopt the whole number rule), and “dosing devices that are not substantially larger than the largest recommended dose of the medication.” Further it is stated:

The CDC recommends using only milliliters as a measure for liquid medications to avoid confusion between teaspoons and milliliters and avoiding relatively unfamiliar measures such as drams (a holdover from apothecaries). The CDC wants the dosing device with the appropriate unit of measurement included with the medication to avoid caregivers using a kitchen spoon or other implement that uses a different unit of measurement. Further, the enclosed device should only have the recommended doses labeled on it to make it even easier and safer to use.

 The ISMP (Institute for Save Medication Practices) goes further than the CDC recommendations and argues for expressing a patient’s weight only in kilograms. The “ISMP, explained that because there are 2.2 kg per pound [sic], switching back and forth can lead to 2-fold errors in dosing of medications by weight.”  Once again, in an echo of the 1902 JAMA column they point out that over the counter medications need to conform to these voluntary recommendations. The article also argues against the use of dual-scale dosage devices.

The article goes on:

Stephen C. Mullenix, RPh, senior vice president of public policy and industry relations at NCPDP, said the white paper is “a call to action” for pharmacists to make sure dosing is correct. They can verify with the prescribing physician to ensure they understand the dosing for a particular drug.

I’m sure the authors of the 1902 JAMA column also saw their words as a “call to action.” The big difference between then and now is the Meyer Brothers backed John Shafroth’s bill for mandatory metrication.

The article ends with the problems encountered when using electronic prescriptions. The example cited is of a doctor prescribing in milliliters and when it arrives electronically, the pharmacies software has a default setting to teaspoons. The article ends with a familiar modern refrain:

Converting all dosing and patient weights to metric is going to take time, Cohen acknowledged. But already he noted that soda cans and many other types of packaging already use metric units and that people will learn the conversions over time. “This isn’t something that is going to happen overnight,” he said.

The lack of mandatory metrication in this country is making people sick, costing our economy financially, and showing us for what we are, a nation that is willing to sacrifice people on an altar of ideology rather than acknowledge and engage with reality. Given our history, I suspect that in another 100 years we may still be waiting patiently for these voluntary recommendations to adhere.

Meyer Druggist April 1922