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 10^{23} 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 10^{3} to 10^{24}. The current set of magnitude reducers is also eight which covers 10^{-3} to 10^{-24}. The entire range is 10^{-24} to 10^{24}. 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 10^{9}. 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}, 10^{3}, 10^{6}, 10^{9}. 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:

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 10^{20} joules 800 Exajoules

2) 1 x 10^{21} joules 1000 Exajoules

3) 8 x 10^{20} joules 800 Exajoules

4) 1.75 x 10^{21} 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 10^{18}. 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 10^{100} 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:

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.

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Updated 2014-05-30

Related essay:

I agree with you on the usefulness of engineering notation and its relation to metric prefixes. I know it wasn’t taught in college when I was there; I never heard of it until I bought a HP-45 several years later. It SHOULD be taught as a variant of scientific notation that ties engineering notation, metric prefixes, and engineering calculations. (NB: But beware of mass, where the base unit is kilogram.)

I think scientific notation is mostly a holdover from slide rule days. The slide rule gave you 3-4 significant figures but you had to do order of magnitude in your head. Most science classes either review the metric system or assume the student knows it; but it is probably taught poorly in the review.

To nitpick at your table, I think some of the fossil fuel values are wrong, but I blame the source table. However, on natural gas, you may have computed value per cubic meter, not per kilogram. A DOE table that I frequently refer to gives 52 MJ/kg (it also gives BTU per pound, cubic foot, or gallon, but at least it includes metric as an afterthought). Tables are a specific exception to the “rule of 1000” and I would have used 0.211 MJ/kg for the batteries as I think it gives a better comparison. The uranium is so far out of range I might have used 81 TJ, but that is a judgment call. However, I would NOT have used commas in writing 81 000 000. I don’t get upset by commas in financial numbers as they are not SI, but it is WRONG to use them as thousand separators when using SI; space or thin space!! Since there are multiple commas, I suppose I know they aren’t decimal commas, but this is a potential point of serious confusion in International articles.

As a final thought, some sources strictly recommend unit symbols with number symbols (Arabic numerals) and spelled out units with spelled out number words, e.g. 20 MJ or twenty megajoules, but never mixed. Of course, the AP Style Guide generally hates symbols, assuming their readership is innumerate, and that is only one of many areas where I disagree with them on metric, but why journalists do such a bad job on science articles.

To follow up on what the Maven wrote about exa- and what John wrote about in the last paragraph of his posting, consider this from the lead article in this past Tuesday’s ScienceTimes in the NYTimes, a good article on H. Sebastian Seung’s work on brain cells:

“…would equal a zettabyte, a term only recently invented when the amount of digital data accumulating in the world required new words. It equals about a trillion gigabytes, …”

A term only recently invented? Hasn’t the prefix zetta- (Z) been in use for over 20 years? Apparently, and unfortunately not surprisingly, the NYTimes is not concerned with the benefits of metrology prefixes as the Maven describes well in his essay here.

Furthermore, going a step further in line with the Maven’s last paragraph, the headline of the Times’s article was “All Circuits Are Busy”, with subheadline “85 billion brain cells, …”

Thus, maybe one day we’ll see something like “85 gigacells of the brain”, and most readers will immediately understand such.

Unrelatedly, since John brings up that AP Style Guide, it should be pointed out that according to such writing “the Maven” would be considered taboo. Afterall, this Guide has been telling us for decades that “the President” should be written “the president”, which, unfortunately, is “style” the NYTimes also adopted about 15 years ago…

Finally. Maven, something like 0.211 megajoules should be singular.

The energy table should not list each item in the list as being “kilogram of X”. The first column should just name the item, and then the next column should express the specific energy as J/kg.

Maybe scientific notation would be easier to read if it were “big-endian”, placing the power of ten before the significand, as 10^9 * 7.2 for the approximate number of humans now on Earth.

Before HTML and Unicode, computers and calculators commonly replaced “x10^” with “E”, i.e., 7.2E9 persons. Some used a space before the E, some didn’t.

Getting back to scientific vs. engineering notation, I like the engineering for its relationship to metric prefixes. However, there is one advantage to scientific. Looking at one of the Maven’s examples, 8 x 10^20 J vs 800 x 10^18 J (800 EJ), (sorry, I don’t know how you did the superscripts) the scientific is very clear on its implied precision, one significant figure. Is 800 rounded to the nearest unit, ten, or hundred? Engineering usually involves three or more significant figures, so not the end of the world, but slight advantage to scientific.