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