Expanding The Metric Vocabulary

By The Metric Maven

My Grandfather in Montana read an amazing number of books in his lifetime. Many of them were science and science fiction. He was one of the last of the US blue collar autodidacts.  A spare bedroom contained the books he had finished reading. There was little room to move, and the books were stacked almost from floor to ceiling in open half-high cardboard boxes. This arrangement placed the spines upward, which made the titles easy to read. He often gave books he read to charities, and always was generous with them. I could take any of the books I wanted. There was one exception.   My Grandfather had a very small shelf where he kept his favorites, with which he refused to part, come hell or high water. One I recall was Isaac Asimov’s Only a Trillion. I inherited another, which I believe made it to his 500 mm long literary shrine, it is Mathematics in Everyday Things by William C. Vergara (1923-1994). Vergara was an Electrical Engineer.

I ran across my Grandfather’s fragile paperback copy recently, and began to page through it. The book has short questions and answers. One which attracted my attention is: “Why is the color of incandescent light different than sunlight?” This caught my fancy because electromagnetism (EM) is my specialty, and also because it made me think of James Clerk Maxwell (1831-1879). It was Maxwell who first explained light mathematically, and suggested it be used to scientifically redefine the meter, which it was in 1960. Light is a wave which travels at approximately 300,000 kilometers per second. These waves are often explained with a simple wave diagram—even though they are more complex than this. Below is Figure 43 in Vergara’s book:

Vergara shows the number of wavelengths which pass by a stationary observer in one second. The diagram shows a wavelength of one foot—sigh. The frequency of this wave is 984.25 MHz. Had he chosen 300 mm for a wavelength, this would have produced a value nicely rounded to 1000 MHz for its frequency. The higher the frequency of a wave, the shorter its wavelength. Light waves are usually expressed in wavelengths and not frequency. Vergara then gives a table which details the wavelengths of different frequencies of EM waves:

I suspect some of you are cringing at the way the table describes lengths of different frequencies of light. I’ll get the complete horror over quickly, I will show you the next table, which gives the perceived colors of the rainbow and their wavelengths:

I’m sure by now you are expecting me to chastise Vergara for his sloppy use of the metric system. Clearly you probably expect me to first rewrite the first table using metric prefixes kilometers, millimeters, micrometers, nanometers, picometers and femtometers thus:

Radio, television, communications…….1000 km to 10 mm
Infrared……………………………………………300 µm to 760 nm
Visible Light……………………………………..760 nm to 400 nm
Ultraviolet………………………………………..400 nm to 13 nm
X-Rays……………………………………………10 nm to 10 pm
Gamma rays……………………………………100 pm to 500 fm

And you would be partially right.

One might have tried to spell them all out, as some readers might not be familiar with the prefix symbols.

Radio, television, communications…….1000 kilometers to 10 millimeters
Infrared……………………………………………300 micrometers to 760 nanometers
Visible Light……………………………………..760 nanometers to 400 nanometers
Ultraviolet………………………………………..400 nanometers to 13 nanometers
X-Rays……………………………………………10 nanometers to 10 picometers
Gamma rays……………………………………100 picometers to 500 femtometers

This looks rather clear for a popular audience. Another option is to not use metric prefixes and instead express all the values in meters, multiplied with appropriate engineering notation power of ten exponents. Honestly, I think spelling out the prefixes probably works best for a popular audience—or even an engineering or scientific one. One tends to look at the mantissa (significand) and the exponent is then later noted. This tends to obscure the interpretation of the magnitude of the values presented:

Radio, television, communications…….1000 x 103 to 10 x 10-3   meters
Infrared……………………………………………300 x 10-9 to 760 x 10-9  meters
Visible Light……………………………………..760 x 10-9 to 400 x 10-9  meters
Ultraviolet………………………………………..400 x 10-9 to 13 x 10-9    meters
X-Rays……………………………………………10 x 10-9  to 10 x 10-12    meters
Gamma rays……………………………………100 x 10-12 to 500 x 10-15   meters

The second table with perceived colors and their range of wavelengths, might be better shown in a modern way as:

Color                              Wavelength in nanometers (nm)

Violet…………………………….. 400-420
Blue………………………………. 420-490
Green……………………………..490-570
Yellow……………………………..570-590
Orange……………………………590-650
Red…………………………………650-760

Certainly the tables as originally written are not very clear, or cognitively easy to access, but it’s probably not Vergara’s fault. So why is it that I’m not blaming Vergara for the incredibly poor use of metric prefixes, crazy decimal expressions, and millionths of a centimeter?—and longtime readers know how much I dislike the centimeter. Because Vergara was restricted in his vocabulary of accepted prefixes. The copy of the book I have has a 1959 copyright. In 1959, the prefixes micro, nano, pico, and femto were not officially accepted as SI prefixes. The first three would be adopted in 1960 and femto added in 1964. Americans, fixated on the pseudo-inch British version of the metric system, known as the cgs system, latched onto the centimeter as a replacement inch, whether it was a good idea or not, and shoehorned it in. This remains far too prevalent in the US and is counterproductive.

Even though the prefixes had not been officially accepted, clearly there was some usage of micro by Vergara, but not the best use. On page 270 of his book Vergara has:

We think of one sound being so many times as loud as another, whereas, we would be hard put to say that the former is 43 micromicrowatts per square centimeter more intense than the latter.

Yes, he used micromicro (µµ) as a prefix which is the same as picowatts. Pico had not been accepted just yet either. Electrical engineers of this time period even had a slang term for micromicro, it was mickey mouse. Vergara’s first table of approximate wavelengths even had another option, which I’m glad he did not exercise: the angstrom. An angstrom Å is 10-10 meters, or one ten-billionth of a meter. This completely does not fit with the 1000 or 103 separation of modern metric prefixes, and destroys any logical consistency. Throwing angstroms into the mix has the potential to make the table even worse. Its usage is now discouraged, and I discourage it also.

The refinement of the metric system and its usage is an ongoing project. There were at least three different versions of the metric system at one time, and thankfully they were finally distilled to SI. The metric vocabulary was again increased in 1975, and 1991 when it was needed, but unfortunately the prefix cluster around unity has not been eliminated. The metric system becomes sleeker and sleeker, whereas the medieval Ye Olde English Arbitrary Grouping of Weights and Measurement units used in the US, linger, remain stagnant, and become more and more irrelevant to describe the modern world. There is no microbarleycorn. The use of Olde English units retards the maximum understanding of science by the general US population, which was the very group for whom Mathematics in Everyday Things was targeted. The refusal to adopt the metric system as the sole measurement system in the US, makes the modern world recede into incomprehensibility, and move toward explanations more congruent with magic than with engineering and science.