decimal dust

[Ken Greenwald]

In the current issue of Newsweek, I read the following in an article on the nursing shortage in the U.S.:

<2005 “And congress budgeted only $150 million or so for financial aid [[to nursing students]], with the result that 82 percent of federal-loan applicants were rejected. ‘The public sector needs to take this on ,’ says Peter Buerhaus, a health economist at Vanderbilt University. ‘We’re talking DECIMAL DUST compared to the billions spent in Washington.”—‘Newsweek,’ 12 December, page 80>

DECIMAL DUST – I love it. In my over 40 years of being involved in science and science education, I’ve never heard this one, and wish I had, because I would have used it over and over again, in describing the decimal noise that proliferates among both students and other folks who don’t know what they are doing when it comes to the use of significant digits [e.g. the 10 places that a calculator produces appears as an answer on a test when perhaps no more that 3 make any sense].

In physics we try to teach the basic rules of measurement and the propagation of errors in calculations using these measurements, and accuracy, precision, and significant digits are important topics. To ascertain the propagation of errors accurately one must employ differential calculus. However, for such simple operations as addition, subtraction, multiplication, division, squaring, and square roots the rules are simple and don’t have to be rederived every time.

After Kennedy (in 1962) challenged the nation to put a man on the moon by the end of the decade, I actually spent about six months of my life, on my first job out of college (1963), working on this sort of thing using the feedback control system equations of our early inertial guidance systems – the systems that tell a spacecraft where it is and guides it to its final destination. My task, at the company I was working for, was to develop the equations that would be used to calculate the errors incurred in position, velocity, acceleration, etc. due to various inaccuracies in the guidance system – which, miraculously (for me at least – using a pencil, a piece of paper, and a slide rule in lieu of the not-yet-available digital computer), I actually accomplished! Heading the project at my company was a recent defector from Russia, a rocket scientist, who brought many of their guidance secrets with him. Ah, the ‘space race.’ Those were heady times! But I digress.

Since DECIMAL DUST, as I see it, is all around us – an extremely apt expression, not only for an insignificant amount of money but also for general decimal noise, I figured that I ought to find out where it came from, how long it has been around, etc. None of my usual sources listed it, but a Google search produced 1875 hits – a not insignificant number – and with the very first hit on the list I hit pay dirt. WordSpy.com had addressed the question:

WORD SPY

DECIMAL DUST: An inconsequential numerical amount.

Example Citation: “In 1999, Gonzalez received $1.4 million from the National Institutes of Health to compare his enzyme-nutritional therapy with the best chemotherapy now available for the treatment of advanced pancreatic cancer. As a percentage of the fifteen billion dollars that the federal government spends on medical research annually, the grant amounts to what one federal health official described to me as ‘decimal dust.’”—Michael Specter, "The Outlaw Doctor," The New Yorker, February 5, 2001

Earliest Citation: “Critics worry that as partnerships bloom, they could drain billions of tax dollars from the Treasury, and even lead to the ‘disincorporation’ of America. The Treasury Department originally estimated it could raise $665 million over a five-year span by changing the tax status. The Stanger Report dismisses that revenue estimate as exaggerated, saying revised Treasury studies now say the government would gain just $24 million next year if it taxes MLPs as corporations — ‘decimal dust,’ as one senator put it."—Stephen Maita, ‘The Bloom is Off the MLP Market,’ The San Francisco Chronicle, July 27, 1987.

Notes: Something (usually a dollar amount) gets called decimal dust after it has been compared to something much larger. For example, consider the numbers in the citation above: Dividing the $1.4 million received by the doctor with the $15 billion medical research budget, the result is 0.000093. That tiny decimal fraction makes the smaller monetary amount seem almost nonexistent, as though it could be blown away like a speck of dust.
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But, for my personal use, I will further define it as not only being 'an inconsequential numerical amount' but also a ‘fallacious numerical amount,' those extra meaningless digits – that decimal noise – based on the improper use of significant digits in calculations. The truth ain’t just what your calculator happens to spit out!

<1992 “Researchers at AEA Technology’s Harwell Laboratory in Britain recently turned a Cray-2 supercomputer loose for 19 hours. The result: an new ‘largest’ prime that leaves the previous record-holder trailing in its DECIMAL DUST.”—‘Science,’ New Series, Vol.256, No. 5054, April, page 175>

<2003 “And when I started to chase this down, I talked to somebody on the Interagency Council for Homelessness at the White House, and he said to me, hey, Jack, this is DECIMAL DUST [[the $600,000 needed for a rejected homelessness program]] okay?”—‘U.S. House of Representatives Hearing on Veterans Affairs, 6, May>

<2005 “The differences in average scores between [[assessment test series numbers ]] 7 & 8, and 9 & 10 amounts to ‘DECIMAL DUST.’—Assessment Newsletter,’ Owens Community College (Ohio), Spring>

Ken G – December 8, 2005

[dalehileman]

Ken thank you for the neologism. Would you consider "decimal noise" and "decimal dust" to be absolutely interchangeable synonyms, or is there enough shade of difference, say, to warrant separate entiries definitions in a dictionary

No, fellas, I'm perfectly serious. I really wanna know, as goes the CSI ditty

[Shelley]

Since DECIMAL DUST, as I see it, is all around us – . . .

So, Ken -- all we are is decimal dust in the wind?

[Erik_Kowal]

In the grand scheme of things, our significance extends into infinitesimality.

---From beyond our Ken

[Ken Greenwald]

And as Bob Dylan had once tried to explain to his math teacher who asked about the significance of Bob’s solutions to his math test problems, “the answer my friend is Blowin’ in the Wind.” The teacher flunked him. But ‘By a Simple Twist of Fate,’ not feeling any animosity over his decision, Dylan replied, “Don’t Think Twice, It’s All Right,” but thought to himself, this cat better get with it and stop fartin’ around with decimal dust because “The Times They Are A-Changin” and he should ask himself how does it feel, how does it feel, to have to solve for an unknown, on your own, like a, like a . . . – And, of course, the rest is history!

Ken G – December 10, 2005

[Ken Greenwald]

Dale, In my above discussion, I just made up the expression DECIMAL NOISE, but I if I were to offer a definition, my preference would be that it refer to the insignificant (versus the significant) and meaningless digits (and not just after the decimal point but also before) that people mistakenly include in a calculated result. I consider ‘decimal noise’ something objectionable because it makes no sense and is wrong.

On the other hand, according to the way DECIMAL DUST has been defined by Word Spy, and the way it has been used in the above quotes, it does not refer to something that is objectionable, but instead to something that is very small and numerically insignificant. In the 2003 quote above the $600,000 request for a homeless veteran’s program is insignificantly small on the scale of the overall federal budget or even on the scale of our military budget – but there is no implication that it is objectionable. ‘Decimal dust,’ in fact, has the sound of something that mathematicians might sit around and get high on - a positive quantity in the eyes of the dustee. (<:)

To illustrate what I consider DECIMAL NOISE, consider the following. If to calculate the distance traveled, you multiply a measured speed of say 2379 or 2.379 x 10^3 meters/second (this powers of ten scientific notation tells one, more definitely, that there are 4 significant digits) times a measured time of 56 or 5.6 x 10^1 seconds (says 2 significant digits), the correct answer, using a standard rule of thumb of significant digits (no more significant digits in a product than in the factor with the least number of significant digits), the answer should be 133,000 meters. However, if you believe what your dimwitted calculator tells you when you multiply these two numbers together, the answer is 133,224 meters. However, the 224 has little significance and is what I would call ‘decimal noise.’ To see why, calculate the range of possibilities allowed by the factors. Consider that the time of 56 seconds could have been rounded up from 55.5 and the speed could have been rounded up from 2378.5, which would have produced a minimum possibility of ~132,000. However, if the 56 seconds had been rounded down from 56.4 and the speed had been rounded down from 2379.4, that allows for the possibility of a maximum of ~134,200. Thus the answer could actually have been anywhere from the possible minimum to the maximum, a range of uncertainty (slop) of 2,200 meters. So, logically, if we don’t even really know the answer to within more than about 2,200 meters, is there really any point in providing an answer that includes that 224 meters? – it is of no significance! If we include it we are claiming a more accurate knowledge of the answer than we really have. And notice that the rule of thumb gives us an answer that is approximately right, about somewhere midway between the two extremes – 133,000 meters. So we really have no justification for claiming anything more, based on the given data.

Note: A more rigorous analysis might ask the folks who took the original measurements for more precise error bounds on their readings (e.g. 2379±3 meters/second and 56.0±.7 seconds). Armed with this information one could then use differential calculus (partial derivatives) to show - or just accept the standard result from error analysis for the propagation of errors – that the percent error in a product is equal to the sum of the percent errors in its factors.

Incidentally, just out of curiosity, I checked Google for the term DECIMAL NOISE to see if it had ever been used before and I got 74 hits. But in every case I checked, ‘decimal noise’ had mistakenly been used for DECIBEL NOISE (~ 14,500 hits), a term I don’t much like – I would just say ‘noise level’(~1, 470,000 hits) instead. The ‘decibel’ (dB) is a measurement of the intensity of sound (e.g. vacuum cleaner - 70 dB, power mower - 80 dB, rock concert - 120 dB, nearby jet - 150 dB, rocket pad launch - 180 decibels, end of the world – ???) and ‘decibel noise’ is, I suppose, a term that is used for noise intensity by a few. Hmm. Wonder if there is a good name for the likes of those 72 erroneous hits. If there isn’t, how about HIT NOISE. And that could include all those dead links that one wishes someone would vacuum up and throw in the Dumpster. But, alas, there is probably no money in the dead link elimination dodge – any volunteers out there willing to spring for it? (<:)

Ken G – December 10, 2005

[dalehileman]

Ken thank you for the confession and clarification. I would nominate your meaning as snigletic. Besides, it does after all have some support:

Bob Brozman, National Guitar Master Interview - Rhythms Magazine
... system and it actually involves, instead of three over two, it's
three-point-da-da-da-da-da over two-point-da-da-da-da-da, it's a lot of decimal noise. ...
http://www.bobbrozman.com/inter_rhyth.html

[Ken Greenwald]

Dale, I missed that Brozman quote among my 74 hits – since it was a music topic, I dismissed it without checking it more closely and I assumed it was another misuse of ‘decibel.’ But you’re right. He does use DECIMAL NOISE in the way that I defined it (though his noise serves double duty) in explaining the ‘tempering’ of the 12 tone diatonic scale where the ratio of each half tone to its immediate neighbor is the 12th root of 2 or 1.059463094. . . . and thus, all half tones are created equal!

Bach invented this compromise tuning realizing that in the old ‘pure’ system that if, for example, one tuned for the key of C major, while the major fifth C-G would be exactly a ratio of 2 to 3 or 240 to 360 Hz, as it should be, the other fifths would not be (e.g. D-A would be 270 to 400 or 2 to 2 + 26/27 or 2 to 2.9629. . . – not quite 2 to 3. But this and other discrepancies, some a bit worse, were enough to grate on a musician's ear. In addition, under the old system, the instruments (those that were tunable) had to tune to the key of the composition they were playing so that the important intervals in that key would be closest to being right – and this would be a real pain if a number of pieces in different keys were to be performed. So Bach decided to invent a system in which no interval except the octave was exactly right, but now more intervals would be closer to being right than before. Thus, over 12 half tones, an octave, the ratio would be the 12th root of 2 raised to the 12th power, which is precisely 2, exactly an octave. The major fifth C-G instead of being precisely 2 to 3 would be 2 to 2.9966. . ., but the D-A fifth would now have a ratio of 2 to 2.9966. . . instead of 2 to 2.9629. . . – closer to the ideal than before. But we are talking DECIMAL places of difference here or DECIMAL NOISE, as Brozman and I have called it, and to the untrained ear this would hardly be detectable, although to trained musicians with a good sense of pitch it would be. Bach first demonstrated this use of ‘equal temperament’ in his composition ‘The Well-Tempered Clavier’ (1722).
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Ken – December 11, 2005