Bulldog Edition

A seminal advance that allowed for the development of the modern world seems never to be consciously noted by most persons. That advance was when people began to identify groups of objects with the abstract idea of numbers. The concept that 1, 2, 3, 4 and so on could be mapped with a one-to-one correspondence to a group of skins or fruit or whatever, took a considerable amount of abstraction, and can be counted as one of the great conceptualizations humans embraced.

Francis Galton (1822-1911) is reported to have made this observation about a tribe in Southwest Africa:

When inquiries are made about how many days’ journey off a place may be, their ignorance of all numerical ideas is very annoying. In practice, whatever they may possess in their language, they certainly use no numeral greater than three. When they wish to express four, they take to their fingers, which are to them as formidable instruments of calculation as a sliderule to an English schoolboy. They puzzle much after five, because no spare hand remains to grasp and secure the fingers that are required for units. Yet they seldom lose oxen; the way in which they discover the loss of one is not by the number of the herd being diminished, but by the absence of a face they knew.

I don’t know if Galton felt sanctimonious about this lack of numeracy, but if he was, it was not particularly justified. Isaac Asimov (1920-1992) was the person who led me to this realization. In his book *Fact and Fancy* he states:

It was 1300 A.D. before the word “million” was invented. Until then, the largest number word was “myriad,” which was Greek for 10,000. Even Archimedes, in calculating the number of poppy seeds in the entire Universe as he knew it, used expressions meaning “myriads of myriads of myriads. …

It was only seven centuries ago that a word to describe 1 000 000 was created, which is equivalent to the modern metric prefix Mega—and adopted only in 1960.

In 1871, Fredrick A.P. Barnard (1809-1889) gave a speech which was expanded into a book called *The Metric System of Weights and Measures*. Barnard had done his best to make the metric system the exclusive system of measure in the US, but Charles Davies managed to derail this initiative. In his book Barnard presents this table:

The largest metric prefix used with a meter is a myriametre. The myriad apparently continued to remain an upper value limit for the everyday person. This is true even though the word million existed at that time, and Friedrich Bessel (1784-1846) determined in 1831 that the star 61 Cygni was about 98 Petameters from Earth, or 9 800 000 000 000 myriametres. The early metric system was provincial and still mired with numerical magnitudes from the time of Archimedes. It would not be until 1960 when a prefix larger than myria would be adopted, which is of course Mega. The prefixes Giga and Tera were also adopted that year. The use of myria as a prefix, despite its ancient origin, seems to have been eschewed without much difficulty, and the better by 1000 approach began. The seldom used prefix cluster around unity has proven much more difficult to eradicate.

While the numerical value for the word million was agreed upon after 1300 A.D., a fork occurred when English words were chosen for larger numbers. The word billion is used for Giga in the short scale; in the long scale, the word milliard is used. The word billion is used for Tera in the long scale, but it is called trillion in the short scale. The history of the changing values of words used to describe large numbers is considerable and meanders. This prefix table provides succinct documentation of the differences between short and long scale words:

Douglas Adams’ *The Hitchicker’s Guide to The Galaxy* has a computer called the Milliard Gargantubrain, which shows that long scale usage is still with us. The use of the short and long scales has allowed for very poor literary numerical expression, which was long ago forbidden in the metric system, but is ignored when long and short scale words are combined with metric prefixes. The statement that a celestial object is one million Kilometers from Earth does not meet with objection, but if it is a KiloMegameter away, that would be laughable. The proper term Gigameters would probably also cause heartburn for the provincial literary crowd who are proudly literate and willfully innumerate, but it is the most succinct prose expression, and does not contain any numerical ambiguity. If we have a billion meters, it could be a Gigameter or Terameter. Carl Sagan (1934-1996) used the term billions and billions. But which billion? Alternatively he could have said milliards and milliards, which would have been unambiguous by comparison—even if it is a metaphor.

The long or short scale words are not prefixes, they are improper prefixes which at worst have dyadic values (i.e. two different values for billion and billionth) and at best are redundant. Long and short scale words should not be combined with metric prefixes to present a numerical value. They only promote innumeracy. Does a milliardth of a nanometer improve one’s understanding of magnitude? Attometer is shorter and clearly and unambiguously defined and has a metric context. When I see a headline like:* 35 years and 18 billion kilometers later, NASA’s Voyager approaches exit from solar system*, I can only do a face-palm. Is this 18 Terameters or 18 Petameters? 18 000 Gigameters is probably more meaningful. Until we are able to reform our prose, and make them all metric, we have little justification to believe that we are clearly more numerically advanced than the Southwest African tribe described by Dalton in the 19th Century.

I certainly agree we should use proper metric prefixes, not large counting words, with metric units.

On long-scale vs short-scale, it may be a little less confusing than you indicate. Carl Sagan was an American, he meant 109. The US has used short scale since the mid 1800s or earlier. The UK used to use long scale but the UK government officially adopted short scale in 1974, followed by the media and the rest of the Commonwealth (see Wikipedia article on long vs short scale). Per Wikipedia, the US never used “milliard” and the UK mostly used “thousand million.” It is common in some other languages but only adds confusion in English.

It is now primarily a translation problem as a number of foreign languages use words similar to billion and trillion, but they mean long scale in those foreign languages. English has no modern long-scale usage. However, in dual language countries (like Canada), you can have situations where English-speaking Canada uses long-scale and French-speaking Canada uses long-scale.

Mega- and micro- were in use prior to formal adoption in the initial publication of the SI in 1960. Merriam-Webster traces the first use of megohm to 1867, megacycle (per second) was in fairly widespread use before 1960. The micron was approved as a unit in 1948 by the CGPM (later abrogated) and microfarad was in common use for capacitors prior to 1960.

Since the html didn’t work, that should have been 10^9 above.

Billions or billiards, decimal point or comma, and so on – will there ever be an agreement on a common, worldwide standard, for all this? Who knows…

BTW, another, more or less well-known example is the mathematical interval: for example, for 0<=x<1, in "American" notation, [0, 1); while, in "European" notation, [0, 1[…

Lots of equally good – or bad? – options, thus: but which one to choose, eventually?

Who knows, again: BTW, the short and long scales both have elements of rationality and irrationality, so it's difficult to say which one would be the best; none of them is perfect, anyway.

(Talking about (often sadly absent, in the so-called real world) logic: RIP, Leonard Nimoy; Spock would probably have had a sensible solution, for all this… 🙂 🙂 )

I’m not sure I understand the European notation. Is the second left-facing bracket taken to mean the 1 is NOT included in the interval? What if the zero was not included, 0<x<1, all points between zero and one, but both endpoints excluded. What if both end points are included, 0 ≤ x ≤ 1?

Anyway, the SI Brochure accepts either the decimal point or the decimal comma, and rejects both as a thousands separator, requiring a space (thin space) instead. For the SI, the matter is settled. I don't see them reopening the can of the worms.

An example of European vs. American mathematical interval notation:

[0, 1] = [0, 1]

[0, 1[ = [0, 1)

]0, 1] = (0, 1]

]0, 1[ = (0, 1)

(The European notation has the marginal advantage that the open interval cannot be confused with an ordered pair of numbers; otherwise, they are equivalent.)

The “myriad”, meaning as mentioned 10,000, still exists as a main part of one Big languiage: Mandarin Chinese

In such,

For ten thousand, the word is yi-wan [literally “one-ten thousand’].

For one hundred thousand, the word is shi-wan [literally “ten-ten thousand’].

For one million, the word is yi-bai-wan [literally “one-hundred-ten thousand”].

For ten million, the word is yi-qian-wan [literally “one-thousand-ten thousand”].

For one hundred million (10^8), the word is yi2-yi4 [literally “one-hundred million” (the 2 and 4 indicate the spoken tones of the same sound are different)].

So, in Chinese, it’s ones, tens hindreds, thousands, ten thousands (10^4); ten-ten thousands(10 x 10^4), hundred-ten thousands (100 x 10^4), thousand-ten thousands (1000 x 10^4), hundred million (1 x 10^8); and so on, and so instead of the usual SI (or engineering notation) beloved by us, it would be a world of “wans” (10^4ths). [Wonder what the Maven would do if he were Chinese and spoke Mandarin as his primary language?]

With that let me simply say to all here “wan-shi-ru-yi” to [literally something like “ten thousand [numerous] things go your way”, implying “may all your wishes come true”, which in English would be equivalent to wishing someone “All the best!”.

Don’t forget India with it lakh and crore. Very confusing.

Lakhs and crores are kind of cool: they help fill gaps. I’ve written code something like this in a computer program:

ones = num mod 10

tens = floor(num / 10) mod 10

hundreds = floor(num / 100) mod 10

thousands = floor(num / 1000) mod 10

myriads = floor(num / 10000) mod 10

lakhs = floor(num / 100000) mod 10

millions = floor(num / 1000000) mod 10

One can argue that the high degree of innumeracy and scientific/technical ineptitude in the English speaking world can be attributed to two failures only partially corrected in recent times. That is the use of the short scale for counting and imperial/USC for measuring. Only in recent times has the majority of the English speaking world metricated.

Because of the short scale used primarily in the US, an understanding of large numbers is almost impossible. The short scale makes it difficult if not impossible to discern the number of zeros associated with number name. The long scale does not have this problem and those living in countries using the long scale have a greater understanding of numeracy.

The long scale is simple to understand as the names of the numbers whose “prefix” are derived from Latin names for numbers are used to determine the value of the number.

The word million was originally meant to be a “large thousand” as the base prefix “mil” means thousand. With the long scale, a number such as octillion is easy to determine its numerical value. The word octo is Latin for eight and when eight is multiplied by the number six (the number of zeros in a million), the result is 48 and in the long scale the octillion is a 1 followed by 48 zeros. The octilliard then becomes the “large octillion” by adding an additional 3 zeros for a total of 51 zeros. All this is done mentally without either a calculator nor a pencil and paper.

This simple trick does not exist in the short scale and as a result most people only know of million, billion and trillion. If a number larger than this is encountered, it must always be explained and no one knows how it was derived, they just except it but later forget it and must be continuously shown.

It is no surprise that countries like Germany who use only the metric system and count using the long scale are better suited to understanding advanced technology, excel in engineering and are rising fast in the modern world, while the US, clinging to its short scale and USC is sinking like a lead balloon.

Americans will always defend their desire not to conform with that which is better, even to the point of being proud about being technically ignorant. But for how long? It is naive to think that you will always be on the top, especially when everyone Else is seeing you tumble. The short scale and USC seem insignificant as a means for a nation to fall, but it is the little things, that many think insignificant that cause one to stumble and fall.

Thanks for another good posting, Ametrica.

By capitalizing a word in your last paragraph, you reminded me to remind the Maven to shun something that has made many people sort of sick, which we find in this sentence from the Maven’s otherwise good prose:

“The concept that 1, 2, 3, 4 and so on could be mapped with a one-to-one correspondence to a group of skins or fruit or whatever, …”

Whatever what, Maven? Now let’s use that word from Ametrica’s posting:

The concept that 1, 2, 3, 4 and so on could be mapped with a one-to-one correspondence to a group of skins or fruit or whatever else, …

Much better!

I think that much, perhaps most, of our innumeracy here in the U.S. is due to two things:

1) Calculators cost next to nothing, and most of us have a calculator with us at all times in the form of our cell phone. It is easy to fall out of practice. In fact, I know one woman who used to work as a bank teller, but she didn’t know how to do subtraction using pen and paper. That’s right, subtraction. The machine had always done it for her, of course.

2) Elementary school education. Those in charge of teaching arithmetic are among those least comfortable with arithmetic. Too much material is introduced too early. In particular, fractions are introduced too early and in confusing ways, and again, by people who themselves are not so comfortable with arithmetic.

And yes, I agree that our “-illion” words are not a good thing. Millions, billions, and trillions all sound alike.

We could immagine something like this: if the prefix mil- is 1000^1, bil- could be 1000^2, tril- 1000^3, quadril- 1000^4, and so on; but that would require to reconsider the million to be a thousand, in order to rationalize the short scale in a new way (by “shifting” it more sensibly, so to speak).

Or, alternatively, we could use the SI prefixes also for expressing the “real” numbers: for example, a thousand could become a kilon, a million a megon, a billion a gigon, a trillion a teron, and so on.

Well, of course, only sci-fi, sofar…

Sounds like a very nerdy, very sci-fi lingo… I like it!

For billion, trillion, etc, let n be the number root of the word (bi = 2),

then the short scale value is 1000 x 1000^n. Short scale has an equally simple rule and doesn’t “oscillate” between -lion and -liard.

More important, the English-speaking world (plus some other languages and countries) has commonized on short scale. In English, the meaning is clear, whether or not there is confusion in other languages.

But if it were simply 1000^n, besides being more logical, one could also unify it with the SI prefixes, instead of having a different set of names (i.e., kilo-, mega-, giga-, tera-, and so on): for example, 1 (new) million could thus be 1 000, and, e.g., 1 000 meters could simply be 1 milliometer; 1 billion meters, 1 billiometer; 1 trillion meters, 1 trilliometer; and so on (and with milli-, billi-, trilli-, etc. etc. for 1000^-n).

Much simpler than today, and infinitely expandable, without the need for inventing new names for new prefixes.

Anyway, personally, I really don’t know if to prefer the long or short scale: so I simply tried to imagine something that could supersede them both, at the same time.

Of course, nothing of all this will change, at least for the time being, so it’s only wishful thinking, so to speak…

The long scale is adding unnecessary prefixes to the prefixes, just for the sake of making them long.

Therefore, by the rule of K.I.S.S. (keep it simple, stupid), only the short scale usage is acceptable.

So let’s always assume the numbers are using short scale and, if anyone complain, let’s laught at their faces until they go running away while crying the tears of eternal shame.

Just today cracked.com had an article (one of the reader-submitted “photoplasty” articles) which stated “your brain’s memory has the potential capacity to hold 2.5 million gigabytes of data.” At first I failed to notice the million and thought “2.5 gigabytes isn’t very impressive.” Then I realized it should’ve said “2.5 petabytes.”

Computers will be in the Peta range soon. We may as well start using the prefix.

Another proposal, based on Greek prefixes (à la SI):

http://www.unc.edu/~rowlett/units/large.html

The real problem is probably that, unless there’s a worldwide cultural revolution, nothing is sadly likely to change, on the standardization and optimization front…

When I was taking science classes in school, I often had to do calculations with prefixed units, including conversions between different prefixes. I found it expedient to simply strip off the prefixes (example: 20 km becomes 20 000 m), perform the calculations, and then attach the appropriate prefix. No prefixes means one less thing to keep track of.