Bruce,
I like the use of a physical example, such as you have devised here with a
cubic meter.
But, as stated by Arnold Arons (_A Guide to Introductory Physics Teaching_, on
the topic of "scaling"), many people have difficulty grasping and relating
ratios between areas or especially between volumes. They tend to see them
linearly, that is, as the ratio of the sides. Hand someone two balls, one with
twice the diameter of another, and ask the ratio of their volumes. Answers will
often include 2 from those who see the diameter predominately, 4 from those who
see the (cross-sectional) area, 8 from those who get it right, and anything
else from those who don't have a clue. Spheres of course are trickier than the
cube that you proposed. However, experience tells me that the use of a linear
example is always more compelling and much easier to visualize. The comparisons
are more dramatic, too. A cube whose volume is doubled doesn't seem to many to
be twice as large.
I would compare a millimeter and a kilometer for an example of one million.
Anyone (who knows how large a kilometer is) can find an object 1 km away. They
can then look at that distance and the interval of 1 mm on a scale ("ruler") in
their hand and more clearly visualize the ratio.
For one billion (10^9), find two places that are 1000 km (1 Mm) apart. One can
use maps.google or maps.yahoo to do this. These two places should be ones that
the visualizer has driven between or at least can imagine doing so based on
similar trips. Then 1 mm can be compared to that drive. Those in Washington, DC
might choose to "drive" to Augusta, ME or to Atlanta, GA. Those distances are
not exactly 1 Mm but they are close enough for this purpose. Consider that you
are one billion millimeters from Six Flags Over Georgia, Bruce.
For one trillion (10^12) we need to compare 1 mm to 1 Gm (1 000 000 km). The
latter distance is represented by 250 trips around the Earth.
Some folks respond well to temporal scaling. We all have a rough sense of the
duration of 1 second of time. There are one million seconds in 11 days, 13
hours, 46 minutes, and 40 seconds. In other words, roughly eleven and a half
days. There are one billion seconds in about 31.7 years. At the age of about
two billion seconds, we can draw early social security benefits.
Of course, we all know that there are 10pi megaseconds in a year, give or take
a few pies. Make mine blackberry or chess pies.
Jim
>-----Original Message-----
>From: Bruce Barrow [mailto:[EMAIL PROTECTED]
>Sent: Thursday, May 3, 2007 09:55 AM
>To: 'Dennis Brownridge', [email protected]
>Cc: [EMAIL PROTECTED]
>Subject: Re: "Billion" and larger names
>
>To all,
>
>Thanks, Dennis, for your erudite summary. We surely will continue to
>deprecate billion and higher terms in SI 10. There seems to be little
>ambiguity in U.S. usage, but the same is certainly not true internationally.
>
>As to whether people know what the term means, that is another question. I
>asked my brother, an accountant, how many units were in a billion, and got
>the "correct" answer of 1000 million. I asked my wife, a Ph.D. in
>psychology, and got a wrong answer! On the basis of this extensive study I
>conclude that half of educated Americans don't really know, which agrees
>with other contributors to this dialog. No wonder Congress spends a billion
>here and a billion there without much notice from the public.
>
>How to visualize an American billion? Well, median house price in my zip
>code (suburban Washington) is $700k.
>That works out to 1400 homes, not too hard to visualize -- "It takes a
>village," as Hillary said in another context.
>
>Or visualize a one-meter cube on your living room floor -- big, but not
>huge. Then visualize a one-centimeter cube -- small, but not tiny. Well,
>it takes one million of those small cubes, tightly packed, to fill the big
>cube. Now visualize a one-millimeter cube -- tiny, but not microscopic;
>larger than a grain of salt. It takes one US billion of those tiny cubes,
>tightly packed, to fill the big one. But what a job!
>
>Thanks to all of you for setting me straight on national and international
>usage of billion.
>
>Bruce
>
>
>
>
>----- Original Message -----
>From: "Dennis Brownridge" <[EMAIL PROTECTED]>
>To: <[email protected]>
>Cc: <[EMAIL PROTECTED]>
>Sent: Wednesday, May 02, 2007 11:31 AM
>Subject: RE: [USMA:38570] Re: "Billion" and larger names
>
>
>>I can assure everyone that "billion" and larger number-names are indeed
>> confusing English-speaking people. I find great confusion among my
>> foreign-born high school students, for example. These terms should
>> certainly be deprecated in SI-10.
>>
>> The French invented both systems and, like the British, have switched
>> back and forth, so the term "French system" is a misnomer. While 19th
>> and early 20th century French dictionaries define "billion" as 10^9
>> (same as the U.S.), current French dictionaries define it as 10^12 (same
>> as what you have called the "German system"). Furthermore, current
>> French dictionaries say that a French "billion" equals a British
>> "billion" but an American "trillion," so their authors believe that the
>> British are still using the French-German system, even though the
>> British have recently adopted the U.S. system for at least monetary
>> purposes.
>>
>> Dennis Brownridge
>
