kw: technical information, numbers, large numbers
When I was in college a classmate told me of something his Fourth-Grade teacher had done: She cut up about 20 sheets of "millimeter paper", the kind of graph paper with a millimeter grid that has 5- and 10-mm highlights, and taped them together into a 1,000x1,000 sheet, one meter square. This she hung on the wall with a sign above, "This is What a Million Looks Like."
I had occasion to remember this recently. It got me thinking. Most of us can't easily think of numbers such as a million or billion, or even several thousands. Yet we live in a world in which large numbers like that are bandied about: "93 million miles (or 150 million km) to the sun", "4 billion dollars" for such-and-such a system of highways, "7 billion people on Earth", "20 trillion dollar national debt", and so forth. What does a billion or a trillion even mean any more, when you can get a pocket-sized external hard drive with 1TB or 2TB of storage, or even more, for a hundred dollars or so? (Folks, a TB is a TeraByte, or a trillion 8-bit computer "characters").
Let's first be clear whose billion and trillion we mean. These days, even the English and other Europeans have pretty much surrendered to the American system of large numbers, in which a billion is 1,000 million, which is a 1 followed by 9 zeroes, and a trillion is a million million, or a 1 followed by 12 zeroes. But when I was young, the British and others still clung to an older system in which a billion had twelve zeroes and a trillion had eighteen. Some used the French term "milliard" for 1,000 million, the American billion. I remember reading a humorous article, "Why there will never be a British Billionaire", that made this vocabulary stick in my head.
Now we can start to think first of the humble Million. The King James Bible has 8/10 of a million words, or 783,137 if you don't count chapter headings and other auxiliary items. So consider the time it might take you to read the whole thing, add about 1/4, and that's the time you'd need to read a million words. I read novels at about 600 wpm, and nonfiction, if it is any good, at about half that speed. Thus I could read a million words of fiction in some 28 hours (so reading the Bible in a year isn't all that hard, no more than 4 minutes daily) and a million words of nonfiction in twice that time.
I once downloaded The Papers And Writings Of Abraham Lincoln, Complete from Project Gutenberg. In plain text (UTF-8) it comes to 3.1 MB, from which I infer about half a million words. That is Abraham Lincoln's lifetime out put of text. About half a million words, or some 64% of the King James Bible in volume. Now, you know how long reading that would take, but imagine writing those half million words longhand, with a quill pen. Writing with a good mechanical pencil I cannot exceed 20 wpm, and I am pretty sure that even a fast writer could seldom exceed half that using a quill. So Lincoln put a lot of time into his writing, perhaps the equivalent of a year or two of full time work. Several percent of all the minutes that he lived.
OK, let's talk about one billion. That teacher with her million tiny squares on the classroom wall would be hard put to show the children a billion tiny squares. All spread out, it would be larger than 30x30 meters. On some reasonable set of surfaces, such as a long stretch of 8-foot (2.4 m) wall, the paper would extend more than 415 meters, just a bit over a quarter mile. A stack of 1,000 1x1m sheets, had she the patience to make them, would be compact enough, about 10 cm thick (4 inches).
So let's consider something a bit easier to put in a bucket, such as sand. I have on hand some sand from Imperial Beach, California, that I collected about a year ago when I was visiting family there. It is from the southern end of the beach, near the Mexican border, where they don't dump a lot of dredged sand to replenish the beach; thus, it is the "natural" sand from that beach. After some examination with a low-power microscope, and counting the grains in a few milligrams of sand, I found that the average grain diameter is 1/3 millimeter and a gram of the sand would contain about 26,500 grains. That means that a billion grains would weigh 37.7 kilograms (about 83 pounds). The volume comes to about 20 liters (porosity is about 40% because the sand is rather angular and poorly sorted), or 5.3 gallons. That's about two buckets of sand; our household buckets are just under 3 gallons' capacity.
How about something smaller? I'd like to have a billion of something I can conveniently, and without strain, hold in one hand. Considering a weight of a kilogram or less, let's start by assuming a specific gravity similar to water and work backwards. A billionth of a kilogram is then a mass of one microgram, and a cube of ice with such a weight would be 0.1 millimeters on a side. In this size realm, the micron (micrometer for purists) is a convenient dimension. A cube 100 microns on a side is about the size of a mammalian fat cell, so a kilogram of fat contains, very approximately, a billion cells. The volume of that kg of fat is one liter (just over a quart).
Fat cells are larger than average. Another familiar cell type is the buccal cell, those you can gather by the hundreds by lightly scraping the inside of your cheek with a soup spoon. Their diameter is about 25 microns and their mass about one-eighth that of a fat cell, so a billion of them would weigh 125 g and fill 1/8 of a liter (about 4 fluid ounces). That's about the size of a golf ball.
For a big step into smallness let's burrow inside. We all have within us trillions of microbes. Most of them make up our "intestinal flora". They are called "flora" because something like a century ago bacteria were thought to be some kind of plant life. Now we know they are a kingdom of their own. But the term remains. What size are they?
They come in quite a range of sizes, because there are thousands of species. But the most common, the now-familiar Escherichia coli ("E coli" in the Press), also known as "coliforms", have a cell volume close to 2 cubic microns, and with a density just a little greater than that of water, a mass of about 2 trillionths of a gram. Whoa! We've already entered a realm in which it isn't hard to imagine a trillion of something. Two grams of E. coli bacteria contain a trillion cells! The volume would be about that of a thimble.
Now, bacteria are small, but viruses are smaller yet. Let's pick the "familiar" influenza virus. They have a modest range of size, but average 100 nanometers (nm). That is 1/1000th the size of the fat cells we mentioned above. The virus particles are flexible enough to pack together with little porosity, if you can gather a large number of them. So one billion of them, packed together, would have the same volume as one fat cell. And a trillion of them would have the volume of 1,000 fat cells; if packed into a little cube it would be one millimeter on a side. That same volume would hold half a million cells of E. coli.
I don't know how much this might help anyone think about the quantities million, billion or trillion. The meter-square piece of "millimeter paper" is easy enough to imagine, and not too hard to make. You could try holding a golf ball and thinking, "A billion of the cells that line my cheek would just fill this ball". Then, pluck a thimble from the nearest sewing kit and, holding it like a cup, say to yourself, "Fill 'er up with E. coli, and that's a trillion." I can't think of any convenient artifact that would hold "only" a trillion influenza virus particles. One cubic millimeter is pretty small!
Well, this was fun to write, and satisfies a "wild hair" I had a couple of hours ago.
Showing posts with label numbers. Show all posts
Showing posts with label numbers. Show all posts
Sunday, January 22, 2017
Wednesday, November 27, 2013
Why are there so many important numbers?
kw: book reviews, nonfiction, science, numbers, short biographies
A subject that is exercising many physicists and cosmologists is why so many peculiar numbers are needed to define the physics of the Universe, and why they are so seemingly unrelated. Even more, some of them, according to the current theories, need to take rather precise values or the Universe cannot exist, or if it can, it cannot support carbon-based life.
For example, the efficiency of conversion of hydrogen to helium in stars like the sun is very nearly 0.007. (A proton weighs 1.00739 AMU, where the C12 nucleus is defined to weigh 12.0, and a helium nucleus weighs 4.0015 AMU; 4×1.00739 = 4.02956; subtracting 4.0015 gives 0.02806; dividing by 4.02956 yields 0.00696). Were the efficiency as low as 0.006, to quote James D. Stein, "The neutron and proton would not bond to each other, deuterium would not form, and the Universe would consist of nothing but hydrogen" (We'll get back to the error in this statement later). And were it a little higher, at 0.008, "…it would be far to easy for protons to bond together," and the "big bang" would seemingly have gone on to bang away all the Universe to helium and heavier elements in short order: no hydrogen means no water, and any life that forms would need a different fluid.
Given that nobody has yet determined some tiny (five or fewer) set of really fundamental constants, from which everything else can be derived, we have quite a number of them. The recent discovery of the Higgs boson was supposed to pave the way for a more fundamental physical theory, but that seems about as far off as it did before. My most recent printout of the CODATA list of "important" constants runs to several pages.
The book is Cosmic Numbers: The Numbers That Define Our Universe by James D. Stein, a mathematics professor at CSU Long Beach. Out of the zoo of CODATA constants, he has chosen 13 to explain to us, and even better, he presents short biographies of the scientists whose work led to an understanding of each of them.
Some numbers have dimensions, meaning that their numerical value depends on the system of measurement. Such is Avogadro's Number, 6.0221413×1023, the number of atoms in 12 grams of the carbon-12 isotope. It is the ratio of the gram to the AMU. By extension, it is the definition for a mole of any substance, where a mole is the weight in grams equal to the atomic or molecular weight of the atoms or molecules. Thus, one mole of pure isotopic iron as Fe56 is 56 grams (or, strictly speaking, 55.9349393 grams, because the atomic weight of that isotope of iron is 55.9349393 AMU). Now, suppose instead of grams, we had in history defined a unit mass to be something else, call it a marg, with a mass about 1.66 times as large. Then Avogadro's Number would be, nearly exactly, 1024, and it is likely that scientists would lobby hard to get the marg redefined to make that number exact. Something similar happened fifty or so years ago, when the inch was redefined to be exactly 25.4mm.
Other numbers are dimensionless, such as absolute zero. This is an extrapolated temperature, defined according to the ideal gas law, at which no more heat can be extracted from a substance, and the atomic or molecular motions that define what we mean by "temperature" would cease completely, except for the tiny gyrations needed to avoid violating Heisenberg's uncertainty principle. The "temperature" 0K (K for "kelvins", which have the same size as Celsius degrees), AKA 0R (in which a Réamur is equal in size to a Fahrenheit degree, but the scale begins at absolute zero), needs no units. Zero is zero.
Another dimensionless number is 1/137, the Fine Structure Constant, initially derived from spectroscopy in a magnetic field. Its actual value is 1/137.036 and about six more digits. Though it can be derived from more fundamental constants such as the unit charge and the speed of light, all the units cancel out, so it is the same numerically in all possible systems of units. This isn't one of Dr. Stein's examples. He presents only two dimensionless constants, Avogadro's Number and the efficiency of hydrogen fusion, discussed above. In the latter chapter (Chapter 10), I was surprised at a number of errors that the physicists among his reviewers ought to have caught.
One was the fusion of proton with neutron, mentioned above. Highly energetic P-P collisions are required for the protons to physically approach close enough for one to emit a positron and become a neutron. Then the strong force can take over and fuse the two. The value of actual interest here is the efficiency of P+P→D+e+ conversion. A deuteron weighs 2.01355 AMU, so the conversion efficiency is 0.00061. I suspect it is this number, not the 4P→He++ efficiency, that matters most. Another error was quite a long discussion of the mechanics of the P-P chain, in which the text uses "electron" instead of "proton" throughout. Electron collisions don't matter in the core of a star. The substance is a plasma. In essence, it is a mass of colliding protons (and deuterons and other nuclei) in a thin soup of unbound electrons, where there is nearly (or entirely) no impediment to P-P collisions except their own positive electric charge. At much lower energies (temperatures up to a few hundred degrees rather than tens of millions), H-H collisions that occur are primarily mediated by interactions between the electron clouds of the H atoms.
OK, gripe over. I confess to being rather staggered by that, but the rest of the book is a delight. We learn, not just the scientific endeavors of Boltzmann or Newton or Boyle, but their lives and something of their personalities. Science is a human activity, and one could say it is the most human of activities: figuring out how things work is our stock in trade (even if we devote our adolescence to figuring out how the opposite sex works!). The beauty of the numbers Dr. Stein has chosen lies in the sheer brilliance needed to first see the requirement for such a quantity, then to ferret out a way to determine what it is. We may see them as obvious in hindsight, but, for example, prior to Newton's insight, a law of common gravitational attraction just didn't fit in anybody's head.
A story is told of Napoleon, challenging his generals to make an egg stand on end. After they'd all given up, he held the egg and rapped it gently on the tabletop, enough to crush the end just a little. Then it stood. One general protested, "Well, that is obvious!", to which Napoleon replied, "It wasn't obvious before you saw me do it."
A subject that is exercising many physicists and cosmologists is why so many peculiar numbers are needed to define the physics of the Universe, and why they are so seemingly unrelated. Even more, some of them, according to the current theories, need to take rather precise values or the Universe cannot exist, or if it can, it cannot support carbon-based life.
For example, the efficiency of conversion of hydrogen to helium in stars like the sun is very nearly 0.007. (A proton weighs 1.00739 AMU, where the C12 nucleus is defined to weigh 12.0, and a helium nucleus weighs 4.0015 AMU; 4×1.00739 = 4.02956; subtracting 4.0015 gives 0.02806; dividing by 4.02956 yields 0.00696). Were the efficiency as low as 0.006, to quote James D. Stein, "The neutron and proton would not bond to each other, deuterium would not form, and the Universe would consist of nothing but hydrogen" (We'll get back to the error in this statement later). And were it a little higher, at 0.008, "…it would be far to easy for protons to bond together," and the "big bang" would seemingly have gone on to bang away all the Universe to helium and heavier elements in short order: no hydrogen means no water, and any life that forms would need a different fluid.
Given that nobody has yet determined some tiny (five or fewer) set of really fundamental constants, from which everything else can be derived, we have quite a number of them. The recent discovery of the Higgs boson was supposed to pave the way for a more fundamental physical theory, but that seems about as far off as it did before. My most recent printout of the CODATA list of "important" constants runs to several pages.
The book is Cosmic Numbers: The Numbers That Define Our Universe by James D. Stein, a mathematics professor at CSU Long Beach. Out of the zoo of CODATA constants, he has chosen 13 to explain to us, and even better, he presents short biographies of the scientists whose work led to an understanding of each of them.
Some numbers have dimensions, meaning that their numerical value depends on the system of measurement. Such is Avogadro's Number, 6.0221413×1023, the number of atoms in 12 grams of the carbon-12 isotope. It is the ratio of the gram to the AMU. By extension, it is the definition for a mole of any substance, where a mole is the weight in grams equal to the atomic or molecular weight of the atoms or molecules. Thus, one mole of pure isotopic iron as Fe56 is 56 grams (or, strictly speaking, 55.9349393 grams, because the atomic weight of that isotope of iron is 55.9349393 AMU). Now, suppose instead of grams, we had in history defined a unit mass to be something else, call it a marg, with a mass about 1.66 times as large. Then Avogadro's Number would be, nearly exactly, 1024, and it is likely that scientists would lobby hard to get the marg redefined to make that number exact. Something similar happened fifty or so years ago, when the inch was redefined to be exactly 25.4mm.
Other numbers are dimensionless, such as absolute zero. This is an extrapolated temperature, defined according to the ideal gas law, at which no more heat can be extracted from a substance, and the atomic or molecular motions that define what we mean by "temperature" would cease completely, except for the tiny gyrations needed to avoid violating Heisenberg's uncertainty principle. The "temperature" 0K (K for "kelvins", which have the same size as Celsius degrees), AKA 0R (in which a Réamur is equal in size to a Fahrenheit degree, but the scale begins at absolute zero), needs no units. Zero is zero.
Another dimensionless number is 1/137, the Fine Structure Constant, initially derived from spectroscopy in a magnetic field. Its actual value is 1/137.036 and about six more digits. Though it can be derived from more fundamental constants such as the unit charge and the speed of light, all the units cancel out, so it is the same numerically in all possible systems of units. This isn't one of Dr. Stein's examples. He presents only two dimensionless constants, Avogadro's Number and the efficiency of hydrogen fusion, discussed above. In the latter chapter (Chapter 10), I was surprised at a number of errors that the physicists among his reviewers ought to have caught.
One was the fusion of proton with neutron, mentioned above. Highly energetic P-P collisions are required for the protons to physically approach close enough for one to emit a positron and become a neutron. Then the strong force can take over and fuse the two. The value of actual interest here is the efficiency of P+P→D+e+ conversion. A deuteron weighs 2.01355 AMU, so the conversion efficiency is 0.00061. I suspect it is this number, not the 4P→He++ efficiency, that matters most. Another error was quite a long discussion of the mechanics of the P-P chain, in which the text uses "electron" instead of "proton" throughout. Electron collisions don't matter in the core of a star. The substance is a plasma. In essence, it is a mass of colliding protons (and deuterons and other nuclei) in a thin soup of unbound electrons, where there is nearly (or entirely) no impediment to P-P collisions except their own positive electric charge. At much lower energies (temperatures up to a few hundred degrees rather than tens of millions), H-H collisions that occur are primarily mediated by interactions between the electron clouds of the H atoms.
OK, gripe over. I confess to being rather staggered by that, but the rest of the book is a delight. We learn, not just the scientific endeavors of Boltzmann or Newton or Boyle, but their lives and something of their personalities. Science is a human activity, and one could say it is the most human of activities: figuring out how things work is our stock in trade (even if we devote our adolescence to figuring out how the opposite sex works!). The beauty of the numbers Dr. Stein has chosen lies in the sheer brilliance needed to first see the requirement for such a quantity, then to ferret out a way to determine what it is. We may see them as obvious in hindsight, but, for example, prior to Newton's insight, a law of common gravitational attraction just didn't fit in anybody's head.
A story is told of Napoleon, challenging his generals to make an egg stand on end. After they'd all given up, he held the egg and rapped it gently on the tabletop, enough to crush the end just a little. Then it stood. One general protested, "Well, that is obvious!", to which Napoleon replied, "It wasn't obvious before you saw me do it."
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