This is the third of four lectures on a rather difficult subject -- the theory of quantum electrodynamics -- and since there are obviously more people here tonight than there were before, some of you haven't heard the other two lectures and will find this lecture incomprehensible. Those of you who have heard the other two lectures will also find this lecture incomprehensible, but you know that that's all right: as I explained in the first lecture, the way we have to describe Nature is generally incomprehensible to us.
Richard P. Feynman, QED, The Strange Theory of Light and Matter, p. 77 [Princeton University Press, 1985]
In 1913 Niels Bohr proposed that the lines in the spectrum of Hydrogen could be explained if electrons could only assume certain energy states in the atom, states which corresponded to quantacized values of angular momentum. The classical definition of angular momentum is mass times velocity times distance from the center of circular motion (kg*m/s*m = m2*kg/s). This turns out to be the units of Planck's Constant (h), which can also be expressed in units of energy times time (J*s).
Thus, each electron will only have angular momentum (l) values that are an integer times Planck's Constant divided by 2 (the number of radians in a circle; h/2 is often expressed as the "reduced Planck's Constant," -- a symbol used by Paul Dirac). The spectral lines of hydrogen result when an electron drops from one angular momentum state to a lower one and releases energy. The energy is then emitted as a photon, a quantum of electro-magnetic radiation, as explained by Albert Einstein in 1905 (the "photo-electric effect"), at a frequency and wavelength proportional to its energy, according to Planck's equation, E = h = hc/, where c is the velocity of light, is the frequency (1/s, Hz), is the wavelength (m), and c = . Bohr's equation for hydrogen is as follows, where n and n' are integer values for the level that the electron is leaving (n) and the level to which the electron is falling (n').
The additional constants are the electrostatic force constant (k) and the mass (me) and charge (e) of the electron. Evaluating the equation simply gets us a quantity of 912 times the factor , with the quantum integers, in units of length as Angstroms ( = 10-10 m).
When electrons drop down to the lowest quantum level, where angular momentum is zero, they emit photons in the ultra-violet part of the electromagnetic spectrum. This is the "Lyman" series of spectral lines. As n becomes indefinitely large, will tend to unity. Thus, an electron falling into the atom, to the lowest energy state, will emit of a photon of 912 . On the other hand, an electron already at the lowest energy state will be knocked completely out of the atom by a photon of 912 . A hydrogen atom is thus "ionized" (H+), and the energy of a 912 photon is therefore the "ionization energy." That would be 2.179 x 10-18 J (E = hc/), but it is usually expressed in "electron volts": 13.6 eV. By comparison, to break apart a proton and a neutron bonded together (a "deuteron," the nucleus of a "deuterium" hydrogen atom) by the strong nuclear force, requires 2.224 MeV (3.563 x 10-13 J), or 163,529 times as much energy. A photon that energetic would have a wavelength of 557 fm, well into the gamma radiation () part of the electromagnetic spectrum.
When electrons drop down to the second lowest quantum level (n' = 2), they emit photons in the part of the electromagnetic spectrum that we perceive as visible light. This is the "Balmer" series of spectral lines. The lowest energy jump (3 to 2), produces a photon with the wavelength of 6560 , which is a bright red line in the spectrum of Hydrogen, usually called the "hydrogen alpha" (H) line. This is one of the conspicuous colors of the universe, since atoms of hydrogen glow with this red color when excited. That contributes a bright red line to the spectra of most stars; and "bright" nebulae in galaxies tend to be red from the excited hydrogen, blue from scattered starlight, or purple from a combination of the two. As n becomes indefinitely large, will tend to n'2, or simply 4. Thus, an electron falling into the atom, to the second lowest energy state, will emit of a photon of 3650 , just over in the UV-A part of the spectrum.
The "Paschen" and "Brackett" series both produce spectral lines in the Infrared, as electrons drop down to the third (n' = 3) and fourth (n' = 4) energy levels of the hydrogen atom. As n becomes indefinitely large, will tend to n'2, which means simply 9 and 16, respectively, which gives us the ionization wavelengths from the respective energy levels.
While Bohr's model of the atom could account for spectral lines, it still could not account for why electrons had quantacized angular momentum in atoms and why electrons could be in orbit in atoms, which would involve acceleration around the nucleus, without radiating away all their energy, which accelerated electrical charges do. The atom could simply not be a little solar system, based on charge rather than gravity, since electrons, on classical principles, would lose energy and fall into the nucleus.
An answer to these questions was offered by Prince Louis de Broglie in 1923 with the theory that, as Einstein had introduced the idea that light could behave like both waves and particles, perhaps particles of matter could also behave like waves. Thus, electrons in an atom were not moving in orbits but filled an orbit as standing waves. Familiar electromagnetic radiation, like light, exists as traveling waves, moving through space at the velocity of light. A standing wave does not move, but vibrates between fixed points, like the string on a violin. The sine wave at right represents a whole wavelength. It has a portion with a positive magnitude, a portion with a negative magnitude, and a node, which has zero magnitude. The ends of the wave, which also have zero magnitude, are usually not considered to be nodes. Half a wavelength would have no nodes; one and a half wavelengths, two nodes; and two whole wavelengths, three nodes. A one dimensional wave has nodes that are points, and it vibrates into two dimensions. Similarly, a two dimensional wave, like a wave of water on the ocean, has nodes that are lines, and it vibrates into three dimensions. A three dimensional wave, which is what electrons in an atom would be, has nodes that are surfaces. Such surfaces can be planes, cones, or spheres. By analogy, we might want to say that a three dimensional wave would vibrate into four dimensions, but this aspect of the matter does not seem have been much discussed or explored. In electron waves, each non-spherical node represents a quantum of angular momentum. Thus, a half wavelength, with no non-spherical nodes, is 0 angular momentum; a full wavelength, with one non-spherical node, is angular momentum; a wavelength and a half, with two non-spherical nodes, is 2 angular momentum; etc.
At it happens, the atom turned out to be a bit more complicated than Bohr's original atom of 1913. Each level of energy contains, not only a new integer quantum of angular momentum, but all the smaller quanta of angular momentum as well. Each energy level in an atom is distinguished, however, by the absolute number of nodes, spherical and otherwise. Thus, the first energy level, with no nodes, has only one form, with 0 angular momentum. Since the shape of the standing wave is spherical, it is called an S orbital. The Pauli Exclusion Principle allows two electrons into the orbital, one with positive spin and one with negative. In the Periodic Table of the elements, that completes the first row, with Hydrogen and Helium filling up the energy level.
The second energy level has one node. This can be either spherical, which means a spherical standing wave with a hidden spherical node inside (another S orbital), or a plane. A plane node gives us an angular momentum of . The plane node cuts the orbital in two, separating a side of positive magnitude from a side of negative magnitude. Such a wave is then called a P orbital. Now we get a further complication. An asymmetrical node produces an angular momentum vector. In classical physics, that vector can assume any orientation; but, as we might suspect, this doesn't happen the same way in quantum mechanics. The vector is quantacized and can only assume certain orientations: 2l + 1 , which is all integer values from +l to -l. These different vectors make a physical difference when an atom is placed in a magnetic field. The orbiting electrons produce a magnetic field, where the angular momentum vector produces a magnetic vector, which then assumes different orientations in an ambient field. The different vector orientations are thus called "magnetic substates" of angular momentum. In a P orbital, the magnetic substates are +1, 0, and -1. In the 0 substate, the vector is conventionally regarded as perpendicular to the z axis, and the node is thus the xy plane, symmetrical around the z axis. In the +1 and -1 substates, the nodes are the xz and yz planes. Since each substate can contain two electrons, the P orbital can contain 6 electrons overall. The second energy level thus has one S and one P orbital, and can hold 8 electrons. This row in the Periodic Table starts with Lithium and ends with Neon, which has an atomic number of 10. That is then the second of the "magic numbers," the atomic numbers where the energy levels have filled up, producing the particular chemical stability that we see in the Inert Gases.
The third energy level is characterized by 2 nodes. This can occur as an S orbital with two spherical nodes, a P orbital with one plane and one spherical node, and a new kind of orbital, the D. The two nodes of the D orbital occur in different ways. To be symmetrical around the z axis in the 0 magnetic substate, two cones are necessary, one below the xy plane, one above. We can also understand these cones to be produced by the rotation, around the z axis, of two lines that pass through the origin. This divides the wave into three lobes, one above the xy plane, one below the xy plane, and then a ring that is wrapped around the z axis in the xy plane. The +1 and -l magnetic substates are then characterized by one plane node that is symmetrical around the z axis in the xy plane and another plane that occupies the xz or the yz planes. The +2 and -2 magnetic substates then have two plane nodes, which intersect each other along the z axis, dividing the wave into four different parts. The first D orbital (3d), which can contain 10 electrons, however, turns out to have an energy comparable to the S (4s) and P (4p) orbitals in the fourth energy level. It fills up, then, only in row four of the Periodic Table, which goes from Potassium to Bromine (where the magic number is 36). This delay in filling the "higher" orbitals is characteristic of the Periodic Table, but it does occur fairly regularly. The D orbitals correspond to the "transition metals" in the Periodic Table.
Only one more kind of orbital occurs with electrons in atoms, the F orbital, which has an angular momentum of 3, allows for 7 magnetic substates, and so will hold 14 electrons. The nodes are all combinations of cones and planes, analogous to the D orbital described above. An F orbital (4f) does not begin to fill until row 6 of the Period Table, giving us the first Rare Earth series. By the time we get to the second Rare Earth series (5f), the elements are so unstable that most only exist artificially. Their chemistry is mostly not really a practical question.
When electrons are thought of as orbiting the atomic nucleus like planets do the sun, what the nucleus itself is doing is a question that may not even occur. However, when we realize that the electrons are not "orbiting" but only occupying energy levels as standing waves, and that electrons in S orbitals, which have no nodes passing through the geometrical origin, can thus be found in the nucleus, we should realize that the protons and neutrons in nuclei must occupy energy levels, and so orbitals comparable to the electrons, themselves. There are some complications with nuclear orbitals, however. Protons and neutrons are different particles and so occupy their own respective sets of orbitals. The nuclear force, by which protons and neutrons are attracted to each other (they are hadrons and baryons), observes "parity," which ends up meaning that each energy level can contain even or odd quantities of angular momentum, not both. The second energy level, therefore, with an angular momentum of , contains the familiar P orbital but no S orbital. This means that much higher level orbitals get filled up for the same number of elements by protons (up to I orbitals, with 6 angular momentum). This goes up even higher with neutrons, which accumulate faster than protons. In the heaviest known elements, neutrons are thus filling J orbitals, with 7 angular momentum. Nuclear orbitals, like the electrons in the atoms, also fill up at different levels than one would expect. This is complicated by two factors: One that it works differently for protons and neutrons; and the other that we have a phenomenon called the "spin-orbital interaction" by which particles with positive spin and particles with negative spin becomes separated from each other and fill at different times, producing very different "magic numbers" for stable nuclei than occur for electrons in nuclei.
The G, H, I, and J orbitals become increasingly complicated. Only the G orbitals, with an angular momentum of 4, and 9 magnetic substates, are shown at right. These are all superimposed on each other, of course, in the atomic nucleus, as the electron orbitals occupy the volume of the atom.
The previous diagrams have illustrated the nodal planes and cones for the different states of angular momentum. The diagram at left illustrates the angular momentum vector for 4, at the magnetic substate of m=+2. This diagram nicely demonstrates the principle that in each magnetic substate we actually have the same quantity of angular momentum. In relation to the z axis, however, there are nine different substates, as only a partial vector appears in that dimension. Since angular momentum concerns circular motion, it may seem a little strange that such motion should have a "vector," which indicates direction. In Classical physics, however, we determine the angular momentum vector with the "right handed rule": if the curled fingers of the right hand point in the direction of the circular motion, the right thumb points in the direction of the vector. For particles with 1/2 spin, they have magnetic substates of either +1/2 or -1/2, where the vectors will simply be up or down. [I have adapted this diagram elsewhere to demonstrate ideas in the metaphysics of the polynomic system of value.]
Two further complications: The waves shown are the "real" part (using real numbers) of the wave function. There is also the imaginary part of the wave function, using imaginary numbers ( -1 = i). What the physical significance of this is is a good question. On the other hand, the physical significance of the real wave function is also a good question. Werner Heisenberg and, again, Niels Bohr regarded the wave function as a "probability cloud": The square of the wave function gives a distribution for the probability of finding the electron as a particle. The wave function collapses into an actual location for the particle when an attempt is made to observe the particle. The idea that the observation creates the reality is Bohr's classic "Copenhagen Interpretation" of quantum mechanics. However, it seems inescapable that the wave function is a real and physical thing, since only a wave phenomenon can account for the interference effects that can be observed with radiation and with particles. That kind of quantum mechanics, which still observes Bohr's principle of Complementarity, but allows for two different levels of reality, seems best accommodated by a Kantian dualism of phenomena and things-in-themselves. The standing wave electrons, protons, and neutrons thus occupy real space and account for the physical size of atoms and nuclei. As long as the atom or the nucleus maintains its integrity, the waves persist; but an experimental or observational intervention to locate individual particles collapses the waves and does produce discreet locations for the particles.
Jerry B. Marion, Physics, The Foundation of Modern Science, John Wiley & Sons, Inc., 1973
Hans Frauenfelder & Ernest M. Henley, Subatomic Physics, Prentice-Hall, Inc., 1974
Roger Penrose, The Emperor's New Mind, Oxford University Press, 1990
P.W. Atkins, Quanta, A Handbook of Concepts, Oxford University Press, 1991
W.S.C. Williams, Nuclear and Particle Physics, Clarendon Press, Oxford, 1991
Kantian Quantum Mechanics
Richard Feynman's Quantum Mechanics
Philosophy of Science, Physics
Philosophy of Science
|The Metric System of 1795|
|The Basic S.I. Units|
|Obsolete CGS Units|
|magnetic flux||maxwell||10-8 Wb|
|Derived Units, S.I. & "Customary" Metric|
The ancient duodecimal (base 12) reckoning of day and night (giving 24 hours in a day) is also still with us and unthreatened. The complications that eventually produced things like the 5280 foot mile came from historical adaptations and the introduction of what seemed like "appropriate" units for different purposes, a process that continues, and not just with binary computer language. Indeed, the duodecimal twelves that turn up in many ancient and customary units, like the 12 inch foot, are arguably better than decimals, since 12 can be evenly divided by 2, 3, 4, and 6, twice as many factors as 10 (evenly divisible only by 2 and 5). This is certainly why twelves started being used, and probably will again. Meanwhile, everyone must do something that was never supposed to happen with the metric system: remember that "kilo" means "1000" in one word and "1024" in another! -- a usage that may die hard even with the introduction of binary prefixes.
Finally, the table at right show a great many official derived S.I. units, and a couple of "customary" metric units, like the metric ton and the hectare. These units falls into four broads categories:
It is curious to reflect that while now in lingustics the value of customary usage reigns supreme, often resulting in the dimissal of educated, elevated, or traditionally grammatical speech as unnecessary, inauthentic, or the classist and oppressive tool of the capitalist patriarchy, just the opposite can be found in discussion of SI units, where even traditional metric units, like the convenient ngstrom (10-10m or 0.1 nm -- about the size of an atom), can be dismissed as archaic or reactionary. This difference is instructive. The variety of customary units resulted from the practice of those dealing with particular materials. Gold was thus weighed (and still is) in (troy) ounces rather than tons or kilograms. It is not often that anyone is going to get a ton of gold together. Although kitchen measures now are available in metric units, teaspoons, tablespoons, cups, etc. provided convenient, integer values for cooking. While 250ml is about a cup, the large integer betrays an origin foreign to the kitchen -- also now using a unit, the liter, that ironically is no longer an "official" SI unit. At the same time, the Celsius measurement of temperature (no longer a basic SI unit either) has no mathematical advantage over the Fahrenheit scale, but a disadvantage for daily usage in that its increments are almost twice as large, which more crudely represents temperatures within the range of meteorological experience.
Those who despise customary units as mediaeval nonsense are thus in the position of the kind of grammatical martinet who tells people who say, "It's me," that they should say, "It is I" (so Louis XIV perhaps should have said, "L'état c'est je"?). What is awkward is when inappropriate units are imposed because of the uncompromising rationalistic cookie-cutter, as in the case of milliliters in the kitchen; but what is dangerous is losing a multi-million dollar Mars spacecraft because the engineers mixed up metric with customary units. The appropriate units for science are the SI ones, and it is as inexcusable (and more) that JPL engineers should be using feet or miles as it is that metric enthusiasts should be disparaging tablespoons or Fahrenheit temperatures in daily usage. Custom, even elevated grammatical usage, is the result of need and usage. The fundamental inspiration of the metric system, however, was rationalistic and dictatorial. In life there is in fact a place for both, and it is wisdom to know the difference.
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Light Years and Parsecs
Babylonian Numbers and Measure
Philosophy of Science
The electromagnetic spectrum is the range of types of electromagnetic radiation from the weakest radio waves to the strongest gamma rays. As described by James Clerk Maxwell (1831-1879), electromagnetic waves are an electric field oscillating in one dimension, a magnetic field oscillating perpendicular to this, and then an electromagnetic wave traveling in the direction perpendicular to the other two at the velocity of light. The velocity of light, c, is now defined as exactly 299,792,458 m/s, where c = : is the frequency (1/s, Hz) and is the wavelength (m). By international agreement, the radio portion of the electromagnetic spectrum is divided into bands based on powers of ten. For this purpose, the velocity of light is taken to be 300,000 km/s, the frequency is used in multiples of 3, and the wavelength then simply becomes multiples of 10. The first band is thus 3Hz to 30 Hz in frequency, corresponding to wavelengths of 100,000 km to 10,000 km.
|band||= frequency||= wavelength||Use|
|1||3-30 Hz||100,000 km-|
|2||30-300 Hz||10,000-1000 km||SLF|
|3||300-3000 Hz||1000-100 km||ULF|
|4||3-30 kHz||100-10 km||VLF|
|5||30-300 kHz||10-1 km||LF/LW|
|6||300-3000 kHz||1000-100 m||MF/MW AM radio|
|7||3-30 MHz||100-10 m||HF/SW|
short wave radio
|8||30-300 MHz||10-1 m||VHF TV/FM radio|
|9||300-3000 MHz||100-10 cm||UHF TV|
30-10 cm Microwave
|10||3-30 GHz||10-1 cm||SHF cm Microwave|
|11||30-300 GHz||10-1 mm||EHF mm Microwave|
|12||300-3000 GHz||1000-100 m||Infrared|
|13||3-30 THz||100-10 m|
|14||30-300 THz||10-1 m|
|15||300-3000 THz||10,000-1000 |
|Light = 7600-4000|
|UV-A = 4000-3150 |
UV-B = 3150-2800
UV-C = 2800-150
|16||3-30 PHz||1000-100 |
|17||30-300 PHz||100-10 |
|18||300-3000 PHz||10-1 |
|19||3-30 EHz||1-0.1 |
|20||30-300 EHz||0.1-0.01 |
|21||300-3000 EHz||1000-100 fm||radiation|
To zero in on the part of the spectrum of most interest to us, visible light in Band 15, we can take advantage of a couple of coincidences of nature:
The table at right, with frequencies and wavelengths, is upside down in comparison to the table of electromagnetic frequencies above. It covers two full octaves above and below middle C. At the top it starts in the Ultra-Violet. Indeed, it includes the entire UV-A part of the spectrum. At the bottom, it is well into the Infra-Red, including the boundary between Band 15 and Band 14. These infra-red frequencies can all be photographed. As a list of music notes, this gives us some sense of how little we actually can see of the electromagnetic spectrum. What is significant about this part of the spectrum, however, is that the particular spectrum of radiation emitted by the Sun peaks right in the yellow wavelengths of visible light. Of all the colors of light, yellow seems to us to be the closest in bightness and transparency to white light itself. This is not a coincidence. The Sun is a yellow star.
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Philosophy of Science
A femtometer (fm), 10-15 m, can also be called a "Fermi," since that scale, the approximate size of a proton, is useful for nuclear dimensions, as the Angstrom ( = 10-10 m) is for atomic dimensions. Indeed, the radius of an atomic nucleus is roughly equal to 1.2*A1/3 fm, where "A" is the atomic mass number, the number of protons and neutrons in the nucleus. Mass numbers (A or B) can be found in the Period Table of the Elements. The Angstrom unfortunately is now passing out of usage, since it is not part of the SI. Nanometers (10-9 m) are coming to be used instead.
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Return to "Electromagnetic Spectrum" text
|gravitational constant||G||6.673(10) x 10-11 N*m2/kg2|
|velocity of light||c||299,792,458 m/s|
|Planck's Constant||h||6.62606876(52) x 10-34 J*s|
|reduced Planck's Constant||h/2||1.054571596(82) x 10-34 J*s|
|Planck Length||(hG/c3)1/2||4.0510 x 10-35 m|
|reduced Planck Length||(G/c3)1/2||1.6160(12) x 10-35 m|
|Planck Time||(hG/c5)1/2||1.3513 x 10-43 s|
|Planck Mass||(hc/G)1/2||5.4560 x 10-8 kg|
|Planck Energy||(hc5/G)1/2||4.9036 x 109 J|
|Planck Temperature||(hc5/G)1/2/kb||3.5516 x 1032 K|
|Planck Acceleration||(c7/hG)1/2||4.9223 x 10102 m/s2|
|Planck Force||c4/G||1.2105 x 1044 N|
|tropical year||y||31,556,925.9747 s|
|Astronomical Unit||AU||149,597,892 km|
|Parallax Second||PC||206,264.806 AU|
|Light Year||LY||9,460,529,744,270 km|
|magnetic constant||0||4 x 10-7 N/A2|
|electrostatic force constant||k||8.987551788 x 109 N*m2/C2|
|permittivity of empty space,|
|1/0c2, 0||8.854187817 x 10-12 F/m|
|charge of an electron||e||1.6021917 x 10-19 C|
|electron volt||eV||1.6021917 x 10-19 J|
|mass of an electron||me||9.109558 x 10-31 kg|
|Boltzmann's constant||kb||1.380662 x 10-23 J/K|
|Stephan-Boltzmann Constant||5.67032x 10-8 W/m2/K4|
|constant in Wien's Law,|
second radiation constant
|c2 = hc/k||0.01438786 m*K|
|Avogadro's number||N||6.02 x 1023 1/mol|
I am intrigued that the "electrostatic force constant," which plays the same role in Coulomb's Law for electrostatic force that the gravitational constant does in Newton's equation for gravity, and which is given in Physics, The Foundation of Modern Science [by Jerry B. Marion, John Wiley & Sons, Inc., 1973], did not seem to be given in 62nd edition of the Handbook of Chemistry and Physics [edited by Robert C. Weast and Melvin J. Astle, CRC Press, 1981]. I thought, "Isn't there some use for Coulomb's Law anymore?" Well, apparently there is, but the law gets written differently and a different constant is used: the "permittivity constant." It is in different units, farads per meter (F/m), but this turns out to be equivalent to C2/N*m2, the reciprocal of the units of the electrostatic force constant (times 4). How that is used, and the rest of the constants, can be seen in Historic Equations in Physics and Astronomy. I now find the "permittivity" constant in the 83rd edition of the Handbook of Chemistry and Physics [edited by David R. Lide, CRC Press, 2002] called simply the "electric constant."
The "Planck" units are of significance (1) as "natural" units of measurement, based on the Planck Constant, the velocity of light, and the Gravitational Constant (with Boltzmann's Constant used to convert the Planck Energy into the Planck Temperature), and (2) for the recent and promising physics of Strings and Super-Strings. The Planck Length is regarded as the smallest physically significant distance, below which is a quantum chaos. It is also regarded as the length of the strings in String theory. The Planck Length is the scale, and the Planck Time the age of the universe, at which gravity is thought to act like all the other forces of nature. Note that while the Planck Length and Planck Time are very small, other units, like the Planck Energy, the Planck Temperature, and the Planck Force, are rather large. A 100 Watt light bulb would expend the Planck Energy in 568 days. What this may mean is that at very small scales, high energies are needed (experimentally) to reveal the structures. The Planck Temperature may have some connection to the temperature of the Universe itself when, shortly after the Big Bang, its size was at the Planck Length.
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Philosophy of Science
For both binary and hexadecimal systems, the only prime factor is, of course, 2, while for decimal counting the only prime factors are 2 and 5. The prime factors of 12, however, are 2 and 3, which means that the base is evenly divisible by 2, 3, 4, and 6. The base 10 is only evenly divisible by its prime factors.
The practical effect for duodecimal counting is especially to be seen in the fractions, at right and left (where, by analogy with hexadecimal counting, d10 = ddA & d11 = ddB). Common fractions like 1/2, 1/3, 1/4, 2/3, and 3/4 are all easily expressed as single digit duodecimals, without the repeating decimals [the repeating groups are here shown underlined] that plague the student who simply wants to deal with a third or two-thirds quantities. We get double digit duodecimals for 1/8 and 1/9.
On the other hand, the simple decimals, 1/5 (= d0.2), and d1/10 (= d0.1), do get us repeating duodecimals, as 1/5 (= dd0.2497) and 1/A (= dd0.12497); but it is a good question how much more frequently the student or other calculator needs to express a one-fifth quantity rather than a third or a quarter. The frequency of fifths and tenths that we do have may largely be an artifact of the use of the decimal system itself. It may, indeed, be easier to remember the repeating group for 1/3 in decimal (d0.3) than for 1/5 in duodecimal (dd0.2497), but the same group also works for 1/B (dd0.12497).
|Powers of Sixty|
"Factorials" multiply together successive integers. Since more integers are multiplied in duodecimal than in decimal counting, we get rounder numbers for factorials in duodecimal than in decimal, as shown at right. Factorial 5 displays the nice touch that each base shows the base for the other system, with 12 (= dd10) in decimal and A (= d10) in duodecimal.
Indeed, it is very unlikely that duodecimal counting will ever replace decimal counting. It would be hell on those who, like me, still have to use their fingers occasionally -- though those with the genetic trait of six fingers on a hand would certainly feel vindicated. It would also be necessary to memorize a larger multiplication table, though I understand that students are no longer even expected to remember the decimal multiplication table -- since some students do this better than others, and this damages their self-esteem and fosters elitism among the better students.
A compomise with sexagesimal counting might be a base thirty, whose factors are still 2, 3, and 5. All this loses is one of the 2's from 60, which was not that important anyway. Still, the requirement to have thirty symbols, and not just ten, twelve, or even sixteen (for the hexadecimal), would still be seriously cumbersome. But the number thirty does remind me of another historical counting system, that the Mayans used the base twenty (vigesimal counting). This is unique in the world, and does accompany an unusual interest in mathematics and numbers on the part of the Mayans. It does not, however, confer the sort of advantages that Babylonian counting did. The prime factors of 20 are still only 2 and 5, so Mayan numbers would still have as many repeating decimals as decimal counting. Thirty is the smallest number that is evenly divisible by the three smallest primes.
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Philosophy of Science, Mathematics
Philosophy of Science