letters Numbers and "Reality"
To the Editor: The letters in the August 1991 issue of this Journal by Canagaratna and by Wadlinger regarding the units of T in the exoression sin T miss a critical mint about trieonometry, namely, that it isn't real. This circumstance is quite analogous to the discovery by Riemann and others during the 19th Century that Euclidian geometry was not real. That is, it was found that a rational mathematical system having every single one of the properties of Euclidian geometry could be developed without once using the words or the concepts of 'line", "point", "plane", "intersection", and so forth. It was an accident that this fundamental mathematical system appeared to us first in the guise of somethine real. This discoverv. of course, led to the so-called non-kuclidian ge~metrie~iand to relativity and to atomic bombs and to the Cold War and to the end of Russian imperialistic Communism, as James Burke might say.) It is entirely possible to develop a mathematical system which has all the properties of trigonometry without ever mentioning the word "angle." Many of the elementary trigonometry books now used in high school and especially in colleees teachine a hieh-school level course do so. Some years ago I taug& out gf a book that relegated angle trigonometrv to a verv few Dazes at the end of one chapter. Most modern calcu~us"texts'~i~e a treatment similar k~the one below. A quite rigorous discussion can be found in T. W. Komer's Fourier Analysis (Cambridge University Press, 1988.) The moper s . . understandine.< of trieonometrv. b e- ~ n with the usual (x,y, coordinate plane containing a circle centered on the on& and havinga radius of 1 unit (the "unit circle"). Now t&e a number s&le having the same size intervals as the x-axis and place this scale on the x-axis so that the zero of this scale and the point (1,0), which is of course on the unit circle, coincide. Wrap this number scale around the circumference of the unit circle: the positive scale is wrapped in an anticlockwise manner and the negative scale is wrapped in a clockwise manner (this is called "the wrapping function"). We now have the entire set of real numbers wrapped around the circumference of the unit circle. We can define the sine of the real number 1, which is found on the number scale wraDDed around the unit circle, as the y-coordinate of the poil;t'on the number scale correspondingto the real number 1.The cosine of the real number 1is thexsoordinate. The tangent is the ratio v l x . and so forth. Note that the notion of "anele" is combletely absent in this re-statement of trigonometry. All of the facts of trieonometrv are as true in this svstem as in the old angle system. N& also that there is absolutely no unit to the arrmment of the function, that is, the Tin sin T has no units.it's a pure number. We can now, to suit our practical needs, define the angle subtended by the number 1on the wrapped scale when a line is d r a g from it to the center of theunit circle and the positive x-coordinate, as an angle of 1radian, or 57.2958', or 387 Klingons, or whatever other thing we please, and go on to work our practical problems from there. The units we choose to use &e irrelevant, since the mathematics is entirely a system of ideas. It just happens that this system is useful to us in solving real problems, such as the length of the diagonal of a cubic unit cell in a crystal, and so forth.
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Journal of Chemical Education
When I want to figure out the area of a rectangular object measured in meters. I am ~erfectlvfree to ienore the units. use logarithms to manipulate the numbers, and stick the units back in at the end. Loearitbms are another ~ r o ~ e r t v of certain real numbers that are pure ideas. ~ogariibm; are also not real. Strauee contortions of inventing some sort of hypothetical reference state to get rid of th; units are completelv unnecessaw If you don't need the units, take them out; and stick sohething rational back in at the end of thecalculation Ifneeded Never let the mathematics tell you what the chemistry is. The pH of a given solution is properly defined as the reading of a healthy, calibrated (!I, pH meter, not -log [some equilibrium property of the hydrogen ion concentrationl. (We've been lying to our students for decades about the real defmition of pH.) If your calculations give you a number different from the meter reading, the assumptions leading to your calculation or the calculationitselfis just plain wrong. Chemistry is the reality. Mathematics is just one of the ways of talking about reality. Mathematicians, if they pay any attention at all to us scientists, and I mean particularly chemists, find us ludicrous. The areuments that we science-tvnes get ourselves into regarding the units of arguments iffnn&ons are incomorehensible to them. Thev don't know what is the matter &th us, since we are not saying anything meaningful. I think that the two letters mentioned above are an example of the worship of unit-factorization gone mad. We've forgotten that we own all this stuff and we can do with it as we please. All of mathematics has notlung to do with science. We find it extremely . helpful - to use mathematics in science. It's just another tool we humans have craRed to use for whatever pleases us. Contrary to an unfortunately popular idea, the-language of science is not mathematics. Mathematics is not a human language a t all. Science is a human activitv. The laneuaee of science is laneuaee. The number "1"is pure idea. 6 e statement "1atom" is something completely different.
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Siegfried N. Lodwig
Science Division Centralia College Centralia, WA 98531
To the Editor: My letter in the August 1991 issue of this Journal concerned the advisability of using the same conventions in mathematics, physics, and chemistry. Thus x in mathematics is a pure number and consequently attempts to invest it with units would create problems without yielding substantial advantages. I also stated that in the expression sin T, T is a pure number. Lodwig is in agreement with this point of view. I agree with Lodwig that mathematics can only be used to represent 'reality". One way to do this is to make a correspondence between "numbers" (the magnitude of various quantities )and the elements of a mathematical system. In this method we keep track of the units in our head, and after the manioulation is over. out the units back. This is perfectly valid: but prone to error. But this is not the only valid way we can make the correspondence, as Lodwig
