MATH POWER

Scientific knowledge is truth, isn't it? Right. But history has a way of delaying truth. It's called societal inertia. Theories start with a guess; then if we're lucky the guess gets properly refined as time goes by. But sometimes it takes lots of time. And the first guess is seldom right.

These are the facts of life:*
Item: Astrology is a guess. Astronomy its scientific spinoff.
Item: Ptolemy guesses at an earth-centered universe. Galileo says, "No, it is sun centered." He's jailed for that; not pardoned 'til late 20th century.
Item: Newton fathoms the laws of gravitation assuming an infinite speed of propagation. Einstein revisits them, recognizing that the speed of propagation is a universal finite constant.
Item: Leibniz champions infinitesimals as teeny-tiny numbers. Newton says, "I work only with ratios." Euler says, "Infinitesimals are really all

zero." Robinson shows in 1960 that Euler knew what he was talking about.
Item: Young guesses that human color vision is trichromatic. Helmholtz says, "No, we sense brightness, hue, and saturation; not red, green, & blue as such." Helmholtz is not jailed, but the Royal Society throws its considerable weight behind Young; and today we have the "Young-Helmholtz trichromatic (RGB) theory," not the "Helmholtz BHS theory." Late 20th-century psychophysical experiments support Helmholtz' view over Young's; show BHS has a simple, evo- lution-friendly theoretical basis integrating day, night, and color vision.
Item: Mathematicians proclaim the delta function a monstrosity but Heavi- side uses it anyway. He's vindicated by Dirac & Schwartz in 1950 & 1955.

Journal editors also have a hand in delaying and even molding the truth. History is rife with examples of great men having their pioneering efforts rejected by those in control who thought they knew better. Newton, Mendel, Planck - all were subject to such abuses. An accounting is given by James V.Bradley in "Editorial Overkill," Bulletin of the Psychonomic Society, 1982, Vol.19, No.5, pp.271-4. Bradley reports that Ziman (1980) referred to science as "Design by Committee."

It remains the fashion today to say that infinitesimals are really, really small - "But not zero!" By contrast, Euler saw them as zero but multiples of one another just the same. In 1755 he described infinitesimals this way:[1]

...There exist infinite orders of infinitely small quantities, which though they all = 0, still have to be well distinguished among themselves, if we look at their mutual relation, which is explained by a geometrical ratio."

(Was Leonard suggesting that we might divide by zero? Oh my!)
Mathematicians en masse discounted his words, saying "Even the good Euler sometimes sleeps." But Robinson recently showed that Euler was not asleep.

Abraham Robinson showed in 1960 that in the realm of standard real numbers, infinitesimals are zero,[2] while in the larger realm of nonstandard hyperreal numbers they are not all the same zero. Although 20 = 0, still 20 is twice 0 in the sense that 2x is twice x; and Euler clearly saw the importance of that distinction while most others did not. The 20th-century historian Dirk Struik wrote, "We cannot, however, be enthusiastic about Euler's way..." but not everyone agrees with Struik's assessment.

It is found that "Euler's Way" opens doors long thought to be forever closed. Widder wrote,[3] "...the reciprocal of an infinity is an infinitesimal ..." in agreement with Euler's Way and in conflict with the prevailing view in 1947, and still today, that infinitesimals are real nonzero numbers and infinities are not numbers at all.

With Euler's Way strictly adhered to, it is found that useful and amazing things begin to happen in the world where denominators are zero.

Euler's Way allows x/x to be replaced with 1 and 2x/x with 2 even when x is 0. We have actually been doing that since Newton's time, justifying it by invoking l'Hospital's rule. (But it seems Newton's way was simply to cancel x before letting it be zero!) Anyway the fact of the matter remains, supporting Euler's Way with infinitesimals.

[1] From Dirk J.Struik, A Concise History of Mathematics, 4th rev.ed., Dover Publications, 1987, ISBN 0-486-60255-9, page 125. (The first three editions stop at the end of the 19th century; the 4th edition continues to 1950.)

[2] "0 is the only standard infinitesimal," p.25, A.E.Hurd & P.A.Loeb, An Introduction to Nonstandard Real Analysis, Academic Press,Inc., 1985, ISBN 0-12-362440-1.

[3] David Widder, Advanced Calculus, Prentice-Hall, 1947, p.231

In a recent issue we speculated (as many have before) that gravity may be simply electricity in disguise. It may come as news to hear that Einstein drew a similar parallel although he is most famous in that field for geometrizing gravity in his general theory of relativity. From Brian:*

Asked what evidence sustained his hunch that the forces of the universe had a single source, he [Einstein] said, "The relation between electricity and gravity must be very close. Light waves, heat waves, radio waves and gravitation all go with the same velocity. In the final conception I believe these phenomena cannot be separated.

That does not mean electrical effects can be parlayed into gravitational ones. It only means an effect in one system may have an analog in the other. Thus if a particular particle motion is known to make a cloud of electrons mimic a cloud of positive charges, then a similar motion of ordinary masses might generate an antigravity field.

Dial m for slope:
Homer: Do you know why we use lowercase "m" for slope of a line? -Robin Mayer

According to Marvin Bittinger & David Ellenbogen: "This usage has its roots in the French verb monter, to climb." Intermediate Algebra, 5th Ed.,p.90,1998.

The sign of 0:
This letter was received 27 Feb'98 in response to a precursor to "The ideal element h: Evaluating functions of 1/x" (Dec.2000 issue).

Mr. Tilton: I've looked at your paper, and the main point seems to be the existence of numbers whose sign is "like" the sign of 0. But I find that idea problematic. The reason 0 has no sign is that numbers come in three flavors; negative, zero, and positive. Zero is neither positive nor negative, but that's because it's zero. All other numbers are either positive or negative, and so I have difficulty understanding what number could be the value of sgm(0). It is not 1 (a positive value) nor -1 (a negative value) nor 0. So what is it? Have you invented a third class of numbers that are neither positive nor negative? ... Sincerely yours,
...David Meredith, Prof.of Mathematics, San Francisco State University

Dear David:
Thank you for commenting; and while you ask good questions, they dance all around the point showing I've made a poor argument. Also recall that sqrt(-1) is neither 1 nor -1 nor 0. The reason you give for 0 being neither positive nor negative, "because it's zero," is only Popeye logic, "I yam what I yam." When you say "0 has no sign" I assume what you mean is that 0 is neither positive nor negative, which is one textbook characterization.* I do not propose a nonzero real number devoid of any possible sign. Nor do I rule out 0 having a third kind of sign whose nature remains to be appreciated, and I shall feel free to continue using the phrase "the sign of 0" in a quite obvious way.

Finally it seems clear that Arctan(1/0) has the absolute value (1/2) and the sign of 0 since the sign of Arctan(1/x) is the sign of x. [Even though 1/0 is not a real number, |Arctan(1/0)| is. 1/0 is only a placeholder symbol there.] So if we can agree that a number is the confluence of its absolute value and its sign there should be no need for further argument. No doubt this situation is uncommon which is good not bad, because it is the uncommon that is often most interesting and useful.

On the other hand if we would choose a different route and decide to outlaw 1/x at x=0 in an effort to avoid the uncommon, then forms like x(1/x) could no longer be evaluated there. But virtually all agree that x/x=1 even when x=0 by l'Hospital's rule. And once this exception is made, then it seems we must allow Arctan(1/x) for x=0 and follow where the path leads.

Please also see "The ideal element h: Evaluating functions of 1/x" in the Dec. 2000 issue. / Best wishes, ...HBT

The symbols "sgm" and "sgn" (lower case) have the same meaning. The use of "sgm" is being discontinued. When the computer signum is meant instead, it will be in all upper case as in SGN(0)=0. Note that sgn(0) is not 0. In all other respects SGN and sgn are the same. Pertinent definitions are: sgn(x) = (2/)[Arctan(1/x) + Arctan(x)], sgn(0) = (2/)Arctan(1/0) = sgm(0) = h, and |h|=1.

While there may be disagreement over whether the statement "0 has no sign" is literally true, it seems all can agree that 0 is neither positive nor negative, exclusively. But what is the sign of |0|?

The definition |x|=sqrt((x²)) is finding its way into more texts, but it remains common to see (often in the same book) absolute value defined as { |x|=x when x>0, |x|=-x when x<0, and |0|=0 }; but that definition requires the sign of |0| to be the same as the sign of 0. The following analysis shows it is not.

Leonhard Euler showed that arctan(1/x) can be evaluated without first finding 1/x, so the fact that 1/0 is undefined is of no consequence; and |Arctan(1/0)| = Arctan(1/|0|) =(1/2), a positive number, meaning |0| is positive.

So it cannot be true that |0|=0; for the number on the left is positive but the number on the right is not. Some may say, "Oh, hey, we're talking about distance here, and distance doesn't need a sign"; but such sophistry sacri- fices rigor for expediency. It ignores the fact that |0| "becomes" positive as soon as we grant Arctan(1/x) membership in the society of functions and see that it has a nonzero, finite absolute value for x=0.

While we cannot correctly write |0|=0, we can write h|0|=0 where h is a number with the sign of 0 and the magnitude of 1. Why not a magnitude of 0 so we can write 0|0|=0? Because that would violate Euler's Way with infinitesi- mals and erroneously imply that x|x|=x. Note that |h|=1=h² and 1/h=h; h is the unit hysteretic number. And yes, h is a "new" kind of number.

Conclusions

It follows from Arctan(1/|0|)=(1/2) that sgn(±|0|)=±1 where sgn is the ideal signum function defined this way: sgn(x)=(2/)[Arctan(1/x)+Arctan(x)]. Also: sgn(x)|x|=x, 1/sgn(x)=sgn(x), sgn(1/x)=sgn(x), |sgn x|=1, and sgn²x=1 even for x=0; and sgn(0)=h.

It is concluded that |0| is positive; and that it would be imprecise to write |0|=0. It is also concluded that sqrt((0²))=|0| is positive, and that 0² is positive because x² can be written |x|².

Aftermath
When the identities sgn(0)=h and sgn(±|0|)=±1 are compared, it is seen they have the same absolute values (namely 1), as do the two arguments of sgn (namely |0|). Last month we wrote <±>1 for h, with the ± sign bracketed to signify a dynamic directional process.* Since h|0|=0, the meaning of the symbol <±> is such that <±>|0|=0, <±> is effectively the sign of 0, and it is as if 0 has two "faces" or is the dual quantity +|0| and -|0|. A lesser conclusion would violate Occam's razor.

Algebra texts continue to use inconsistent terminology in connection with imaginary and complex numbers. But taking the Argand complex-number diagram as a given, the following things become clear:*

1. Imaginary numbers and real numbers including 0 can properly be called 'complex' because they lie in the complex number plane.

2. A complex number such as 5+4i cannot properly be called 'imaginary' (any more than it can be called 'real') because it does not lie on the imaginary axis (or the real axis) of the Argand plane.

3. 'Pure imaginary' is without any meaning beyond 'imaginary.'

4. In a+ib, a is the real part, but b is not the imaginary part. Both a and b are real, and b is properly called the coefficient of the imaginary part.

5. "+i" & "-i" act like signs halfway between + and -. Therein lies the value of complex numbers to indicate direction; and if we study sgn(z)=z/|z|=exp(i) where z is a complex number, we can come to appreciate what that means. Note that exp(i) is not a function of r, and |exp(i)|=1 identically.

Which came first: imaginary or complex numbers? As with other areas of mathematics, the history is incomplete and often confusing.

David Eugene Smith's A Source Book in Mathematics (1959, Dover) starting on page 46 gives excerpts from John Wallis's 1673 book, Algebra, in which he speaks of "Imaginary Roots" of quadratic and cubic equations. And starting on page 55, Smith gives excerpts from Caspar Wessel's 1797 paper concerning complex numbers, "On the Analytical Representation of Direction; An Attempt." Euler (1707-1783) introduced the notation i for sqrt(-1), according to historian Petr Beckmann. Other names connected with imaginary and complex numbers are Argand (1768-1822), Cauchy (1789-1857), and Steinmetz (1865-1923). But there are many others, too.

Recommended reading: Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton U.P., 1998,ISBN 0-691-02795-1.

Visualize this: A semi-infinite field of green on the right directly abuts a semi-infinite field of red on the left, the boundary being straight "up and down." That is simply the two-dimensional Cartesian coordinate plane where positive x-values are green, and negative ones are red. All points in that plane are either green or red; there is no line of a third color separating them — no separate "zero land." The Y-axis has no separate existence. It is an idea; a locale or location, an abutment, a transition, an edge, a demarkation, a boundary or border. It is not a thing; not a locus or line as those terms are generally understood.

Now imagine yourself to be an infinitely small GREEN point situated just to the right of the boundary.* You move steadily to the left. When are you not GREEN anymore? Certainly not until you cross the boundary. Only when you have crossed the border moving westward from Kansas to Colorado can you properly say, "We're not in Kansas anymore." A like scenario applies to a point that starts out RED and moves right; it does not toggle from RED to GREEN until it crosses the boundary.

So what color is a point when it is "directly on" the boundary, "at zero"? Actually that question is a nonstarter,** but it is regularly asked anyway and demands an answer. A reasonable answer is, "It is GREEN when the boundary is reached from the right, RED when reached from the left." In this picture may be seen the nature of the "third sign," the sign of 0, <±>. Note that this view does not violate the dictum that "0 is neither positive nor negative," for that dictum only means 0 is not exclusively either positive or negative, and is meant only to affirm its neutrality.

* "GREEN point" is to be read "point in the GREEN field." These are not physical colors; they are idea colors not subject to wavelength limitations.
** Like inquiring about the status of a steel ball "balanced" on a razor's edge. It is a point of instability attainable only momentarily by moving across it.

Exercise 1 -
Prove that x/|x| = sgn(x) everywhere. Do not take sgn(x)|x|=x as a given.
HINT: Consider it in the GREEN and in the RED then at x=0.

Exercise 2 -
Prove sgn(0)=h by showing that (2/)Arctan(1/0)|0|=0. Assume Ex.1.
HINT: Show that (2/)Arctan(1/x)|x|=x holds when x=0.

Signum by clipping -
The definition of sgn in terms of Arctan given earlier depends on the product of two irrational numbers, and 1/; and so its usefulness in an algorithm is perhaps limited. A "better" way is described now.

It is found that |u+1|-|u-1| = 2sgn(x) if u=1/x+x. When evaluating that for all RED and GREEN points (see "A point out standing in its field," above), we find that 1/x appears only in the form 1/x-1/x, and so its undefined nature at x=0 is of no consequence; for just as x/x is everywhere 1, 1/x-1/x is every- where 0. And just as 1/x does not have to be evaluated first in Arctan(1/x) neither does it have to be evaluated first in |u+1|-|u-1|.

Words of the month -
nonstarter - something that is not productive
sophistry - subtly deceptive reasoning or argumentation

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Written reader comments are invited on all material. Those intended for my attention must be submitted by U.S. Mail to my Tucson address. All such comments are subject to being published unless requested otherwise. They may also be subject to editing. ... Homer Tilton, Editor

MATH POWER is published monthly. It is published and edited by Homer B.Tilton under the auspices of Pima Community College, East Campus, 8181 E.Irvington Rd. Tucson AZ 85709-4000. All material is copyrighted by Homer B.Tilton unless otherwise noted. A limited number of copies may be made at educational institutions for internal use of faculty and students. For more extended copying or to request additional copies contact the Editor at the above address. Letters and editorial material are welcome. All submitted material may be published in MATH POWER, and edited, unless specifically requested otherwise.