Tuesday 28 July 2009

Rediscovered, or Did He Cheat?

In a recent post I commented on the "discovery" by a British High School student,Anthony Bayes, of a graphic relationship that allowed every point on the coordinate plane to represent a quadratic equation of the form x2+bx+c=0.

I was impressed that a high school age student would come up with the idea of a transformation from a quadratic to a point on the plane. THEN.... a few days ago I came across a post from the Mathematical Gazette, January of 1913, entitled "A Graphic Solution of the Equation xn-px+q=0, much of which dealt with the idea that the curve 4y=x2 (which the author called the Discriminant Curve) would serve as a graphic solution approach for any quadratic of the form x2-px+q=0.
Note the use of -p, a step up from what young Anthony had used, making it a slightly more useful approach, and perhaps testifying to the independence of his discovery. The author of the piece, A. Lodge, (with no other information given), did not claim it as original, nor site another use.

So the solutions to the equation x2-px+q=0 are found by the points where the tangents to 4y=x2 intersect the x-axis.

The article also pointed out, and I admit I had never thought of this, that the solutions to x2-px+q=0 are the same as the solutions to $x^2-\frac{p}{k}x+\frac{q}{k^2}=0$. It is easy enough to prove, but for the use of a nomograph like the approximations using the graph of 4y=x2, it allows you to work on a much narrower range of x and y values.

The article also referred to solutions of higher powers of trinomials that could be solved in the same way, but my little mind has not grasped it enough to explain it yet, so I save that for another day...

Saturday 25 July 2009


Just trying out a new method that allows latex on blogger, ignore for now...


$\sum_{i=1}^n x_i\$

John D. Cook, who writes a really great blog over at The Endeavour pointed out that the Latex method I use does not show up on the Google reader screen as math, but Latex script. He also sent me a link to the "TeXer" at the Art of Problem solving in one of his comments (below) that is also a nice way to clip a picture and put it into your web page.

You go to the site and type in the code and press submit, then right click to save the picture and post to Blogger in the usual way (Not as quick, but I think it will show up in reader)..

John also pointed out that you can do a lot of math-typing with straight html, and sent this link as well from his web page (thanks John).

I do NOT know LaTex well, and I know many other HS teachers do not either, but I want to learn more without taking a concentrated course.... one way to learn, and compose LaTex script is to use a widget I have on my Google home screen called "Sitmo"

You can use the character and symbol screens to construct the symbol, then copy the LaTex code, or copy the image.. and of course, its also free.

A footnote for those keeping score: I just realized this was my 250th post on Blogger....

Monday 20 July 2009

A Mandelbrot-like Set for Quadratics

The Mandelbrot set can be thought of as a catalog of closed Julia Sets over the complex plain. Each point on the plain representing a complex number, and the coloring representing whether the Julia set for that point had a closed or open orbit.

The equation y= 1/4 x2 on the coordinate plain can, in similar fashion, serve as a catalog of all possible quadratic equations. The method was discovered by a high school age British student by the name of Anthony Bayes around 1955 as reported in the Mathematical Gazette in September of that year by H. Gebert, the young man's teacher.

If every point (b,c) on the coordinate plain is representative of a quadratic equation x2+bx+c=0, then the set of all points in the cup of the parabola have only complex solutions [for example, the point (2,5) would represent the equation x2+2x+5=0 which has only complex roots]. Those points beneath the curve have two real distinct solutions, and the points on the curve specify equations which have a double root[ again, (2,1) would represent the perfect square trinomial x2+2x+1=0]. For any point on the curve, the solutions are -1/2 the x-coordinate. In the case of (2,1) the double root of x2+2x+1=0 is at -1/2 of 2 = -1.

Any line tangent to the curve passes only through points representing equations which share at least one solution. The line y=x-1 which is tangent to the curve at the point (2,1). Since the solutions for x2+2x+1=0 are both -1; all the points on this tangent represent quadratic equations which have x=-1 as one solution. As a case in point, (or a point in this case), (5,4) is on the tangent line, and the solutions of the quadratic equation x2+5x+4=0 are x=-1 and x=-4. If you drew the other tangent to the parabola passing through the point (5,4) it would contain the point (8,16) on the parabola representing the quadratic with a double root at x=-4, that is, the quadratic x2+8x+16=0.

The reflection of this tangent in the y-axis gives the tangent through the point representing the equation x^2-2x+1, with the double root at x=+1.  At this point your students should know the point of intersection of the tangent at (8,16) and the tangent at (-2,1).

This would also allow a simple approximation method to find the solutions of any quadratic with reasonably small values using only a straightedge and a printed graph of the curve y=1/4 x2. Simply pick the point (b,c) corresponding to the values of the equation, and then lay a straight edge through the point and tangent to the parabola. The solutions would be -1/2 the x values of the points of tangentcy, as given above.

Saturday 18 July 2009

"It could easily be shown..." Probability and Pi and the Riemann Zeta Function

Beware of articles that begin, "It could easily be shown..." It is like arm wrestling a two year old, if you win, so what, and if you LOSE??? Yow....
I know this, and yet, I still proceed foolishly to read them. The one currently on my mind was a "Note on Pi" by R. Chartes in the March 1904 Philosophical Magazine, my current old document of choice. It pointed out that ICEBS "that if two numbers are written down at random, the probability that they will be prime to each other is 6/pi2."
Here it is from Wolfram Mathworld:

This is the reciprocal of the famous answer to the Basel problem evaluate by Euler.

The fact that the probability that two random numbers are relatively prime was equal to this value was discovered by M. Cesaro and J. J. Sylvester in the same year, 1883. Sylvester gives a proof in a footnote to a paper I found in his collected works (page 602)

Ok, yeah, that is easy, and I should have figured it out...The proof is easy to extend to the probability that three numbers are relatively prime is the reciprocal of the sum of the reciprocal of the cubes (if that seems hard to read, try to write it). Strangely, the discovery (by Sylvester) is nested in work he was doing with Farey Fractions.

The image at the top shows a pair of coordinate axes with a point (x,y) painter black if GCF(m,n)=1, and white otherwise.

I think one of the really nice things that can be done with younger students studying common factors and slope (can I say in Alg I?) is to show them that the greatest common factor of m and n is the number of lattice points on the line from (0,0) to (m,n)....[not counting (0,0)] Here is a graph of the segments to (4,10) and (12,3)

Bromine Stinks, of course

My wife was reading something from William James that sent me running to the dictionary. He used the term "fuliginous mist", and I had no clue about its meaning. The definition (in case you didn't know either) is :""Fuliginous" is a word with a dark and dirty past -- it derives from
"fuligo," the Latin word for "soot." In an early sense (now obsolete),
"fuliginous" was used to describe noxious bodily vapors once thought to be
produced by organic processes. The 'sooty' sense, which English speakers have
been using since the early 1620s, can be used to describe everything from dense
fogs and malevolent clouds to overworked chimney sweeps. "

It reminded me of something I read recently about the discovery of Bromine

I Came across this on Elementymology & Elements Multidict by Peter van der Krogtl, and couldn't resist posting...

Bromine was discovered by two scientists working independently.
Antoine-Jérôme Balard (1802-1876), who was working in a pharmacy school in Montpellier, studying the brown seaweed Fucus, at that time Iodine was manufactured from ash of calcinated Fucus. Balard isolated a new substance. At first he thought that it was a Chlorine or Iodine compound. As he could not isolate the compound, he suggested to have found a new chemical element. Balard suggested the name muride, from the Latin word "muria" for brine.

The French Academy of Science, in turn, proposed the name brome from the Greek word bromos meaning stench (note) to indicate its strong irritating odor. In English the suffix -ine was added, since this suffix was previously used for other halogens

Almost simultaneously, in the Autumn of 1825, student Carl Löwig (1803-1890) took a bottle of a reddish liquid with an unpleasant smell to the Laboratory of Medicine and Chemistry of Prof. Leopold Gmelin (1788-1853), at the University of Heidelberg. Löwig told Gmelin that the liquid, of mineral origin, resulted from the treatment with gaseous Chlorine, thus explaining the red color. Gmelin realized that this was an unknown substance and encouraged Löwig to produce more of it so they could study it in detail. Unfortunately, winter exams and the holidays delayed Löwig's work too long. In the mean time, in 1826, Balard published his paper describing the new element.

The Japanese name has the same meaning. For the writing they use the two Chinese characters 臭 shuu kyuu = smell, stink, emit foul odor, and 素 "so" (elementary, principle, naked, or uncovered).

Thursday 16 July 2009

Another New "Old" Term

Adjusting some notes in my web page the other day, I was editing a page with a note on the Electron.
Here is some of it:

Our solar system is often called heliocentric because the sun is at the center. The purpose of that sentence was not to teach you science, but to point out the three different word roots that are used to refer to the star that gives life to our planet, sun, helios, and sol.
In ancient Greek myths Hyperion was the god of the sun, but eventually this became more associated with his son, helios. An eclipse in India in 1868 offered an opportunity to do something never before done, pass light from the Sun's atmosphere through a spectroscope. When light passes through a spectroscope it breaks into bands of unique colors that represent pure elements. One color was at a position never found on Earth. Assuming that the element only occured in the Sun, Astronomer Norman Lockyer named it Helium, "sun element". Later it was found on earth, but the name stuck.

When the Romans conquered the Greeks, their fascination with the Greek culture led to retaining many of the same myths, with the major characters replaced with a related Roman god. The Roman god associated with the sun was Sol, and thus we get words like solar system, parasol , and solarium (sun room). The para in parasol is not the same as the para in parabola and parallel, but comes from the Latin parare, which means to prepare. A parasol is thus preparation for the sun.

The word sun, itself, may come from the same Indo-European root that gave us sol, or perhaps there is an old Norse or Tutonic god out there I haven't found yet. However it started, it made its way from the Germanic into the Middle English as sunne to become our sun of today.

Another Greek word for our closest star was elector, which meant something loosely like "bringer of light". Things that reminded them of the brillance or color of the sun, such as amber, were called electrum. Even the early Greeks were aware of the effects of magnets and recoginzed that, when rubbed, amber acted like a magnet to some small light objects. In 1600 when English Scientist William Gilbert wrote De Magnete Magneticisque Corporibus, the similarity of the behavior of magnets to the effects of amber inspired him to create the word electricity from the Greek word for amber. The unit of magnetomotive force in the CGS system is called the Gilbert in his honor. Gilbert is buried in St. Johns College Chapel in Cambridge, Uk.

By now you've probably figured out that the name for the electron also is related. When JJ Thompson discovered the particle we now call the electron while studying the mysterious cathode rays in tubes, he called them corpuscles. The Word, Electron had been coined earlier by G. Johnstone Stoney in 1891. Stoney used the word to denote the unit of charge found in experiments that passed electric current through chemicals. He had written about the unit of charge as early as 1874, making him one of (if not the) first to recognize that, in his words, "For each Chemical bond which is ruptured within an electrolyte a certain quantity of electricity traverses the electrolyte, which is the same in all cases." He first referred to the unit quantity as E1.

In the same sense the term was used by Joseph Larmor, J.J. Thomson's Cambridge classmate. Larmor devised a theory of the electron that described it as a structure in the ether . But Larmor's theory did not describe the electron as a part of the atom. When the Irish physicist George Francis FitzGerald suggested in 1897 that Thomson's corpuscles were really "free electrons," he was actually disagreeing with Thomson's hypotheses. FitzGerald had in mind the kind of "electron" described by Larmor's theory. The idea did not stick, but the name did.

I was reading an old Philosophical Magazine (October, 1894) article by Stoney about the creation of the unit and the term, when I was struck by an unusual notation I had never seen before. He describes the unit of charge of the electron as three eleventhets of the C.G.S electrostatic unit of quantity. This he illustrates to be 3 x 10 -11.
After a little research, I found another article by Stoney, in the November 1899 issue of the same magazine which he had presenter earlier to The Scientific Proceedings of the Royal Dublin Society. In the article he defines several terms which, I assume, he is the creator of (please correct me if you have information on this).

If any of these terms below are still in use by someone, I would appreciate information about the nature of that use.

Tuesday 14 July 2009

A "Happy" Phrase That Didn't Stick

In the July 1912 issue of the Philosophical Magazine I came across a term I had never seen in the review of "Elements of Analytical Geometry", 1911, by Professor G. A. Gibson and Dr. P. Pinkerton.

The review praised the book, "One special feature of the book is the great number of excellent diagrams drawn on squared paper. These cannot fail to be of great instructive value to the pupil."

They went on to describe, "Another excellent feature is the early introduction of what Chrystal [George Chrystal, of whom more later] called the 'freedom equation' of a curve, in which x and y, instead of being expressed in terms of one another, are each expressed in terms of a third variable."

Then they conclude, "The method is known in some books as the parametric representation; but there is no doubt that Chrystal's distinction between the Freedom and Constraint equations is one that should be early brought to the mind of the student, and the phraseology is particularly happy (does anybody write that way anymore?). In another review where the author did not use the term, it was lamented that he had, "missed the opportunity of calling them by this picturesque title."

The term parametric was far from an established term at the time. According to "Earliest Known Uses of Some of the Words of Mathematics" by Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida.

PARAMETRIC EQUATION is found in 1894 in "On the Singularities of the Modular Equations and Curves" by John Stephen Smith in the Proceedings of the London Mathematical Society. I could not find the document in which Chrystal first used "Freedom equations", and welcome comments from those with more information. It seems like it may have been well before 1911 since an article about Chrystal in the November 2, 1911, copy of Nature indicates his contribution of the term may be little known.

It does seem the term may still have some usage in modern times as I found a definition of it at Math Resources web site, apparently a Canadian site.

Chrystal was a powerful force in Edinburgh Math Society just before 1900, and I mentioned him earlier as one of the contributors to symbols for the sub-factorial notation.

Addendum: In the Spring of 2015, Dave Renfro (one of my most valuable leaning tools) sent me a note about the Gibson/Pinkerton book and described it as "excellent for its coverage of graphing techniques." The book is available on line here.

Monday 13 July 2009

Pushing Back the Date on the Irrational Number Line

In my research on math books, I had been surprised by how recent it seemed that number lines had come into use in textbooks. Prior to this summer, my notes read, "I have not, at this point, seen a book (or article) prior to 1960 which illustrates a general number line with anything other than integers."
But over my summer road trip, I was able to move that date back considerably as a result of some documents I received from Dave Renfro. Although the term, "Number line" did not become common until around 1960, (it does appear as early as 1928) but the use of a number line with rational fractions appears in "A College Algebra" by Henry Burchard Fine in 1904,

and three years later, in 1907, there is an example of a number scale with an irrational (the square root of two) in Albert Harry Wheeler's "First Course in Algebra." I point out with some smugness that Mr. Wheeler is listed as the "Teacher of Mathematics in the English High School at Worcester, Massachusetts. Nice to have a first by a High School Teacher.

For more on my research on Math Books on the Number line The same page also has more general remarks about the development of math texts.

Thursday 9 July 2009

Dorothy we Definitely ARE in Kansas

Scott City, Kansas, to be exact..

folks who read my blog know I love unusual signs...

the sign above was in a room at the Lazy R Motel, just to the north side of town.... and yes, we still took the room....

but I did not clean any Pheasants in the room....(and my pheasant was SOOO looking forward to a bath)... I'm your basically obedient guy...

Not Your Average Hole in the Ground

My beautiful wife and the Grand Canyon....

The Canyon visit was



Overwhelmingly beautiful....
If you have never seen it, pick up you keys and start driving...

If you have, Frank Sinatra was right, it is lovelier the second time around..

Tuesday 7 July 2009

From the surface of Mercury to Mars Hill

New Horizons. August 2001. Artwork commissioned for the New Horizons mission to Pluto. Pluto's horizon spans the foreground, looking past its moon, Charon, toward the distant, star-like Sun. Painting by Dan Durda..

I had come west from Roswell to Phoenix to visit my son in July, against his good advice. He referred to Phoenix in the summer as "The surface of Mercury"... OK, he was right, IT WAS HOT...but after a few days visit he accompanied us north to Prescott (which I just learned means priests cottage) for another family visit, and then up to Flagstaff.
While I was in Flagstaff I stopped by the Lowell Observatory to visit. This is the site H.P. Lowell had created as an observatory to study Mars and also to search for a planet he predicted was outside the other planets from the effects on the outer planets, He called this not-yet-found planet, Planet X.

In the late 19th and early 20th century, observers of Mars drew long straight lines that appeared on the surface between 60 degrees north and south of the martian equator. Italian astronomer Giovanni Schiaparelli called these lines canali, which became canals in English. Lowell extended this observation to a theory that Mars had polar ice caps that would melt in the martian spring and fill the canals. He even extended the theory to include intelligent life on Mars that had designed the canals.

Eventually it became clear that there were no martian canals, but Mars hill went on to be the sight where a self educated Kansas schoolboy found his dream of working in astronomy in 1929, when the observatory director, V M Slipher,"handed the job of locating Planet X to Clyde Tombaugh, a 23-year-old Kansas man who had just arrived at the Lowell Observatory after Slipher had been impressed by a sample of his astronomical drawings." [keep astronomical drawings in mind, it matters to the story].
On the nights of Jan 23 and 30th of January, 1830, he found a planet in the images that he thought was the Planet X. "The discovery made front page news around the world. The Lowell Observatory, who had the right to name the new object, received over 1000 suggestions, from "Atlas" to "Zymal".[21] Tombaugh urged Slipher to suggest a name for the new object quickly before someone else did.[21] Name suggestions poured in from all over the world. Constance Lowell proposed Zeus, then Lowell, and finally her own first name. These suggestions were disregarded.[26]

The name "Pluto" was proposed by Venetia Burney (later Venetia Phair), an eleven-year-old schoolgirl in Oxford, England. Venetia was interested in classical mythology as well as astronomy, and considered the name, one of the alternate names of Hades, the Greek god of the Underworld, appropriate for such a presumably dark and cold world. She suggested it in a conversation with her grandfather Falconer Madan, a former librarian of Oxford University's Bodleian Library. Madan passed the name to Professor Herbert Hall Turner, who then cabled it to colleagues in America.
The object was officially named on March 24, 1930."

As I sat in the theater watching a video at the institute, a familiar name rolled across the credits...

The picture at the top of the blog is one of his commissioned works. Dan had been a student in my class the first year I taught advanced HS math. To say that I taught Dan would be a stretch, he had his head in the clouds long before he met me, and never lost his vision. Today he works at the Southwest Research Institute in Boulder, and is an expert on Asteroids. He even has one named for him, Asteroid 1992 YC3 named 6141 Durda.

Sometimes with the best ones, all you can hope to do is hold the light on the path they have already set for themselves, and help them go past.... Dan's dream was once to go cave diving on one of the porous asteroids he studies... I hope he makes it.... You Go DAN!

Sunday 5 July 2009

A Change of Focus

Had a moment to cruise some other blogs this morning and came across the video below at Casting Out Nines. Just a personal note.... I totally concur with Professor Benjamin's remarks....

History of Science Teaching in England

"The introduction of science teaching into the Universities and into the schools, both secondary and primary, in England is of comparatively recent date, and was not achieved without considerable opposition from those who regarded science as a purely utilitarian subject without cultural value."
The above paragraph is from a review of the book, History of Science Teaching in England by Dorothy M. Turner (printed in 1927) in the Philosophical Magazine in June of 1928. The rest of the review should provide all the reason any modern teacher of math/science needs to use the book to become more aware of the history of science education. Many of the comments, both pro and con for science teaching could be read in modern education journals and newspapers. A limited preview is available for free at Google Books. This is actually a 1981 reprint of the original, which is still available from Amazon

"It is interesting to note that the demand for science teaching resulted, in the first half of the nineteenth century, in the establishment of the Mechanics Institutes and night schools in which science teaching of a kind was given long before it was adopted as a general subject in the school or University curriculum and before any discussion had been given to the question why science should be taught. The author discusses in an interesting manner different views which have been held as to the reasons for teaching science."

"All students of pedagogy or of the history of science will find much to interest them in this volume."

Thursday 2 July 2009

No Elk Left Behind

North of Phoenix in the Tonto National Forest (does the Lone Ranger have a national forest???) I came across the sign above... testing taken to another un-natural extreme... the sad part is, over half the elk have failed the math section of the test...
Ok, I think (from a hasty assumption based on what I saw as I drove through) they have set up a motion sensor on one of the major elk trails where it crosses the road and when they move it starts a light flashing on the road... come on... how fast is an elk moving...wouldn't it make more sense to put a sensor on the highway for the cars and put a flashing light up to warn the elk???

Signs of Civilization

Still on the road....
We came out of Roswell along the not too dramatic Hondo River, then turned along the Rio Benito into the Capitan Mountains at just the right time. A rainstorm the day before had set the high plains ablaze in colorful blossoms.

This is the land of Billy the Kid and Smokey the Bear. Smokey was found as a cub clinging to a burnt out tree after a forest fire in the Capitan Gap. There is still a marker near the spot on Hwy 380 near Lincoln.

Back in 1881, three years after the close of the Lincoln County War (Oh.. the Ranchers and the Farmers should be friends...) and less than three months before his death, Billy the Kid made his last, and perhaps his most spectacular, jailbreak from the Lincoln jail.

Lincoln was founded in 1860 after the US Army had "brought the Mescalero Apaches under control" (Apparently they had been naughty indians). Originally settled by Spanish farmers, with the name Los Placitas del Rio Bonito (much prettier) the town was renamed Lincoln when the county was formed, and named Lincoln in 1869.

Stopping along the road to take pictures of the flowers, we found the sign posted on the front of a locked gate. We decided not to stay and visit with the Thomases in spite of their obvious hospitality, but I did wonder how the US Army in 1860 would have responded if the Apaches had put out such a sign..."Sgt, blow retreat, it's back to the fort for us.. we can't intrude on private land..." (or maybe not)..

I suppose if I really wanted the land I could go in and challenge them to a fight for it, as the precedent for using force had been clearly established, but except for the flowers, the land held little that seemed worthy of a fight. SO, we took off down across the high plains, perhaps the same direction by which Billy the Kid escaped. Off toward Soccorro and the Very Large Array Observatory, and along the way we passed the Trinity site, where the Atomic Bomb was first tested.... Civilization....

Wednesday 1 July 2009

A Good (old) Read

"...our school Curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled many a find mind. It is the aim of this book to restore this cultural content and present the evolution of number as the profoundly human story which it is"
Such were the comments of Tobias Dantzing about his book, Number, The Language of Science, the book of which Einstein said, "This is beyond doubt the most interesting book about the evolution of mathematics which has ever fallen into my hands.
If you have never read it, peruse the preview at Google Books, and maybe pick it up at Amazon... In my mind, it is a book for teachers to read every summer.