Wednesday, 29 November 2023

Too Nice to Ignore, Gravity and Bernoulli's Lemniscate

 

From a 2009 post with additional material added.


*Wik
Two stories intersect here, one a famous event from the history of calculus that most folks are familiar with, and one that seems not to make it much into classrooms and that I only learned about today. 
Almost every student of mathematics will sooner or later come across the beautiful problem of the Brachistochrone, which in Greek means "shortest time." It is the path that will carry a point-like body from one place to another in the least amount of time under the force of constant gravity. Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve. The curve does not depend on the body's mass or on the strength of the gravitational constant.

The story goes that Johann Bernoulli posed the problem to readers of Acta Eruditorum in June, 1696. He published his solution along with four other solutions from Newton, Jakob Bernoulli, Gottfried Leibniz, and Ehrenfried Walther von Tschirnhaus. ( l'Hôpital also seems to have had a correct solution, but it was not published).

Newton historians claim that Newton received the problem in the mail one afternoon after returning from his job at the mint (so this would be after he was older). The story goes that he solved it overnight, and posted it the next morning. Since it took weeks for some of the others to solve it, we may assume that Newton was still a pretty good mathematician well after his known prime.

A footnote to this story, in the epic novel Moby Dick, "Ishmael thinks about them while cleaning the try-pots (giant cauldrons in which whale blubber is rendered) on the deck of the Pequod.  It was in the ...trypot with the soapstone diligently circling around me, that I was first struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, ...,will descend from any point in precisely the same time."  
How would Melville's modest education bring him this incredible math fact.  A possible answer.....Joseph Henry, you know, the guy for whom the unit of inductance is named.  
It is almost certain that the limited public school education of Melville would not include this fact.  Most high school students today would never be introduced to it.  But in Melville's brief time at the Albany Academy it was said that Herman excelled in "ciphering" and won the school prize.  Perhaps his interest in geometry and such was inspired by an outstanding teacher, and former alumni of the Albany Academy, young Joseph Henry.

(Once Upon A Prime by Sarah Hart)  


Ok, so that's the one everybody knows about. But I was just fixing up some notes on the life of Gian Francesco Malfatti, whose date of death was, well, today, in 1807. Now Malfetti did some nice stuff, too. He did some really important work on fifth degree equations; but he is best known for a geometry problem about three mutually tangent circles inscribed in a triangle. He posed the problem of as how to
,
*Mathworld
carve three circular columns out of a triangular block of marble, using as much of the marble as possible. He thought the solution was described by the triangle problem. The Geometry problem of constructing three circles each tangent to each other and two sides of the triangle is now generally known as Malfatti's problem, even though he didn't do it first, Japanese geometer Chokuen Ajima beat him too it.  (Both the earliest solutions, I'm told, used a combination of geometry and algebra, but it seems Steiner did show how to do it with pure geometric methods). What's worse is that it wasn't even the solution to the physical problem of the columns he was trying to show. Around 1930 someone proved that it wasn't always the best solution, and then in the 60's, M. Goldberg showed it was NEVER the best solution.

Then while checking some dates on his Wikipedia entry, I noticed something else he had done, the something I had never heard of, and this time he was right. He was working with the lemniscate (it means ribbon) first described in 1694 by Jakob Bernoulli, and he noticed an interesting gravitational relationship about it. If you draw a chord (pick a chord, any chord... ok; sorry) through the center and any other point on the lemniscate, then a point acting under the influence of gravity will reach that point of intersection at the same moment, whether it travels down the chord, or around the lemniscate. Now that just seems to nice to have ignored in math and calculus classrooms. If you are a teacher, maybe the next time you talk about the brachistochrone, (or maybe tomorrow when they need a diversion) you should point out this little beauty, also.

Tuesday, 28 November 2023

Strange Connections, The Thesaurus Guy

 


From a 2011 post with some additions.

I recently came across a note that the Peter Mark Roget whose name is associated so closely with the thesaurus was a scientific type.  More particularly, in 1815 he invented the log-log scale (the logarithm of the logarithm of the number on the C and D scales... ) on the slide rule which facilitated finding powers and roots of numbers. 

I looked into his history and found that his education was in medicine and his work on the thesaurus was part of a lifelong coping mechanism to fight depression.  Roget described his thesaurus in the foreword to the first edition in 1852:

"It is now nearly fifty years since I first projected a system of verbal classification similar to that on which the present work is founded. Conceiving that such a compilation might help to supply my own deficiencies, I had, in the year 1805, completed a classed catalogue of words on a small scale, but on the same principle, and nearly in the same form, as the Thesaurus now published."

But before he actually got around to publishing his great book, he not only invented the slide rule scale, but following the death of Sir John Herschel, Roget was secretary of the Royal Society from 1827 to 1848. He also made observations in sight and persistence of vision that influenced the development of movies. 

From"Cheshire Antiquities,© Craig Thornber, Cheshire, England, UK.
"His observations were based initially on looking at the world through a series of slits such as one might have in a vertical Venetian blind or palisade. A rotating cartwheel viewed through such as system gives an optical illusion. The spokes at the top and bottom appear straight but those at the sides appear to bend downwards. Roget worked out the path of the light to show how this happened. He went on to explain a phenomenon that often perplexes devotees of Westerns, a hundred years before the invention of film. At certain speeds, the cartwheel appears to stop or go backwards. Roget's observations were made by viewing through vertical slits but he showed the position of each spoke in the wheel at each glimpse and how this could lead to the optical illusion of stasis or backward motion. The same phenomenon is observed when a film is made with a cine camera. In 1820, Roget worked with Michael Faraday and Joseph Plateau in a series of experiments on vision leading to Roget's paper to the Royal Society on the Persistence of Vision. Roget's work showed that an image persists in human perception for about one sixteenth of a second and this forms the basis on which animations, film and television are based. "
(Wikipedia)

On 9 December 1824, Roget presented a paper entitled Explanation of an optical deception in the appearance of the spokes of a wheel when seen through vertical apertures. ...
While Roget's explanation of the illusion was probably wrong, his consideration of the illusion of motion was an important point in the history of film, and probably influenced the development of the Thaumatrope, the Phenakistiscope and the Zoetrope.


Amazon offers a paperback (20 pages) book from Roget's paper.



Roget also was involved in the creation of University of London and the precursor to the Royal Medical Society.    A busy Guy...





-------------------------------------------------------

William Ensign Lincoln invented the definitive zoetrope in 1865 when he was about 18 years old and a sophomore at Brown University, Providence, Rhode Island. Lincoln's patented version had the viewing slits on a level above the pictures, which allowed the use of easily replaceable strips of images. It also had an illustrated paper disc on the base, which was not always exploited on the commercially produced versions. On advice of a local bookstore owner, Lincoln sent a model to color lithographers and board game manufacturers Milton Bradley and Co.

The name zoetrope was composed from the Greek root words ζωή zoe, "life" and τρόπος tropos, "turning" as a translation of "wheel of life". The term was coined by inventor William E. LincolnW.E. Lincoln's U.S. Patent No. 64,117 of April 23, 1867 




While I was searching this out, I came across the fact that the invention of the Ln scale (for finding e^x) was by an 11th grade high school student. 
From a post by Robert Adams:


The Ln scale was invented by a high school student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent (in the range 0 to 2.3) on the Ln scale and read e^x on the C (or D) scale and e^(-x) on the CI (or DI) scale. Pickett and Eckel were given exclusive rights to the scale in the early sixties. Later, Stephen Cohen created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of their slide rules.



And just one more footnote, I always found it a little quirky that the log scale on the slide rule was linear.

After I wrote this, several readers commented about people you might not know were science/math folks.  Steven Colyer offered three offhand :

Art Garfunkel studied Mathematics at Columbia University.

Mick Jagger went to The London School of Economics.

August Ferdinand Möbius of Möbius strip fame was primarily an Astronomer.

I added that actress Teri Hatcher (Lois Lane, Desperate housewives...) studied math and engineering at  De Anza College. Not an unlikely choice since her mother was a computer programmer at Lockhead-Martin and her dad was a nuclear physicist.

Steve reminded me that Winnie on "Wonder Years" long ago, Danica McKellar, studied math at UCLA and later wrote several books, including "Math Doesn't Suck" to motivate young women towards math science studies.

And in more modern time several writers on The Simpsons, and Mayim Bialik on The Big Bang Theory have impressive Sci/math credentials.

Surprise me with your list of strange connections.


 

Sunday, 26 November 2023

Political Polls and Margins of Error????

 This was written just before the election in 2008, and most of the polls had settled into a 53-45 spread for Obama, which, for all my nay-saying in this blog, was very close.  




Well, we are almost to the election, which means an end, finally, to the interminable projection polls. Ok, I actually like statistics, but I'm not sure I accept that political polls are not playing a little fast and loose with the assumptions that are needed to compute confidence intervals. I love it when the election goes the wrong way and they have to come up with scenarios for WHY they blew it. Of course with so many of them out there making 95% confidence intervals, about five percent of the ones you hear SHOULD be wrong... but I think there is more to the problem than just that.

I came across a blog from Iowahawk ( I didn't provide a link because my students come here and some of his language is not the sort of thing I display for my students..they know all the words anyway, but they won't hear them from me) that had a nice expression of what I felt, so I stole parts of it shamelessly...


Statisticians love balls and urns. A typical Stats 101 midterm, for example, usually includes a question along these lines:


"You take a simple random sample of 1000 balls from an urn containing 120,000,000 red and blue balls, and your sample shows 450 red balls and 550 blue balls. Construct a 95% confidence interval for the true proportion of blue balls in the urn."
From this the typical Intro stats student can deduce that they are 95% certain the real proportion of blue balls in that urn is 55%, plus or minus 3.1% .

"This is, for all intents and purposes, how political pollsters compute the mysterious "margin of error," which has everything to do (and only to do) with pure mathematical sampling error. If you look at the formula above and round it just a smidge, you get a simple rule of thumb for the margin of error of a sampled probability:
Margin of Error = 1 / sqrt(n)

So if the sample size is 400, the margin of error is 1/20 = 5%; if the sample size is 625 the margin of error is 1/25 = 4%; if the sample size is 1000, it's about 3%.


"It works pretty well if you're interested in hypothetical colored balls in hypothetical urns, or survival rates of plants in a controlled experiment, or defects in a batch of factory products. It may even work well if you're interested in blind cola taste tests. But what if the thing you are studying doesn't quite fit the balls & urns template?"



What if 40% of the balls have personally chosen to live in an urn that you legally can't stick your hand into?

What if 50% of the balls who live in the legal urn explicitly refuse to let you select them?

What if the balls inside the urn are constantly interacting and talking and arguing with each other, and can decide to change their color on a whim?

What if you have to rely on the balls to report their own color, and some unknown number are probably lying to you?

What if you've been hired to count balls by a company who has endorsed blue as their favorite color?

What if you have outsourced the urn-ball counting to part-time temp balls, most of whom happen to be blue?

What if the balls inside the urn are listening to you counting out there, and it affects whether they want to be counted, and/or which color they want to be?

If one or more of the above statements are true, then the formula for margin of error simplifies to

Margin of Error = Who the heck knows?

Saturday, 25 November 2023

The Dead Grandmother Syndrome

 


Everyone knows the stress of testing can be detrimental to students, but it seems that some college professors have observed that it can be even more hazardous to their family members. Recently I came across a (very tongue-in-cheek) article by Mike Adams of the Biology Department at Eastern Connecticut State Univ. about the strength of this effect.

"It has long been theorized that the week prior to an exam is an extremely dangerous time for the relatives of college students. Ever since I began my teaching career, I heard vague comments, incomplete references and unfinished remarks, all alluding to the "Dead Grandmother Problem." Few colleagues would ever be explicit in their description of what they knew, but I quickly discovered that anyone who was involved in teaching at the college level would react to any mention of the concept. In my travels I found that a similar phenomenon is known in other countries. In England it is called the "Graveyard Grannies'' problem, in France the "Chere Grand'mere," while in Bulgaria it is inexplicably known as "The Toadstool Waxing Plan" (I may have had some problems here with the translation. Since the revolution this may have changed anyway.) Although the problem may be international in scope it is here in the USA that it reaches its culmination, so it is only fitting that the first warnings emanate here also."

"
The basic problem can be stated very simply: A student's grandmother is far more likely to die suddenly just before the student takes an exam, than at any other time of year.

"While this idea has long been a matter of conjecture or merely a part of the folklore of college teaching, I can now confirm that the phenomenon is real. For over twenty years I have collected data on this supposed relationship, and have not only confirmed what most faculty had suspected, but also found some additional aspects of this process that are of potential importance to the future of the country. The results presented in this report provide a chilling picture and should waken the profession and the general public to a serious health and sociological problem before it is too late."

The rest of the article goes on to "document" the existance of the effect, and prescribe potential interventions."


Friday, 24 November 2023

Typing Monkeys

  

More observations stimulated by John Barrows book, "One Hundred Essential Things You Didn't Know You Didn't Know."  Everybody has heard the suggestion that a million (or some other number) of monkeys typing continuously for many millennia would eventually produce a) Shakespeare, b) all of known science, c) the bible, d) all of the above). 
It began with Jonathon Swift and Gulliver's Travels, 1872, according to Professor Barrow. In the tale "a mythical professor of the Grand Academy of Lagado who aims to generate a catalog of all scientific knowledge by having his students continuously generate random strings of letters..." (I think, see emphasis in the excerpt below, that it was random strings of words).. Anyway, according to the good Professor Barrow, the story was embellished in different forms until French Mathematician Emile Borel{there is a street and a square named for him in the 17th District in Paris} suggested that random typing monkeys could duplicate the French national library. A few years later(1929), Arthur Eddington Anglicised that to "books in the British Museum." By 1972, Arthur Koestler writing in The Case of the Midwife Toad, New York, 1972, page 30, refers to Monkeys typing Shakespeare as "proverbial":"Neo-Darwinism does indeed carry the nineteenth-century brand of materialism to its extreme limits--to the proverbial monkey at the typewriter, hitting by pure chance on the proper keys to produce a Shakespeare sonnet." Ok, so eventually someone had to put this to a more scientific test, and they did. "A website entitled The Monkey Shakespeare Simulator, launched on July 1, 2003, contained a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from Henry IV, Part 2, reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:"RUMOUR. Open your ears; 9r5j5&?OWTY Z0d... " Even more impressive, to me, is the fact that 'in another part of that book, Swift tells of how the astronomers on the flying island of Laputia had: "discovered two lesser stars, or satellites, which revolve around Mars, whereof the innermost is distant from the center of the primary exactly three of his diameters, and the outermost five: the former revolves in the space of ten hours, and the latter in twenty-one and a half". Swift wrote this in 1726, but it was not until 1877 that Asaph Hall discovered the two moons of Mars.'... I just did a little checking on the orbit and periods he "predicted?" and the actual periods are about 7 hours for Phobos, and about 30 for Deimos... and their distance from the planet were about 9 x 103km and 23.5 x 103km. Mars has a diameter of 6.794 x 103km so they are closer to 1.5 and 3 radii away it seems, but wow, for 100 years before the actual discovery??? Don't you wonder what made him use Mars instead of Venus or ??? Wait, maybe authors typing randomly can describe the true nature of the universe (with some limits of error)... ------------------------------------------------------------------------------------------ Ok best guess,  Swift had heard of a misinterpretation of Johannes Kepler. He had interpreted the anagram that Galileo Galilei sent to him in 1609 to tell him about the discovery of the phases of Venus as the discovery of two moons of Mars. 

 If you have your copy, here is what I found in Chapter Five of Gulliver's Travels. The whole thing is available at the Guttenburg Project. "The first professor I saw, was in a very large room, with forty pupils about him. After salutation, observing me to look earnestly upon a frame, which took up the greatest part of both the length and breadth of the room, he said, "Perhaps I might wonder to see him employed in a project for improving speculative knowledge, by practical and mechanical operations. But the world would soon be sensible of its usefulness; and he flattered himself, that a more noble, exalted thought never sprang in any other man's head. Every one knew how laborious the usual method is of attaining to arts and sciences; whereas, by his contrivance, the most ignorant person, at a reasonable charge, and with a little bodily labour, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study." He then led me to the frame, about the sides, whereof all his pupils stood in ranks. It was twenty feet square, placed in the middle of the room. The superfices was composed of several bits of wood, about the bigness of a die, but some larger than others. They were all linked together by slender wires. These bits of wood were covered, on every square, with paper pasted on them; and on these papers were written all the words of their language, in their several moods, tenses, and declensions; but without any order. The professor then desired me "to observe; for he was going to set his engine at work." The pupils, at his command, took each of them hold of an iron handle, whereof there were forty fixed round the edges of the frame; and giving them a sudden turn, the whole disposition of the words was entirely changed. He then commanded six-and-thirty of the lads, to read the several lines softly, as they appeared upon the frame; and where they found three or four words together that might make part of a sentence, they dictated to the four remaining boys, who were scribes. This work was repeated three or four times, and at every turn, the engine was so contrived, that the words shifted into new places, as the square bits of wood moved upside down. Six hours a day the young students were employed in this labour; and the professor showed me several volumes in large folio, already collected, of broken sentences, which he intended to piece together, and out of those rich materials, to give the world a complete body of all arts and sciences; which, however, might be still improved, and much expedited, if the public would raise a fund for making and employing five hundred such frames in Lagado, and oblige the managers to contribute in common their several collections. He assured me "that this invention had employed all his thoughts from his youth; that he had emptied the whole vocabulary into his frame, and made the strictest computation of the general proportion there is in books between the numbers of particles, nouns, and verbs, and other parts of speech."

The Monkeys, well, they showed up a little later.  (Félix-Édouard-Justin-) Émile Borel (7 Jan 1871; 3 Feb 1956) was a French mathematician who (with René Baire and Henri Lebesgue), was among the pioneers of measure theory and its application to probability theory. In  1913, in one of his books on probability, he proposed the thought experiment that a monkey hitting keys at random on a typewriter keyboard will - with absolute certainty - eventually type every book in France's Bibliothèque nationale de France (National Library). This is now popularly known as the infinite monkey theorem.

In 2024 a report of a computer simulation testing how long it would take was posted on ARXIV by Ergon Cugler de Moraes Silva, of Sao Paulo, Brazil which gives, "In essence, inscribing ‘In the endeavor to generate the entire phrase ‘To be, or not to be, that is the Question’, the magnitude of the challenge becomes staggering. It would necessitate approximately 2.68×1069 attempts, consuming a colossal time span of 2.95×1066 seconds, equivalent to 8.18×10^62 hours or 9.32×10^55 years. To put this temporal scale into perspective, it’s crucial to emphasize that 9.32×10^55 years is roughly 6.75×10^45 times greater than the estimated age of the universe, which stands at 1.38×10^10 years.."

Monday, 20 November 2023

Notes on the History of the Pigeonhole Theorem

 

This is an update of several posts I wrote as late as 2009, and some additional information acquired since then.


Sometimes problems that seem very hard, can be very easy if they are viewed in the right way, and one of those easy ways to make some hard problems manageable is the Pigeon-Hole Principle. Over the last few weeks seems like lots of problems invovling this idea have shown up, so I thought I would bring it to you.
The basic idea is so easy any sixth grader would agree; if you have two boxes, and you are going to put three balls in the boxes, then at least one box will get more than one ball..... "well, Duh!" they answer... and yet... it seems easier to apply than it might be. Now that you know the secret, try these two problems. I'll post the answer down lower on the page where you must not look until you take a few minutes to ponder the problems.
Here is the first from a recent blog I read: "39 people are attending a large, formal dinner, which must of course occur at a single, circular table. The guests, after milling about for a while, sit down to eat. It is then pointed out to them that there are name cards labeling assigned seats, and not a single one has sat in the seat assigned to them. Prove that there is some way to rotate the table so that at least two people are in the correct seats."
This one seems tougher, but really isn't, it just requires a different way of thinking. "Suppose you pick six unique integers from 1 to 1000. Prove that at least two of them must have a difference that is a multiple of five.

I'll give you the proofs of each of these, and then get to the main topic of the history of this important theorem in discrete mathematics

Ok, The Proofs... for number one... Suppose you handed each person a number that was how many seats they needed to move to the right to find their assigned seat. Since no one is at the right seat, the number can not be zero or thirty-nine. SO each of the people has a number between 1 and 38...wait, there are 39 people...two of them (at least) must be the same distance away from their assigned seats.... admit it…..that’s pretty cool.
For number two it is sort of the same idea, but you have to think about how much each number would have for a remainder if you divided them by five. The only possible choices are 0, 1, 2, 3, or 4... but there are six numbers, so two of them have the same remainder...and two numbers that have the same remainder on division by five, are a multiple of five apart.... think of 1,6, 11, etc for remainders of one. If you want to read more about how remainders can play a part in solving problems, see my blog on "casting out sevens"


The basic idea behind this mathematical principle is what students would call common sense; if there are n objects to be placed in m receptacles (with m less than n), at least two of the items must go into the same container. While the idea is common sense, in the hands of a capable mathematician it can be made to do uncommon things. Here is a link to an article by Alexander Bogomolny in which he uses the principle to argue that there must be at least two persons in New York City with the same number of hairs on their head. This "counting hairs" approach dates back to the earliest version of the principal I have ever seen.

The same axiom is often named in honor of Dirichlet who used it in solving Pell's equation. The pigeon seems to be a recent addition, as Jeff Miller's web site on the first use of some math words gives, "Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, A partition calculus in set theory, Bull. Am. Math. Soc. 62 (Sept. 1956)" (although they credit Dedekind for the principle). In a recent discussion on a history group Julio Cabillon added that there are a variety of names in different countries for the idea. His list included "le principe des tiroirs de Dirichlet", French for the principle of the drawers of Dirichlet, and the Portugese "principio da casa dos pombos" for the house of pigeons principle and "das gavetas de Dirichlet" for the drawers of Dirichlet. It also is sometimes simply called Dirichlet's principle and most simply of all, the box principle. Jozef Przytycki wrote me to add, "In Polish we use also:"the principle of the drawers of Dirichlet"
that is 'Zasada szufladkowa Dirichleta' ". I received a note that said, "Dirichlet first wrote about it in Recherches sur les formes quadratiques à coefficients et à indéterminées complexes (J. reine u. angew. Math. (24 (1842) 291 371) = Math. Werke, (1889 1897), which was reprinted by Chelsea, 1969, vol. I, pp. 533-618. On pp. 579-580, he uses the principle."


He doesn't give it a name. In later works he called it the "Schubfach Prinzip" [which I am told means "drawer principle" in German]

The idea has been around much longer than Dirichlet, however, as I found out in June of 2009 when Dave Renfro sent me word that the idea pops up in the unexpected (at least by me) work, "Portraits of the seventeenth century, historic and literary", by Charles Augustin Sainte-Beuve. During his description of Mme. de Longuevillle, who was Ann-Genevieve De Bourbon, and lived from 1619 to 1679 he tells the following story:
"I asked M. Nicole (See below for description of M. Nicole) one day what was the character of Mme. de Longueville's mind; he told me she had a very keen and very delicate mind in knowledge of the character of individuals, but that it was very small, very weak, very limited on matters of science and reasoning, and on all speculative matters in which there was no question of sentiment ' For example,' added he, ' I told her one day that I could bet and prove that there were in Paris at least two inhabitants who had the same number of hairs upon their head, though I could not point out who were those two persons. She said I could not be certain of it until I had counted the hairs of the two persons. Here is my demonstration/ I said to her: M lay it down as a fact that the best-fiimbhed (not sure what this word was supposed to be, ..Plumed??) head does not possess more than 200,000 hairs, and the most scantily furnished head b that which has only 1 hair. If, now you suppose that 200,000 heads all have a different number of hairs, they must each have one of the numbers of hairs which are between 1 and 200,000; for if we suppose that there were 2 among these 200,000 who had the same number of hairs, I win my bet But suppose these 200,000 inhabitants all have a different number of hairs, if I bring in a single other inhabitant who has hairs and has no more than 200,000 of them, it necessarily follows that this number of hairs, whatever it b, will be found between 1 and 200,000, and, consequently, b equal in number of hairs to one of the 200,000 heads. Now, as instead of one inhabitant more than 200,000, there are, in all, nearly 800,000 inhabitants in Paris, you see plainly that there must be many heads equal in number of hairs, although I have not counted them.' Mme. de Longuevillle still could not understand that demonstration could be made of the equality in number of hairs, and she always maintained that the only way to prove it was to count them. "
The M. Nicole who demonstrated the principal was Pierre Nicole, (1625 -1695), one of the most distinguished of the French Jansenist writers, sometimes compared more favorably than Pascal for his writings on the moral reasoning of the Port Royal Jansenists. It may be that he had picked up the principal from Antoine Arnauld, another Port Royal Jansenist who was an influential mathematician and logician. Here is a segment from his bio at the St. Andrews Math History site.
-------------------------
He published Port-Royal Grammar in 1660 which was strongly influenced by Descartes' Regulae. In Port-Royal Grammar Arnauld argued that mental processes and grammar are virtually the same thing. Since mental processes are carried out by all human beings, he argued for a universal grammar. Modern linguistic theorists consider this work as the beginnings of the modern approach their subject. Arnauld's next work was Port-Royal Logic which was another book of major importance. It was also strongly influenced by Descartes' Regulae and also gave a first hand account of Pascal's Méthode. This work presented a theory of ideas which remained important in philosophy courses until comparatively recent times. In 1667 Arnauld published New Elements of Geometry. This work was based on Euclid's Elements and was intended to give a new approach to teaching geometry rather than new geometrical theorems."
He was a correspondent of Gottfried Wilhelm Leibniz, and of course Pascal, who wrote the Pascal "Provincial Letters" in support of Arnauld. I enjoyed the quote about him from the Wikipedia bio: "His inexhaustible energy is best expressed by his famous reply to Nicole, who complained of feeling tired. 'Tired!' echoed Arnauld, 'when you have all eternity to rest in?"
I have not been able to find any thing in Arnauld's personal writing at this time to confirm that he was aware of or used the Pigeon-hole Principle. I have also seen a comment that there is a book by Henry (or Henrik) van Etten (pseudonym of Jean Leurechon, who coined the term thermometer) , circa 1624, which uses the method for problems involving "if there are more pages than words on any page" and various other illustrations. The writer suggests that the problem is in the French version but not the English translation. Would love to hear from someone who can confirm, and perhaps send a digital image.

Around five years after I wrote the above, I was advised of a paper published by A. Heeffer and B. Rittaud that mentioned this Leurechon (They give the date as 1622) contained a single line about the principle, and amazingly, that involved the idea of proving two men had equal hairs on their heads.  " “It is necessary that two men have the same number of hairs, gold, and others.”
Later the authors add, "It is now established that an immensely popular work published at Pont-`a-Mousson in 1624 resulted from these disputationes. Entitled R´ecr´eation mathematicque, this French work is commonly attributed to Jean Leure-chon, but there are good reasons to believe that this attribution is wrong."  This book goes on to explain the solution from the idea posed in the 1622 book.  They say there is an English translation from 1633, which is [Jean Appier Hanzelet],Mathematicall Recreations , T. Cotes (1633).


After the fact

Shortly after I wrote the original post, I had a classroom encounter with a student who presented me with another teaching moment.

A young man in one of my classes, obviously trying to improve his A+ by sucking up to the teacher, mentioned that he had read my recent blog on the pigeon-hole principle. He went on to suggest that he really doubted the idea that 39 people could randomly seat themselves and ALL be in the wrong seat. "It just seems VERY unlikely." he suggested.

Rather than tell him the answer, I set him the task of simulating the activity with a deck of cards. Pull out any suit, say the spades, and really shuffle the remaining cards well. Now we need to decide on an order for the remaining suits, so let clubs be the numbers one to thirteen in order from Ace, two, up to King for thirteen. Then the ace of diamonds can be 14, up through the King of diamonds for 26. Finally the ace of hearts is 27 up to the king of hearts for 39. Now turn over the cards and as you do count, one, two, etc... and if you get a card that is where it should be, stop.. they didn't all sit in the wrong chairs. You need not go on forever, just ten or so trials should give you an idea of whether the event is really, really uncommon, or not so very uncommon. (I now realize an easier way to do this would be to have two decks of cards, lay 39 out in one row in order, then from the shuffled deck, lay the cards one at a time under where they should appear.)

I didn't tell him that I knew the probability (or a good approximation), and that he should probably get three or four trials in a string of ten shuffles in which none of the cards landed in the right place. Such a mis-ordering of the cards was just the idea behind the first critical study of the idea we now call derangements by Leonhard Euler, the great Swiss mathematician. Euler was studying the probability of winning in the game of rencontre, now called "coincidences" in his paper "Calcul de la Probabilite dans le jeu de Rencontre", published around 1751.

So what did Euler discover? Well for larger values of N, say 39 or so, the probability of having a perfect mis-sorting of the items approaches 1/e, or about 36.8%, more than a third of the time. It is not an unusual event at all. For smaller numbers you can find the probability by using the idea shown here for six items..
. This can be rewritten more easily using the factorial notation as P= 1/2! - 1/3! + 1/4! - 1/5! + 1/6! which is only a tiny bit above 36.8%, already very close to the 1/e value given above for the limiting value. If the number of items is even, the series will be a little more than 1/e, and if it is odd (and the last term is subtracted) then the probability will be a little below 1/e, with the propbability approaching 1/e as a limit as n gets greater and greater.

My student got two completely mismatched sets of 39, and expressed surprise that it was higher than he would have thought, but he didn't sound convinced that what had occurred was not just an unusual anomaly (or else he thought I might have rigged it somehow?)

I decided to simulate a lot more times than would be practical with a deck of cards, so I cranked up Fathom, a wonderful simulation software by the folks at Key Curriculum, and had it repeat the experiment of seating the 39 people at random 1000 times, and then count how many landed in the right place. The results are shown in the graph below.

It happened that no one landed in the right place 371 times.... Hmmmm, I guess Euler got it right.

Comments about additional sources related to this are always welcomed. 

Wednesday, 15 November 2023

Some Notes on Division, and its History (Including Alien Division for Fractions)

  The word Divide shares its major root with the word widow. The root vidua refers to a separation. In widow the meaning is obvious, one who is separated from the spouse. A similar version of the word was often meant to describe the feeling of bereavement that a widow would feel. The prefix, di, of divide is a contraction of dis, a two based word meaning apart or away, as in the process of division in which equal parts are separated from each other. Notice that the vi part of vidua is also derived from a two word, and is the same root as in vigesimal (two tens), for things related to twenty. An individual is one who can not be divided.

In a division problem such as 24 / 6 = 4 the number being divided, in this case the 24, is called the dividend and the number that is being used to divide it, the 6, is called the divisor. The four is called the quotient. If the quotient is not a factor of the dividend, then some quantity will remain after division. This quantity is usually called the remainder, although residue sometimes is used. The Treviso Arithmetic uses the word lauanzo for remainder. In Frank Swetz's book, Capitalism and Arithmetic he gives, "The term lauanzo apparently evolved from l'avenzo, meaning a surplus, or in a business context, a profit." Swetz also points out that in the 15th Century the term partition (partire in Latin) was synonymous with the word divisision.

In today's schools almost every grade school student learns to divide, so students may be surprised to learn that in the 16th century schools Division was only taught in the University. One of the first arithmetics for the general public that treated the subject of division was Rechenung nach der lenge, auff den Linihen vnd Feder by Adam Riese. Here is how the Math History page at St Andrews University in Scotland described it,
"It was published in 1550 and was a textbook written for everyone, not just for scientists and engineers. The book contains addition, subtraction, multiplication and, very surprisingly for that period, also division. At that time division could only be learnt at the University of Altdorf (near Nürnberg) and even most scientists did not know how to divide; so it is astonishing that Ries explained it in a textbook designed for everyone to use."

I think it is even more astonishing that the sitution described still existed in 1550 in Germany. Perhaps the earliest "arithmetic" to provide instruction in the local vernacular of the common people was the 1478 "Treviso Arithmetic", so named because it was printed in the city of Treviso (the author is unknown) just north of Venice. Frank J Swetz writes about the situation in Capitalism and Arithmetic (pg 10):
From the fourteenth century on, merchants from the north traveled to Italy, particularly to Venice, to learn the arte de mercadanta, the mercantile art, of the Italians. Sons of German businessman flocked to Venice to study...


Early algorithms for division:
By the middle ages there seem to have been five approaches to the process of division.

The first was called the Galleygalea, or Scratch method. This method was efficient in a period of expensive paper and quill pens since it required less figures than other methods. Even the modern long division method requires more figures. The name Galley was used because the resulting pattern after the division left a picture that seemed to remind the early reckoning masters of the shape of a ship at sail. The term “scratch” has to do with the crossing out of values to be replaced with new ones in the process. The ease with which this could be done on a sand board or counting board made it a popular approach in the cultures of the East, and the method is believed to come from the early Hindu or Chinese. For example, Cajori writes, "It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure." He also comments on the popularity of this method, " For a long time the galley or scratch method was used almost to the entire exclusion of the other methods. As late as the seventeenth century it was preferred to the one now in vogue. It was adopted in Spain, Germany, and England. It is found in the works of Tonstall, Recorde, Stifel, Stevin, Wallis, Napier, and Oughtred. Not until the beginning of the eighteenth century was it superseded in England. " Addendum (I just found a note that says that the famous Arab mathematician,  Al-Khwarizmi, whose book laid the foundation for western introduction to the now-common Arabic numbers, used this method in his writing. The Hindu  mathematical calculations were usually done on a  dust covered tablet, and they would wipe out numbers instead of scratching through them so that in the final result, only results appeared.  The scratch-out method in Europe was simply an adaptation for paper or slates. )

Here is an image comparing how the galley method works shown beside the current US Model for long division, which the Italians called a danda.

The page the image is from has a nice step by step illustration of the process. 

In 2005 I acquired a German student "copy book" from 1804 which seems to show the Galley division method and the student's illustration of the ship around the work. 



A second method that was sometimes taught was the process of repeated subtraction. The image below shows an example from Ray's New Practical Arithmetic published in 1877. I have seen this method in an English textbook as late as 1961 (Public School Arithmetic by Baker and Bourne). It also appears in a 1932 US publication of Practical Arithmetic, by George H. Van Tuyl, and perhaps in others .


This method of subtraction grew into what is taught frequently as an alternative use to many modern classrooms, and I believe is the standard method at some levels for a program called "Everyday Math".  Instead of slowly subtracting one divisor at a time, the use of simple multiples is used to group subtract.
For example, to divide 227 by 8, it is easy to see that 10 x 8 or eighty can be subtracted, so they might take out ten groups of eight repeatedly until the remaining part was too small to divide by 80.  So after removing 20 groups of eight,  there would be 67 remaining.  At this point they might recognize that 8x8 is 64 so by removing  8 more sets of eight would leave only three remaining, so the quotient would be 28 with 3 remaining.   
A video of this method is shown here
(I have been informed that the "correct" term for this method is "partial quotients".)

A third method was called per repiego by parts, which I have seen in books into the 20th century. In this method a division was accomplished by breaking the divisor into its factors, and then dividing the dividend by one of the factors, and sequentially dividing the resulting quotient by each remaining factor in turn to get a final quotient. The problem below is modeled on a problem in the 1919 copyright A School Arithmetic, by Hall and Stevens.
divide 92467 by 168 or 4 x 6 x 7


4|92467

6|23116 …. groups of four and 3 units over

7| 3852 ….. groups of 24 (4x6) and 4 foursover

___550 groups of 168 and 2 twenty-fours over



The complete remainder is 2 (24) + 4(4) + 3 = 67




A fourth method
 is presented in the Liber Abaci, by Fibonacci in 1202. After introducing how to divide by numbers of one digit,
and then larger primes, he develops a set of "Composition Rules" for numbers with more than one digit. A composed fraction might look like image at right. Fibonacci used the Arabic method of writing fractions from right to left, and this composed fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation, 67/75 with each part of the numerator being read over the product of all the denominators below or to the right.
The "composition" of 75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the fraction 1/75.

When he divides 749 by 75, he first uses only the first denominator, 3. The quotient of 749 by three is 249 with a remainder of 2. The 2 is placed as a numerator over the three, and the 249 is divided by the second number in the denominator (a five). 249 divided by 5 gives 49 with a remainder of four.

This remainder, 4, is placed as a second number in the numerator over the five in the denominator. Now the 49 is divided by the final number in the denominator (another five) and the quotient is 9 with another remainder of four. This four is placed over the final five and the nine is placed to the right as the quotient. Fibonacci then gives the answer of 749 divided by 75 as 9 and 4/5 + 4/25 + 2/75 or 9 74/75.   



A fifth method, which is similar to what we would now called short division except that the student used a table of division or multiplication facts. The method was called per colona, by the column, or per tavoletta by the table, in reference to the table of facts used. An example of this method appears in another popular American arithmetic by Nicholas Pike, from 1826. The use of tables to aid in multiplication and division were a common practice from the 1400’s up to the early 20th century.



The sixth is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step. The image below shows a typical long division problem with the partial products crossed out and the resulting "Italian method" on the right.

The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 The Common School Arithmetic : prepared for the use of academies and common schools in the United States by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided.


In a Gresham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school."



Later as I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . (It seems when I just checked that the Met no longer allows that link, will replace ASAP) The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.



I recently found a site called The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one
interesting note
, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk, and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.

The Agony and the Obelus, much Ado about Notation

Recently (in 2015) James Tanton posted a short article about problems that are circulating on the internet such as (and this is the one he used) "What is the value of the following expression: 62 ÷ 2(3)+4, and then asked, "Is the answer 10 or is the answer 58?" (my personal choice for historical reasons explained below is 3.6)

I don't care to argue the possible choices, although Professor Tanton does a good job of that in his blog, but I'm more interested in the history of some symbols for division he mentions there, obelus, vinculum, and one he didn't, the solidus. In particular, I'm interested in how the usage may have changed over time.

The earliest of the three terms to appear was the vinculum, and it came to us from the Hindu or Arabic mathematicians between the seventh and twelfth century. Here is how it is described by Jeff Miller's excellent web page on the first use of math symbols
Ordinary fractions without the horizontal bar. According to Smith (vol. 2, page 215), it is probable that our method of writing common fractions is due essentially to the Hindus, although they did not use the bar. Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.

The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).

Several sources attribute the horizontal fraction bar to al-Hassar around 1200.

Now if you read Prof. Tanton's article, in which he ecstatically plugs the use of the vinculum, this is NOT what he is suggesting. The horizontal fraction bar made its way into western culture mostly on the back of Leonardo Fibonacci, who introduced both Arabic numbers, and some of their symbols. He referred to the fraction bar as "uirgula"; which has become the more modern word virgule, something like a wand or small rod. Unfortunately, today the virgule is a term interchangeable with the older term solidus, and you recognize it as the slanted fraction bar, as in 3/5 (and occasionally with an s like bend such as the current symbol for integration), but all that would come much later.

The use of the vinculum that has the professor so excited was introduced around 1452 by Nicholas Chuquet The word is from the diminutive of vincere, to tie. Vinculum referred to a small cord for binding the hands or feet often used to keep cattle from wandering too far afield as they grazed in common areas. The meaning in math is mostly unchanged from that original meaning. The vinculum notation was once used in much the same way we now use parenthesis and brackets to "bind together" a group of numbers or symbols. Where today we might write (2x+3)5 the early users of the vinculum would write 2x+3_5 . Originally the line was placed under the items to be grouped although a bar over the grouping became the more lasting usage (and still is in the symbols for the radical sign for roots, and the repeat bar for decimal fractions). The bar on top seems to have been first used by Frans van Schooten.

 Dr Peterson at the Math Forum disagrees with calling the fraction bar a vinculum and has written, "I find no evidence, by the way, that it has ever properly been called a vinculum, which is a bar OVER an expression and serves to group it as parentheses do today. The fraction bar has something in common with that, but not enough in my opinion to justify the usage. With both vinculum and virgule used for other things, I just call it a fraction bar and am perfectly happy with that term!" (I'm OK with that, too.) Professor Tanton suggest that the vinculum, properly used, would eliminate questions about whether the answer to the question is 58, 10, (or 3.6).

The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking.  Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger and the same name is applied to an editing symbol that looks like a little dagger, . The root also gives rise to the word obelisk for a pointed pillar of stone.
 The symbol(s) was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction, for instance, by the famed Adam Riese as early as 1525, although he did not use it exclusively, intermixing the standard horizontal subtraction bar. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659. 
There has long been a controversy about whether the symbol was introduced to him by John Pell. Cajori in his famous book on mathematical notation says there is no evidence for this, but some later historians, Jacqueline A. Stedall for one, now think it quite probably was Pell's creation. Pell had been Rahn's teacher in Zurich and they communicated on the book. Pell was famous for vacillating over whether he would, or would not, let his name be used on information he shared with others.

Let me make it clear I am not an authority on math history and do not read German,  but as I looked at the examples in Teutsche Algebra, I began to think that Pell/Rahn was not introducing this as a mathematical operator as it is now used. I could find no examples where the books used something like the expression in the problem in Prof. Tanton's blog.  Instead it seems to be used exclusively for a shorthand in explaining the operations used.  

Here is an image from page 76 of the Algebra, and it is using a method of teaching algebra by use of a 3 column format, which is certainly from the work of Pell. Each line contains a line number in the middle, instructions for what is being done to the equation in the left column, and the result in the right column. Today many solutions would simply show the sequence of equations in the right column.


The first two lines describe the given information. In the third line, the swirl is exponentiation and says that equation 1 has been squared on both sides. It is line 8 that provides the interesting note about the ÷ usage. The left column says equation 7 is divided by GG+1, but if you look at the right side, you will see that 7 ÷ GG+1 treats all the material to the right of the expression as if it were included in a parenthetical enclosure. Don't divide by GG and then add 1, but divide by the total quantity GG+1.

Now the two surprises here, for me, is that a) Rahn/Pell intends that the "÷" breaks the operation into two parts, the left and the right side as if they were enclosed in parentheses or marked with a vinculum. But the second, is that he doesn't use the expression as an operator in his expressions. Instead he uses the common horizontal division bar/vinculum common to others. So when did we begin to use "÷" as an operation with numbers. I do not have access to the great libraries that contain the early English arithmetics and algebras that eagerly adopted the obelus (it was almost never used anywhere except in English speaking countries), so I am hoping some of you who have more experience/access/knowledge can share so the rest of us will know. When did expressions like 62 ÷ 2(3)+4 first appear in arihtmetic/algebra books? (At the moment I suspect they are a 20th century creation.)

So what about the Solidus. The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for example 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.
The solidus was introduced as a fraction/division symbol first suggested in De Morgan's Calculus of Functions he proposes the use of the slant line or "solidus" for printing fractions in the text, as in 3/4. In 188 G. G. Stokes put this into practice. Cayley would write to Stokes, "I think the solidus' looks very well indeed . . . ; it would give you a strong claim to be President of a Society for the prevention of Cruelty to Printers."
Stokes, in explaining his choice, says that the slanted bar is already in use for fractions, and simply uses it to expand to algebraic division. Then he states an explanation of the operational use, "In the use of the solidus, it seems convenient to enact that it shall as far as possible take the place of the horizontal bar for which it stands, and accordingly that what stands immediately on the two sides of it shall be regarded as welded into one." He then gives examples that make clear that he intends that a / bc means abc . He even gives a method for a period stop to indicate that the grouping has ended, so a/b.c would mean ab(c)

So when did this end. When did we make the switch to the confusion of PEMDAS or BEMDAS or whatever it is called in your country. Cajori (1929) suggests that when using division and multiplication, "there is at present no agreement as to which sign shall be used first."  So it seems that the advent of memorized mnemonics independent of the symbol seems to have occurred later than that.  Similarly in 1923 the National Committee on Mathematical Requirements of the MAA recommended that the ÷ and : for division be replaced with the / solidus "(where the meaning is clear}."

So I looked on my bookshelf and found a 1939 copy of The New Curriculum Arithmetics, Grade Seven.  The authors are a professor of elementary education, a dean of a school of education, a superintendent of schools, and an elementary supervisor, surely folks who would be aware of the MAA recommendations, and yet, there was the ÷ all through the problem sets.  What was not there was a section on order of operations, or any problems that went beyond " number ÷ number."  No long strings of numbers and operations strung together.

Certainly the question was in the air, but unsettled in 1938 when Joseph A. Nyberg of Hyde Park HS in Chicago wrote in The Mathematics Teacher
 
Read the part in Italics again.... multiplication first, then division, without regard to the order.  That is not what you are telling your students today (I hope).  So maybe they were just working it out.... Nope, here is what N. J. Lennes had written in The American Mathematical Monthly in the article Discussions Relating to the Order of Operations in Algebra in February of 1917, 21 years earlier.

Better, right?  then turn the page, and find
So there is our old friend the obelus used exactly as I suspect Pell and Rahn had intended (if they intended it to be used as an operator at all), and lower down the solidus in the manner that Stokes suggested, but apparently used in a way the users thought distinguished it from the use of the obelus.  And you wonder why your students are confused?

I still have yet to resolve when the first use of the obelus appeared for division as an operator in an algebraic or arithmetic problem.  Anyone who has more information, please share. 
 I will continue my search as time allows and when I find out more I will continue to update this post. Thank you for any information you can share.







Division of Fractions by the Alien Method (and followup)


 I wrote about an experience that happened when I let my kids watch an old science fiction movie in class just before Christmas... The blog, and a followup requested by a teacher who admitted he wasn't really sure why the common "divide and multiply method worked... Here they are as a package...
-------------------------------------------------------
The day before Christmas break one of my seminar students brought in the old (1951) video of "The The Day the Earth Stood Still". I worked at my desk as they watched, and about thirty minutes in they called my attention to ask if the math on the blackboard was "real". The Alien in the movie, Klatu(Michael Rennie), in the company of a young boy who lived in the house where he was renting a room, had entered the home of a professor who was supposedly knowledgeable about Astro Physics. I did not recognize any physics I knew from the brief shot of what looked like differential equations of no particular relation, but that could be my limited physics more than the actual images.
I returned to work, but in a few minutes in another scene, Klatu is helping Bobby with his homework and the only line you hear is "All you have to remember is first find the common denominator, and then divide." My head pops up... what were they doing? "Common denominators" leads to thoughts of fractions, but almost no one teaches finding common denominators as a prelude to dividing fractions (which is sort of a shame because it makes division of fractions work like multiplication...the way kids think it should.) It works in fact, if you do not find the common denominator first, but sometimes the answer is as confusing as the problem.
When you multiply fractions, as every fifth grader learns, you just multiply top times top and bottom times bottom... 2/3 x 5/7 = 10/21. The fact that division works the same way is often missed, or misunderstood because it so often leads to nothing simpler... 2/3 divided by 5/7 is indeed (2 divided by 5) over (3 divided by 7) but that seems not to give the classic simple fraction we seek. For some fractions, it will work out fine... if 4/27 is divided by 2/3, the answer is (four divided by two ) over (27 divided by 3) = 2/9 and that is the answer you get by the method you memorized (but never understood, most likely) in the fifth grade.
But what if we follow the advice of the alien Klatu. If we convert 2/3 and 5/7 to fractions with a common denominator, we get 14/21 and 15/21, and if we divide top by top and bottom by bottom we get 14/15) over 1, which is just 14/15... job done...
I can imagine including some visuals and suggestive images to help it make sense... It is after all, just a reversal of the multiplication process. If we say "3 dogs times 5 = 15 dogs" then by division we should have the equivalent expressions that "15 dogs divided by 5 = 3 dogs." and just as naturally "15 dogs divided by 3dogs = 5" . Students who have learned (I've been in England too long, I just had to edit "learnt") that "eighths" and "fifths" are just units like "dogs" and "kittens" should then understand that 5 eighths divided by three eighths is just as clearly 5/3.


A few days after I wrote that blog I got a response that asked, more or less, "OK, why does the common algorithm work?"
This was my response


I want to make one comment about division of fractions that seems harder to visulaize than for general division, and then I hope to explain in simple terms just why "invert and multiply" works.
For every multiplication problem, there are two associated division problems; A x B = C begets C/A=B and C/B=A. Elementary teachers call these a "family of facts for C" (or did in the recent past.. educational language changes too fast for firm statments by a non-elementary teacher). So if we add units to one or both factors, appropriate units must be appended to the product. So how does this effect operations with fractions? Well if we have length, as in ANON's comment, then the division problem he states, "If we think of it as a piece of wood with length 2/3, then I believe the question is how many 5/7's are there in the piece of wood" he is dividing length by length to get a pure scaler counting how many pieces (or fractions of a piece) will fit into another. In the case he gives, the answer would be only 14/15 of a piece... because the 2/3 unit length is not quite enough to provide a 5/7 unit length piece...
The multiplication associated with this operation is then 14/15 of 5/7 units = 2/3 units... What about the other division in this family of facts... 2/3 units divided by 14/15 (a scaler here, not a length)will give 5/7 units length. What is this situation describing? This seems the one most difficult for teachers and students alike. We all know what it means to divide a length into (by?) two pieces, but what sense does it make to divide it into 1/2 a piece.
We might try to make this clear to students by taking some common length (12 inches?) and see what happens if we divide it into (by) 8 pieces, then four, then two, then one, (each division is by half the previous number) and look at the pattern of lengths. 12/8=3/2; 12/4 = 3; 12/2 = 6; 12/1= 12... I am confident most students could identify the next numbers in the sequence, 12/ (1/2) = 24, and 12/(1/4) = 48.
At this point, using whole numbers as divisors, the pattern for "invert and multiply" seems obvious, but this is far from a why for all fraction problems.
Let's look at one more case where we sneak in a related idea at the elementary level. Given a problem like 3.5 divided by .04, the student is taught to "move the decimal places enough to make the divisor (.04) a whole number. What we do is another problem (350 divided by 4) that has the same answer (87.5)as the original. Another why does that work that is not often explained.
What do the two operations have in common.... multiplication by one. In each case we have a division (fraction) operation and we simply mulitiply the fraction by a carefully chosen version of one that will make it easier to do. If we view 3.5/.04 as a fraction, then every fifth grader knows that multipliying it by one will not change its value. This is the core of what we do to find equivalent fractions... to get 3/5 = 6/10 we multiply by one, but expressed as 2/2... The decimal division problem uses the same approach... we multiply 3.5/.04 by 100/100 to get another name for the same fraction, 350/4.
Now to explain "invert and multiply" we just use the same idea... dividing fractions is simply fractions which have fractions instead of integers in the numerator and denominator. We want to multiply by one in a way that the division problem will be easier. But the easiest number to divide by is one,... so why not pick a number that changes the denominator of the fraction over a fraction to be a one... that is, multiply by its reciprocal. So for 2/3 divided by 5/7 we can write