Math Words, pg 6

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Adjacent The word adjacent is used to represent things that are next to , or close to, each other. This closely reflects the origin of the word. The ad prefix implies near or toward. The major root is from the Latin word for "being thrown down", jacere. Literally then, adjacent angles are "thrown beside" each other. The word is closely related to adjective, which shares the same two roots in the same order, with only a slight variation in the form that indicates their different routes into modern English. An adjective is often syntactically placed between an article and a noun and thus is adjacent to the noun it modifies. The "throw" base of the root is more evident in words like project and eject.

Annulus When two circles share a common center, the area between the smaller and the larger is referred to as an annulus (see figure). It should remind you of the annual rings you have seen on tree stumps.

It is hard to resist the temptation to try to connect annual, annul, and annulus. Any connection that exists must relate to the early root of the word null, nullus, for none or nothing. The symbol for the null quantity is the circular zero, clearly looking like an annulus and an annual ring; and something that is annulled is treated as if it had not happened. However remote the etymology of the three words, they can each help you remember the others if you link them together in your mind.

I had not intended to suggest from the first sentence of the last paragraph that the three words were from a common origin, but that the common sounds could provide a mnemonic for remembering what they all mean. On the other hand, I did omit to provide the correct etymologies. Recently Alberto Tarable from the Politecnico di Torino, Italy sent the following pointing out the correct etymologies and with the admonition that I (we) should not to be lured by the temptation I suggested.

> Well, it is hard, but we should definitely resist, because the three words do not share any common root. In fact:
- 'annul' comes from Latin 'ad + nullare' --> 'annullare', the only one that actually comes from the adjective 'nullus', none;
- 'annual' comes from Latin 'annum', year, and has no connection whatsoever with 'nullus';
- 'annulus' comes from Latin 'anulus' (one 'n'!), which is the same as 'ring'. The reason of the name is quite obvious, because the annulus looks like a ring.
It is better to avoid deriving a name from a root only because they sound alike!
I agree, and will try to follow his good advice.

Bureau Bureau is another word that relates, in part, back to the wide spread use of counting tables in business transactions. Today the word may refer to a dresser table, a writing table, or a department of the government, and how this came to be is our story. The root of the word comes from the Latin burra for a tuft of wool. (as a side note, the Romans used the word to describe clownish or slapstick behavior, and that is the origin of the word burlesque). The French made a rough wool cloth which they called bural. The cloth was used to cover a counting table, and eventually the cloth, table and all became bureau. Eventually the room in which the counting took place, and finally the entire agency were called the Bureau. Records of wills and estates from the middle ages show that these tables, their coverings, and the counting instruments which necessitated the drawers, were passed along as heirlooms. Eventually they must have lost their value as counting tables, but were still serviceable as tables to hold writing materials in the study, or my-ladies fine toiletries in the boudoir. Counting boards also made their way into the kitchen, but here they more often keep a name that was a contracted form of their original purpose, counters.

Check When your teacher tells you to "check your work", they may not be thinking of it as a number related word, but between its ancient origins, and its multiple meanings of today, check has roots in the history of mathematics. The origin of the word began with the Persian word for king, shah. When the ancient game or chess was played, it too was called by this name, because the king was the most important piece. The Latin derivation of the word became scaccarium and became eschec in the French. The board became the chequered board and cloth which bore this pattern of crossed rows of parallel lines began to be called "check cloth". This cloth was a formal part of the official procedures of banking in 13th century England. In 1186 the Royal Treasurer of England, Richard, the Bishop of London, wrote dialogus de Scaccario (Dialog about the chessboard). In it he describes how a "black cloth with parallel lines at equal distances" was placed on the counting board and used in auditing the official records. Official accounts were recorded on tally sticks, and the counting board was used to verify the accuracy. This is where we get the expression, "check your work". The stock of the tally stick was redeemed for money if the tally stick "checked" and provides the source of our word for a negotiable bank instrument called a check. Eventually the name for the British royal treasury began to be called the Exchequer. A name for a king becomes the name for the place where the king keeps his money.

Concurrent Although concurrent is used in common English to mean things that happen at the same time, the mathematical relationship is more about the same point in space rather than time. Two or more lines are concurrent if they all pass through a common point. This idea of "running together" is the basic idea behind the word concurrent. The con is the common prefix that means "together", and current is from currere which meant "to run". A currier is one who runs messages from one point to another. Current, course, and currency (as in money) are also from the same root, as is curriculum. The term cursor, for the symbol that runs ahead of the point of typing on a computer is from the same root and was the Latin term for "runner".

Concyclic is a similarly constructed term which refers to the intersection of two or more circles. Cycle is from the Greek kuklos fromcircle.

Digit is another word that reflects back to the earliest uses of mathematics. The word digit still refers both to the fingers (and toes) as well as the arabic number symbols for 0 to 9. The Indo-European root, deik is related to many other words that reflect back to the use of the hands and fingers to "point" out objects. Index, indicate, dictate, indict, token, dice, judge and teach are all related to the same root.

Divide symbols There are several different symbol names used or associated with division. The most common looks like a close parenthesis with a horizontal bar extending to the right at the top .
The parenthesis was introduced in the early 1500's and over time the bar was added, but when it first occurred is unclear. There was a period where long division was written with a parenthesis on each end, as 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.

The same text uses a combination of a parenthesis and a bar under the number as shown at right with the quotient written below. The bar above proably combined these two advantages. The parenthesis and overbar symbol seems to have simply been called the division symbol, and sometimes the division parenthesis or division radical. The "two parentheses" method shown in Davis' book above was still in use into the 20th Century. In Practical Math for Beginners by Frank Castle, published in 1918, the same setup is used (pg 19) to illustrate the division process. In the following year, however, A School Arithmetic by Hall and Stevens uses a method (pg 10) essentially identical to the ones in current textbooks, except that the bar over the divisor is missing (below)

The symbol "÷" is called an obelus, and was first used for a division symbol around 1650. The invention is often credited to British Mathematician John Pell but I have also seen credit given to J H Zahn, Teutsche Algebra (1659). The colon, ":", was used as a fraction symbol, and later as a division symbol by Liebnitz around 1685 in much the same fashion as the obelus, "8:4=2".

The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: "I do not like X as a symbol for multiplication, as it is easily confounded with x; ... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC · LM. Hence, in designating ratio I use not one point but two points, which I use at the same time for division." [A History of Mathematical Notation, F. Cajori]

The horizontal fraction bar predates all of these symbols, first appearing in Arabic mathematical writing around 1200. Shortly thereafter it was used by Fibonacci and gained permanent status. Many people call the horizontal fraction bar a vinculum.

In 1684, Leibniz introduced the colon, :, as a symbol for division, and it became the most common symbol used on the continent. When printed type came into use, the horizontal bar or fraction type notation was very difficult to print because it used up three lines of space. Printers took to using the colon, or a slanted fraction or division bar that we now see on calculators and computers. This is alternately called a solidus and also a virgule. In his 1917 Recreations in Mathematics, H E Licks described it as a "shilling mark" [before the United Kingdom converted to decimal coinage, the solidus was used to seperate shillings and pence, as in 5/6 to indicate five shillings, six pence] and wrote, "The use of the shilling mark / to indicate division is comparatively recent, it having been first employed about 1860. In this country it was rarely used until after 1890, but is now very commonly found in algebraic notation...". Cajori remarks that De Morgan recommended the use of the / in 1843, and although he continued to use : in his subsequent works, his advice was taken up by Stokes from 1880 and several others. Some years later the National Committee on Mathematical Requirements (1923) opined, "Since neither ÷ nor :, as signs of division, plays any part in business life, it seems proper to consider only the needs of algebra, and teh make more use of the fractional form and (where the meaning is clear) of the symbol /, and to drop the symbol ÷ in writing algebraic expressions."

Explementary I first heard of the word explementary in July of 1999. It was "re-created" by Steve Wells of a company called think3 while working on a new CAD program, thinkdesign. The word was needed to represent the angle required to complete a 360 degree circle. They wanted a word that would be a natural sounding extension of complement, and supplement. The Latin explementum means "filling" or "stuffing" (as reported by Ken Pledger, and other sources) and it is "explement" that is reported to be in the O.E.D. as "that which fills up". This is very much the same meaning as complement and supplement. After a couple of days, he found the word was not as new to mathematics as we had thought. Several days later he wrote to tell me that the word already appeared on the DICTIONARY OF TECHNICAL TERMS FOR AEROSPACE USE (Web edition edited by Daniel R. Glover, Jr.) NASA Lewis Research Center, Cleveland, Ohio. Here is their definition, as sent to me by Mr Wells:
"Explement -- An angle equal to 360 degrees minus a given angle. Thus, 150 ° is the explement of 210° and the two are said to be explementary. See complement, supplement.
Explementary angles -- Two angles whose sum is 360°."
My thanks to Mr Wells for his advice and corrections as much of this content came directly from his letters.

The term conjugate angles is also sometimes used. This may come from the polar representation of complex conjugates. Two complex numbers a + bi and a - bi are called conjugates, and the polar representations using the Argand diagram will have angles that sum to 360o


Johnson's Theorem If three congruent circles all intersect in a single point, then the other three points of intersection will lie on another circle of the same radius. This simple little theorem was discovered by Roger Johnson(1890-1954) in 1916.

Johnson also popularized another problem about circles. In his Modern Geometry he presented two Japanese Sangaku or Temple Geometry problems. One became a very popular example of advanced triangle geometry; "If a convex polygon inscribed in a circle is triangulated by chords from one of the vertices to the others, then the sum of the diameters of the inscribed circles of the triangles in the triangulation is constant, independent fo the vertex chosen."

You can find a brief biographical note of Johnson's life at the web page of Professor Clark Kimberling. William C Waterhouse of Penn State also sent a copy of Johnson's obituary that contains a little more detail about his life.

Dr. Roger A. Johnson, chairman of the Mathematics Department at Brooklyn College until his retirement in 1952, died yesterday in Goldwater Memorial Hospital, Welfare Island, at the age of 64. He had been in the hospital for seven months suffering from a heart ailment. He resided at 35-36 Seventy-ninth Street, Jackson Heights, Queens. >BR> Born in Gardner, Mass., Dr. Johnson attended Amherst College and Harvard, where he completed his doctorate in mathematics in 1913. He taught at Western Reserve University in Cleveland and at Hamline University in St. Paul. Dr. Johnson came to New York in 1926 to join the faculty of the Brooklyn branch of Hunter College, which later became Brooklyn College. He became chairman of the Mathematics Department in 1947.
Dr. Johnson had written many mathematics papers and several books, the best known of which was a college text, "Modern Geomtry," published during the Thirties. He was a member of the American Mathematical Association [sic] and belonged to Phi Beta Kappa and Pi Mu Epsilon, matheamtics honor fraternity.
He is survived by his widow, Elizabeth B. Johnson, whom he married in 1913, and three sons, Roger A., Jr., of Utica, N.Y.; Hayden B. of Poughkeepsie, N.Y., and Duncan G. of Owensboro, Ky.

Julia Set When a point on the complex plane is iterated repeatedly into some function, the magnitude of the function will either grow so large as to be unbounded, or it will continue to forever move inside a bounded region. The set of points forming the boundary between points which are bounded, and those which are not, is called a Julia Set for the point. If the original point selected lies inside the closed region of the Mandelbrot Set, the Julia set will be a connected set. If the point is outside the Mandelbrot Set, then the Julia Set will be a collection of distinct, unconnected points called Fatou Dust . Julia Sets are named in honor of Gaston Julia(1893-1978), a French Mathematician. Julia's most famous work, Memoire sur l'iteration des fonctions rationnelles, was written in a hospital in 1918 at the age of twenty-five. As a soldier in World War I Julia had been severely wounded and lost his nose. He wrote between the several painful operations necessitated by his wounds.


In statistics, kurtosis is a measure of whether the data is more humped in the center or more spread out. The formula for kurotisis is given by . A normal distribution has kurtosis of 0. Data which is more spread out, such as uniform distributions have negative kurtosis, and are referred to as platykurtic. Peaked distributions with most of the data packed near the center have positive kurtosis, and are called leptokurtic.

The root of kurtosis goes back to the ancient Greek word kurtos, which meant bent. The related Greek word krikos for ring is also the Latin antecedent of our present word for circus, but the word originally described the shape of buildings (circular or oval). The Greek root is also closely related to such present-day words as convex, curly, and crisp. It may be that one of the three K's in the KKK stands for circle, as the word was a common one for secret societies formed after the break up of the Confederacy. Although the roots are very old, the word kurtosis itself is relatively new. It seems that the first use was by Karl Pearson around 1905 in Biometrika IV according to the Oxford English Dictionary. Pearson also developed the terms for variations of kurtosis, "Given two frequency distributions which have the same variability as measured by the standard deviation, they may be relatively more or less flat-topped than the normal curve. If more flat-topped I term them platykurtic, if less flat-topped leptokurtic, and if equally flat-topped mesokurtic. " The lepto prefix is pretty common in biology and comes from the Greek root leptos for something that is fine, thin, small, or delicate. The Greek root platus for things that are broad, wide, or flat gives the alternative name. Students know this root best from the broad flat Australian animal named the Platypus.

Lissajous' figures & Bowditch Curves - A figure or graph made by moving the x- and y- axis as independent or parametric functions. In the simplest cases the figures describe a circle or ellipse, but they also will form lemniscates and other complex figures.

They are named for Jules Antoine Lissajous, who wrote about them in 1857. They had been written about earlier by Nathanial Bowditch, and some people suggest they more correctly should be named for Bowditch. Here are some clips from Eli Maor's wonderful book Trigonometric Delights which every high school math student and teacher should read...

"In 1855 he (Lissajous) devised a simple optical method for studying compund vibrations: he attached a small mirror to each of the vibrating objects (tuning forks for example) and aimed a beam of light at one of the mirrors. The beam was reflected first to the other mirror and thence to a large screen, where it formed a two-dimensional pattern, the visual result of combining the two vibrations."...... "But apparently nothing is new under the sun; Lissajous' figures had been discovered long before by the self-taught American scientist Nathaniel Bowditch, who produced them in 1815 with a compound pendulum."

I knew them only as Lissajous figures until very recently. Shortly after I first read about Bowditch I happened to be in Tokyo Visiting the Edo Museum for an exhibit named Worlds Revealed - The Dawn of Japanese and American Exchange. Like others, I had always had the misconception that Commodore Perry opened trade with Japan in 1853, so I was surprised to find that a number of American ships from Salem, Massachusetts, sailing under Dutch charters had traded with the Japanese as early as 1800. The company was called the East India Marine Society, and in 1802 the First Secretary was Bowditch. On exhibit was a much more popular mathematical creation of Bowditch; his book, The New American Practical Navigator, that Bowditch, and the Marine Society had published in 1802. The book was a compilation of the most accurate measures of the positions of major astronomical objects at numerous longitude and latitude coordinates. The book was, literally, a mariner's bible until an accurate sea clock allowed sailors to conquer the longitude problem. Bowditch's position and accomplishments seem even greater in light of the fact that he was almost totally self educated in mathematics.

In August of 2008 I read a post by Milo Gardner on the almost unheard of Wilkes Expedition, which explored the western Americas and the Pacific, and Milo added that "... mathematicians during the early 1800's were assigned to working on Manifest Destiny issues and projects. On the Wilkes Expedition you'll find Bowditch as one of its  navigators. An island in the Pacific is named for Bowditch, since it had not been on any US  or European map prior to the expedition's visit." The island, I found out, is sometimes called Fakaofu, and is located in the Stork Archipelago in the South Pacific.

Students may make the curves on graphing calculators or software with parametric graphing capability. To make a circle use x=5 cos(t) ; y=5 sin(t). For an Ellipse change one of the fives in the circle to a different number. The simple lemniscate shown below is formed by the parametric equations x=3cos(t); y=2 sin(2t)

Million The root of million is the same Latin mille for thousand, that gives us mile and millennium. The ion suffix implies "large" or "great", and so a million is literally a "great thousand". The ancient Greek and Roman life made little demand for numbers of this size and the largest named number of that age was myriad, which stood for ten-thousand. In his A HISTORY OF ELEMENTARY MATHEMATICS WITH HINTS ON METHODS OF TEACHING, Cajori writes,

The first improvement on ancient and mediaeval methods of numeration was the invention of the word millione by the Italians in the fourteenth century, to signify great thousand* or 10002. This new word seems originally to have indicated a concrete measure, 10 barrels of gold. The words million, nulla or cero (zero) occur for the first time in print in the Arithmetica of Borgi (1484). In the next two centuries the use of millione spread to other European countries. Tonstall, in 1522, speaks of the term as common in England, but rejects it as barbarous!
Although the word million seems to have come into use as early as the middle of the 14th century, most mathematicians would use the phrase "thousand thousands" to avoid confusion. Although it did not gain common use in mathematics until the 1700's, it was in print well before that time. The word was sometimes used to describe an exact amount, and sometimes used as we would use uncountable. Million appears in the King James version of the Bible [Genesis 24:60] and in Shakespeare [Hamlet act II scene II] {"for the play, I remember, pleased not the million"}.

Myriad The early Greeks had little need for a number bigger than a kilo (thousand) and sometimes referred to larger numbers as murious, uncountable. When the need, or desire, to create a number to represent ten-thousand came along, a plural of the same root was used, murioi, which the Romans converted to myriad. Today the common meaning of myriad has returned to its original meaning, countless. A general term for many legged anthropods, such as millipedes and centipedes, is myriapods.

Pascal's Triangle The short life of Blaise Pascal was rich with mathematical invention. Unfortunately, most young students know him only through the triangular array of the binomial coefficients which bears his name, but which was NOT his creation. When he spoke of the object, Blaise Pascal used the term "arithmetical triangle" (triangle arithmetique). In many countries it is not called Pascal's triangle. In Italy it is called Tartaglia's triangle. In many parts of Asia it is called Yang Hui's triangle. Yang Hui was a minor Chinese official who wrote two books, dated 1261 and 1275, which use decimal fractions (long before the western use) and which is probably the first written account of the arithmetic triangle. The Chinese triangle appeared in 1303 on the front of Chu Shi-Chieh's Ssu Yuan Yü Chien. This image of the Chinese triangle is from the web site of Peter Lee. (click on the image to see the original). Omar Khayyam, the Persian poet, mathematician, and astronomer also wrote about the triangle and may have preceded Yang Hui, but the exact years of his life are not certain. [see note furthur below for the early contribution of Indian mathematicians]

In a post from James Landau on a math history site I learned that you can use Pascal's triangle to approximate p. As N (the row number) gets bigger and bigger, the middle term in the row divided by the sum of the row comes closer and closer to 2/root(2np). This is a very slow convergence.

At this site you can see photo copies of Pascal's 1654 treatise, Traité du triangle arithmétique. An illustration of the arithmetic triangle is shown on page four.

The Math Forum's Dr. Math FAQ page on Pascal's triangle has information and related links to explore as well as lesson plans for teachers at several grade levels.

A page about the life of Blaise Pascal also has some material on the famous triangle.

This page is focused on the Fibonacci sequence in Pascal's triangle.

In Early July of 2004 I received a note from Matthew Hubbard, the curator of Pascal's Triangle From Top to Bottom. In it he informed me that I had failed to include, in particular, the contribution of India to the study of the arithmetic triangle. A quick visit to his web site led me to :

The idea of taking "six tastes one at a time, two at a time, three at a time, etc." was written down correctly in India 300 years before the birth of Christ in a book called the Bhagabati Sutra, a text from the Jainist religion; this gives the subcontinent of India the distinction of being the earliest civilization to have an understanding of the binomial coefficients in their combinatorial form "n choose k" in a text that survives to this day.
The site contains much additional material about Indian study of the triangle, and other information that makes it well worth a visit.

Matthew also called me to task for my suggestion above that the use of "Pascal's Triangle" was somehow inappropriate. He wrote in justification of the term, " One of the reasons I wrote is the idea of misnomy in mathematics; you put the word (mis)named in front of Pascal's Triangle. While it is certainly true that many, many people had studied the binomial coefficients prior to Pascal, his work is honored because it was read by people who came after him, most notably Monmort and deMoivre, who credited Pascal's Treatise in their works several decades later. Moreover, it is worth reading, as Pascal finds many identities in the triangle that no one before him had written down.
It's too late to get the world to call it Pingala's Triangle, and I fully appreciate the desire of civilizations to honor their own, but I think if anybody's name is going to be linked to this famous array, Blaise Pascal is as good a candidate as any and significantly better than most."

My thanks to Matt for the additional material on the Indian contribution, and for helping to insure a balance of credit where credit is due, and certainly Pascal is due much credit for his exposure of many aspects of the triangle, by whatever name it is called.

A few years later, in April of 2008, Prof. Randy K. Schwartz from Schoolcraft College in Michigan graciously shared a copy of an image from a commentary and translation of section XI of 'Fiqh al-Hisab', a manual written by Ibn Mun'im (Maghreb) between 1207 and 1212. The image was provided to him with materials at a conference on Arabic Mathematical History in 2002. He suggests that more can be found from Victor J. Katz, “Combinatorics and Induction in Medieval Hebrew and Islamic Mathematics”, in Ronald Calinger, ed., Vita Mathematica (1996, MAA Notes no. 40). This is still well behind the time of Pingala, or even Halayudha, so I still want for images from either of those.

Pentomino The word domino was first used to refer to the hooded black cape worn by priests, and later to black masks (does anyone still remember the Lone Ranger's mask, that is the type) worn at masquerade balls. Although people sometimes try to attribute the name to a short form of dues-omino or some such word, it seems much more likely that it was from the Latin root for master, from which we also get dominate. Around the beginning of the twentieth century there were several popular puzzles using shapes made with more than two squares stuck together along their edges. Because there are so few possible shapes with four or less, and so many with six or more, interest centered on the twelve unique shapes that are formed by joining five squares on an edge. The earliest use of what we now call Pentominos in a mathematical recreation was posed by H. E. Dudney as the problem of the broken chessboard in The Canterbury Puzzles published in 1919. The term pentomino was created by Solomon Golomb, as an extension of the word domino. He first used the term at the Harvard Math Club in 1953. I read recently that the word has been copyrighted, so assume there are all those little registered trademark symbols on each word.
The twelve pentominoes shown below are often referred to by the letters they resemble.

You can learn alot more about Pentominoes at a site created by Predrag Janicic where I found the graphic above. It even includes a link to download a Pentominoes game to play against your computer. [this link has been dead the last couple of times I tried it] Here is the link to a puzzle software called "OOG: the Object Orientation Game" from MCM Productions that can be downloaded (it was free when I checked) which allows players to test their problem solving ability with over 300 puzzles, including Tangrams, Pentominoes, Hexagons and more. I found it very entertaining.

The general term for a number of squares joined at their sides, such as the pentominoes (five squares) is Polyominoes. Hexominoes have six-squares, while tetrominoes have only four, etc. Teachers may be interested in some Lesson Plans on Polyominoes developed by Cynthia Lanius. Another good link (because it has lots of other links) with some really sound math in it is at The Geometry Junkyard.

Recursive / Recursion The mathematical meaning of recursive is to describe a process or function for which the output at each level depends on the output at the previous step. Think of a machine in which the output is fed around to the entry to be the input of the next product. The re is the common Latin prefix for "back" and the root currere, to run. The literal meaning is "to turn back" or "to run back", aptly describing the reuse of the output as an input. Other related words from modern language include currency and curriculum.

Score Like compute and tally (see each), score is a reminder of the primitive counting and record keeping technique of cutting notches in a piece of wood or bone. The idea of making one mark for each member of a collection is an ancient idea as evidenced by inscribed marks on ice age bones, and ochre dots on the walls of Magdalenian caves. Because a long string of identical marks is difficult to count quickly, groups were sometimes marked off with a heavier or different mark. The Old Norse word for this practice was skor and the mark was used to make groups of twenty. Many European primitive cultures used a base of twenty and so the word found purpose, and thus persisted in language. The base twenty system is apparent in the language residue of the English (20 shillings to the pound), as well as the Irish, French, and Danish. A base twenty system of counting was also used by the Mayans of Central America and the Ainu, the indigenous people of the Japanese islands. The Indo-European root of score is sker. The word is related to cutting or slicing and is the progenitor of dozens of words sharing this common theme. Some examples include shears, scissors, and skirmish. The Old English version sceort for "to cut" gave us not only the word short, but shirt and skirt as well. The Latin form dropped the "s" to produce words like carnage, carnival, and carnation, a flower named because it was the color of flesh. Today our most popular pastimes remind us of our mathematical beginnings as they report the sports scores, the number of marks for each team.

James A Landau noticed something of a puzzle about the use of score. He writes, "I checked the Oxford English Dictionary, 2nd Edition and found that the first citation for "threescore" was in 1388, for "fourscore" was in 1250, and for "sixscore" was in 1300. There were no entries for twoscore, fivescore, sevenscore, eightscore, or ninescore, which is a little curious. Why would people only start counting by scores at 60 and quit after 120?"

There used to be a unit called a shock for groups of 3 score. The following quote comes from a post by John Conway. "Most of the major European languages had a break after 60, which usually had a special name of its own ; for example, it was a "shock" in English before it became "three score". In Elizabethan times, the standard names for 60,70,80,90 were "threescore", "threescore and ten", "fourscore" and "fourscore and ten" and the other European languages did much the same thing.

Tally comes from the name of a stick on which counts were made to keep a count or a score. The Latin root is talea and is closely related to the origin of tailor, "one who cuts". Many math words have origins that reflect back to the earliest and most primitive uses of number. Compare the origins of compute, score. The first record existing of tally marks is on a leg bone of a baboon dating prior to 30,000 BC. The bone has 29 clear notches in a row. It was discovered in a cave in Southern Africa. It is sometimes called the Lebombo Bone after the Lebombo mountains in which it was found. The exact age of such artifacts is a subject of debate, and their mathematical usage is somewhat speculative. Some sources have stated that the bone is a lunar phase counter, and by implication that African women were the first mathematicians since keeping track of menstrual cycles requires a lunar calendar.

Another candidate for the oldest tally record in history is a wolf bone found in Czechoslovakia with 57 deep notches cut into it, some of which appear to be grouped into sets of five.

In Mathematics Galore by Budd and Sangwin, there is a story of much more recent tally sticks. It seems that until around 1828 the British kept tax and other records on wooden tally sticks. When the system was discontinued they were left with a huge residue of wooden tally sticks, so in 1834 they decided to have a bonfire to get rid of them. The bonfire was such a success that it burned the parliment buildings to the ground. What Guy Fawkes could not do with dynamite the Exchequer did with tally sticks.... The power of math.

The story, as improbable as it seems, is varified by a speech by Charled Dickens 1855. [Charles Dickens, Speech to the Administrative Reform Association, June 27, 1855, in Speeches of Charles Dickens, ed. K.F. Fielding, Oxford: The Clarendon Press, 1960, p. 206, ] The somewhat clipped version below is taken from Number, The Language of Science by Tobias Dantzig (pgs 23&24)

Ages ago a savage mode of keeping accounts on notched sticks was introduced into the Court of Exchequer and the accounts were kept much as Robinson Crusoe kept his calendar on the desert island. A multitude of accountants, bookkeepers, and actuaries were born and died... Still official routine inclined to those notched sticks as if they were pillars of the Constitution, and still the Exchequer accounts continued to be kept on certain splints of elm-wood called tallies. In the reign of George III an inquiry was made by some revolutionary spirit whether, pens, ink and paper, slates and pencils being in existence, this obstinate adherence to an obsolute custom ought to be continued, ..... All the red tape in the country grew redder at the bare mention of this bold and original conception, and it took until 1826 to get these sticks abolished. In 1834 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur ot any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighborhood. However, they never had been useful, and official routine required that they should never be, and so the order went out that they were to be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over-gorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost therof.
Several images of the fire was painted by J.M.W. Turner who watched the fire from a boat on the Thames. I have a clip that I can not credit that says, "The fire of 1834 burned down most of the Palace of Westminster. The only part still remaining from 1097 is Westminster Hall. The buildings replacing the destroyed elements include Big Ben, with it's four 23 feet clock faces, built in a rich late gothic style that now form the Houses of Commons and the House of Lords. These magnificent buildings are still the subject of many paintings, including my own Parliament, with the grand Westminster Abbey on their north." The one below hangs in the Tate Gallery; while another, I believe, is in a gallery in Cleveland, Ohio.

Around 1960 an ancient mathematical record on bone was uncovered in the African area of Ishango, near Lake Edward. While it was at first considered an ancient (9000 BC) tally stick, many now think it represents the oldest table of prime numbers. Here is a link with a picture where you can see and read more about the "Ishango bone"

vinculum, fraction bar, "repeat bar" As a "Math Doctor", one of the most common questions asked by elementary school students is, "What is the name for the bar over repeating decimal fractions?" I always answer that "repeat bar" seems the best name to communicate what it does, but I know they want the classic Latin name for the bar, vinculum. The word is from the diminutive of vincere, to tie. vinculum referred to a small cord for binding the hands or feet. The meaning in math is mostly unchanged from that original meaning, as the purpose of the repeat bar is to bind together the sequence of repeating digits. The term repeating decimal has sometimes been replaced with "recurrent" or "Circulating". The sequence of digits which repeat have also been called the "circulants" and "repetends" (and sometimes "repetents"). 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. 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 roots and the one for division).

It is in the expression for repeating decimals that students are most familiar with the use of the overbar as a symbol, yet it seems to have been the last application of the bar, and seems not to have occured until after 1930 in the US. In F. Cajori's A History of Mathematical Notations (1929) he points out two forms of marking repeating sequences in decimals but does not mention the vincula or overbar. Cajori credits John Marsh [Decimal Arithmetic Made Perfect, (London, 1742)] with being the first to use a symbol to indicate the repeat sequence. Marsh sometimes placed a single dot over the first number in the repeat sequence, and sometimes placed one on the first and last.

Although others used accents to denote the repeating digits, it seems that the use of dots over the digits became the common notation in most of the world. The dots were also used in the US and persisted into the Twentieth century. I have not yet discovered when, or why, the US opted for the use of a bar. In Public School Arithmetic by Baker and Bourne (G. Bell and Sons, London, 1961) in a section on "recurring decimals" (pg 349) he gives an example with 32/41 expressed as .78048 with the dot over the first and last digit. The exact same approach was used in America. In White's Complete Arithmetic (1870) A section on page 289 introduces "Circulating Decimals" as shown below.

He goes on on the next page to define a pure circulate as one that has no figure but the repetend, (for example 1/3); and a mixed circulate as a decimal with one or more figures before the repetend. He then follows with directions for converting both pure and mixed circulate decimals to a common fraction. I also found a use of the dots for repeating decimals in the 1940 College Algebra of Paul Rider of Washington University

The 29th yearbook of the NCTM, Topics in Mathematics for Elementary School Teaching, published in 1964, contains an example on page 320 which clearly uses the bar to show the decimal name for 3/14 with the statement, "...where the bar over 142857 indicates that the group 142857 repeats endlessly." In Dec of 2005 I was sent a copy of page 40 of the 1959 yearbook by Dave Mason, a teacher at South Tahoe High School. The page includes the use of the repeat bar as shown below

In the same year as the 29th NCTM yearbook(1964), Irving Adler obtained a copyright for A New Look At Arithmetic, and on page 220 he writes, "To indicate a repeating decimal with a minimum of writing, it is customary to write only enough decimal places to include the repeating part once, and to identify the repeating part by underlining it. Thus the repeating decimal for 211/990 is therefore represented by .213....". It is worth mentioning that William Oughtred, the 16th Century mathematician indicated all decimals by underlining.

A third example, or rather a hybrid of two of the former, also appeared in a book with a 1964 copyright. A A Klaf's Arithmetic Refresher was published a few years after his death by his family. The book is written in a question and answer style somewhat reminiscent of the classic dialogs of antiquity. On page 188 it asks, "How are recurring, circulation, or repeating decimals denoted?" It then goes on to answer, "b) by dots placed over the first and last figures of the recurring group." This is described exactly like the more common earlier usage, but the figure that follows includes dots, and then an arc above them, similar to what I have shown here. (see also "divide symbol" for more history of symbols). Similar arcs were used over groups of three numbers to indicate the periods (thousands, millions, etc) in some early use of Hindu-Arabic numerals. Gerber(980), who later became Pope Sylvester, referred to them as "Pythagorean Arcs."
Sam Koski of Miami Springs Senior High sent me a note about a 1960 text, Algebra ,ITS BIG IDEAS AND BASIC SKILLS by McGraw Hill. "Repeating decimals are written with ellipsis; .333... I don't see any indication of the "bar" notation."

In spite of the seeminly late date of apperance in textbooks, David L Renfro has found several journal articles dating back to 1920 that used the underline or overbar method for repeating decimals. I have taken snips from his post here:

Repeating decimals are indicated by the use of an overline bar in the following:
[2] Author unknown or n/a, "Problems -- Notes. 16. Skeleton Division", American Mathematical Monthly 28 #6/7 (June 1921), 278.
One of the sentences is: "The repeating digits are shown by a line over the crosses." (The crosses stand for unknown digits whose determination is the problem to be solved.)
[3] Frederik Schuh, Title unknown or n/a, Nieuw Tijdschrift Voor Wiskunde 8 (1920-21), 64.
The previous reference cites this as the source for the problem and, since quote marks are used in the statement of the problem, I would assume that the overline bar usage was also present in this slightly earlier reference. However, I did not have access to this journal, so I was not able to verify for sure that an overline bar usage appears in [3].
[4] David Raymond Curtiss, "I. Solution of a problem in skeleton division", American Mathematical Monthly 29 #5 (May 1922), 211-212.
An underline bar is used in the following for repeating decimals:
[5] James McGiffert, "Intrinsic decimals", Mathematics News Letter 7 #3 (December 1932), 7-10.
An overline bar is used in the following for repeating decimals:
[6] H. T. R. Aude, "Intrinsic decimals", Mathematics News Letter 8 #1 (October 1933), 8-12.
The overline usage is not nearly as noticeable as the underline usage in McGiffert's paper, but it does appear on p. 9.

Where today we might write (2x+3)5 the early users of the vinculum would write 2x+3 5. Cajori traces the use to Nicholas Chuquet in 1484. He attributes the use of a vinculum with the radical to indicate roots to Thomas Harriot in 1631, The more general use of the horizontal bar above the collected items was reinforced by Fr. van Schooten in his 1646 edit of the collected works of Vieta. [When I first wrote that statement, I assumed the use of the vinculum in place of parentheses for general computation such as the distribution of multiplication over addition had been limited to 15th and 16th century at the latest, but then I found the use (see below) in a 1924 edition of Practical Arithmetic by Van Tuyl. An even later example appears on page 20 of Elementary Shop Mathematics, published by the Lincoln Extension Institute in 1943 .]

>Notes on Parentheses and Brackets as Grouping Symbols{These notes are from Jeff Miller's website on the first use of mathematical symbols}.


Parentheses ( ) are "found in rare instances as early as the sixteenth century" (Cajori vol. 1, page 390). Apparently the earliest work Cajori names in which round parentheses are found is General trattato di numeri e misure by Nicolo Tartaglia (c. 1506-1557) in 1556. Round parentheses occur once in Ars magna by Cardan, as printed in Opera (1663) (Cajori vol. 1, page 392; Cajori does not indicate whether the parentheses occur in the original 1545 edition).
Cajori (vol. 1, page 391) says that Michael Stifel (1487 or 1486 - 1567) does not use parentheses as signs of aggregation in his printed works, but that they are found in one of his handwritten marginal notes. Cajori expresses the opinion that these parentheses are actually punctuation marks rather than mathematical symbols.
Kline says parentheses appear in 1544. He presumably refers to Arithmetica integra by Michael Stifel.
Brackets. Brackets [ ] are found in the manuscript edition of Algebra by Rafael Bombelli (1526-1573) from about 1550 (Cajori vol. 1, page 391).
Ball (page 242) and Lucas say brackets were introduced by Albert Girard (1595-1632) in 1629. This information appears to be inaccurate.
Kline says square brackets were introduced by Vieta (1540-1603). He presumably refers to the 1593 edition of Zetetica, which according to Cajori uses both braces and brackets.
Braces. Braces { } are found in the 1593 edition of Francois Vieta's Zetetica (Cajori vol. 1, page 391).

The use of a vinculum to bind the names of points together, , to indicate a line was used by Cavalieri in 1647.

Some people also refer to the horizontal fraction bar as a vinculum as it binds the numerator and denominator into a single value. Leonardo of Pisa, or Fibonacci, introduced the bar to the west when he copied the Hindu-Arabic mathematicians who first used the fraction bar symbol, although he wrote his mixed numbers in the "Arabic" form, from right to left; 1/2 23 for twenty-three and one-half.