Math Words, pg 14





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All models are wrong, Some models are useful. The quote is most often attributed to the statistician George Box. It is ocassionally attributed to W. E. Demming, but I think incorrectly. I recently found a note posted by Shawn Callahan that gives the phrase as a chapter heading by Box in one of his books ; "I thought it was Deming at first but did some searching and found that Box used it as a heading in a book chapter in 1979. Here is the citation: Box, G.E.P., Robustness in the strategy of scientific model building, in Robustness in Statistics, R.L. Launer and G.N. Wilkinson, Editors. 1979, Academic Press: New York.

Arc Sine or Inverse Sine function

Prior to 1729 there was no commonly used abreviation for "The angle whose sine is N". In that year Daniel Bernoulli used the letters AS as a symbol for the inverse of the Sine function. Within a very few years Euler had tried A alone for inverse sine, and then A t for the inverse of the tangent function.

Lagrange seems to have been among the earliest users of arc. sin (note the use of a period) and this emerged as the dominant symbol (or abbreviation) for the Inverse functions in Europe.

Around 1815 in England John Herschel, son of the famous astronomer William Herschel who discovered the planet Uranus, proposed the use of Sin-1N. He justified this as an extension of the use of d2x for the second derivative which had become a common notation in calculus. Cajori in his History of mathematical notations writes


This has became the common notation in England and the US but was less common on the continent. Herschel himself used arc.sin with the period in his Traite de la Luminere in 1829, published in French.



Cardinality

In mathematics when we refer to a finite set, we refer to the number of elements in the set as the cardinality of the set. [See cardinal for related info. For example if we let B represent the set of prime numbers smaller than ten we would know that B={2,3,5,7) and we would say that the cardinality of B is four. But what happens if a set is unbounded, or infinite. Galilleo was the earliest I know of to consider the paradox that there had to be as many even numbers as there were integers since each integer could be paired one-to-one with the even numbers 0-0, 1-2, 2-4, 3-6, etc. Much later, Georg Cantor developed a theory of the cardinality of infinite sets by letting any two sets with a one-to-one relation between their elements have the same cardinality. For the set containing the natural numbers (the counting integers) he used the term Aleph Null, and the symbol -0. Aleph is the first letter of the Hebrew alphabet. Cantor also showed that not all sets had the same cardinality. In 1874 he showed that the real numbers were NOT the same size of infinity as the counting numbers, and he assigned the number Aleph one to represnt the reals. He also showed that there is an unending heirarchy of these sets, and that for a set of any cardinality, a set of larger cardinality can be created, so there is an infinite set of cardinalities of infinite sets..... (go ahead, stop and think about it). Although Cantor assigned the reals the number aleph-1, many set theorist hold that it may be possible for a set to have cardinality between the integers and the reals, so the jury is still out on some questions about the actual assignments of aleph values for some infinite sets.

A nice, but still relativly simple explanation of the levels of infinity are given at this mathacademy link.



Confidence Interval

When statisticians report the interval they predict an unknown parameter will fall into, they usually add some measure to indicate the faith, or confidence they have that the true value is really located in the stated interval. Statements like, "We are 95% confident that the true number of defectives produced by the process is between 55 and 70 defects per thousand." are examples of a confidence interval.

A little about the history of the term came in a note from David Bee to the AP Statistics list:

This year [2004] marks the 70th anniversary of Jerzy Neyman's 19 June 1934 paper before the Royal Statistical Society entitled "On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection."

Relatedly, this paper was the one first presenting the concept of a "confidence interval" (interval estimate).

For a brief verbal coverage of this (and the beginning of the Neyman- Fisher battles), see Chapter 12 ("The Confidence Trick," P117-124) in David Salsburg's book The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century.

-- David Bee

The word confidence comes from the union of the Latin co for "with" or "together with", and the root fides for "faith" or "trust". Other words from the same root include fidelity, the act of being faithful, and infidel, one who is not faithful. Confide, defy, and fiance are also from the same root.

Interval comes from the union of the Latin roots inter, for between, and vallum for ramparts, and which gives our word for wall. An interval, then, is the space (or distance) between two walls, which are usually given mathematically as numerical points.



Cossists

When the early Italian mathematicians began to develop the foundations of algebra using the new Arabic numerals, they used the Latin word cosa, unknown, to represent the unknown quantity in a problem. In the 15th and 16th century with the emergence of the German Rechenmeisters, or master reckoners, the word coss was used in the same role. One of the most influential of these master reckoners was Adam Riese, who played a significant role in the movement toward the use of "Hindu-arabic" numerals and away from the counting boards and Roman numerals that still persisted. His Die Coss in 1524, and his other arithmetic primers were so well received that "nach Adam Riese" is still used in Germany to describe mathematical accuracy and precision. The image at right appeared on a German stamp on the 400th anniversary of his death. It is from a woodcut that appeared on the cover of Rechenung nach der lenge, auff den Linihen vnd Feder . A click on the image will take you to a page where images of stamps celebrating mathematics have been archived. An image of the cover can be found here. The word "cossist" became a term for algebraist throughout Europe during this period. Many of the symbols we now use were developed during this time, including the signs "+" and "-" for addition and subtraction, and the first use of decimal fractions.



Floor Function

The formal definition of the floor function of a real number, x, is the greatest integer less than or equal to x. For students in elementary grades this is simply thought of as rounding down to the next lower integer if the number is not already an integer. The floor function of an integer is equal to itself. For example the floor function of 2.3 = 2. The symbol for the floor function of x is a pair of vertical lines with a "floor" beneath them, . A similar figure is used for the ceiling function which is related . The symbols are not consistently used, but they are becoming much more common.

The floor function is also called the Greatest Integer Function because of the way it is defined. Because this is so long, calculators and computers often use int or Int for the symbol, although the student should be careful as on some calculators there may be an "int" symbol which only removes any fractional part. The only difference is with negative numbers. |_ -2.3 _| = -3, not -2, since -3 is the largest integer less than or equal to -2.3.

According to Jeff Miller's site on the first use of math symbols, the use of the |_ _| notation was created by Kenneth E. Iverson in 1962. The use of brackets [x] to represent the function is still sometimes used, and the function is sometimes referred to by the name bracket x also. Miller dates this back to the work of Karl Gauss in 1808 and also indicates that E(x) was used by Legnedre.



Gallon

In the Roman period the term gallon was used to represent the general idea a capacity or volume of material, and seemed to have been applied to both wet and dry measures. The earliest official attempt to standardize the gallon was during the thirteenth century when Edward I set a standard of eight pounds of wheat; but gallons with eight pounds dry weight (Avoirdupois) and others with eight pounds liquid (Troy weight) coexisted with others that seemed to meet no particular standard at all. In 1707, during the reign of Queen Anne, a new standard was written. The English established a Wine Gallon, which was also called the Queen Anne gallon and was defined to be 221 c.c. and was about the size of the traditional wine gallon in England. This is the size of the US Gallon to this day. A beer gallon also had become more or less standardized by tradition at ten pounds of beer (or ale). This was very close to the dry weight of wheat in the same container and in 1824 the English established a liquid measure, the Imperial Gallon with a capacity to hold 10 pounds (Avoirdupois) of distilled water . Because this was after the US became independent of Great Britain, the two countries have since had different units for a liquid gallon. The US liquid gallon holds about 8.3 pounds of water. The "dry gallon" is seldom used in the US.

The word gallon comes from a Latin root galleta for "wine jug" which may have actually been of Celtic origins.



Geoboard

The geoboard is a common tool in mathematics education. Originally made from pins or nails driven into a block of wood in a rectangular array, today they are mostly made of plastic. You can even work online with a , Java version of a geobard

The topic of who invented the geoboard was answered by Dick Tahta. Here is his post to Historia Matematica.

Geoboards were introduced by Caleb Gattegno when he was working in London in the early 1950s and referred to frequently in articles by him at the time. Writing in French, he referred to them as "geo-plans" and this term appears in articles he wrote for the two remarkable collections of articles by members of the CIEMEM (Commission Internationale pour l'Etude et l'Amelioration de l'Enseignement des Mathematiques) and published by Delachaux & Niestle (Neuchatel, 1958 and 1960).

It may also be worth recalling that the blind mathematician Nicholas Saunderson employed a square lattice of pegs joined by silk thread to form "palpable or tangible symbols" - in this case of arithmetical rather than geometrical 'figures'.
I have found a 1954 article, "The Gattegno Geoboards", using the term (in ATAM Bulletin, 1954) Gattegno was also a founder of a commercial company, The Cuisenaire Company, in England in 1954 to produce the Cuisenaire small rules and the handbooks which accompany them and was director of this company until 1986. The story of how he met Georges Cuisenaire and developed a partnership can be found at the company webpage.

The board seems also to be common in Germany, where it is called the Nagelbrett, and in Italy it may be called geopiano.

The similar device created by Nicholas Saunderson for arithmetic calculation for the blind is described in the book Recreations in Science and Natural Philosophy by Jacques Ozanam. An English translation with a description of Saunderson's method is describe on page 11 and 12. You can see a copy of the book at The Cornell Library Historical Mathematics Monographs



Symbols for Inequalities The use of the symbols > and < for "greater than" and "less than" date back to 1631 and the posthumous publication of Thomas Harriot's Artis analyticae praxis. The Univ of St. Andrews Math History site says that Harriot did NOT use the symbol, and it was created by his editor (unnamed). Although Oughtred had a symbol that persisted, in several variations, in occasional use into the 18th century, the symmetry and simplicity of Harriot's symbol made it more generally used. Wallis seems to have been the first to combine the equal sign with the inequalities. He used several variations, the most common being a bar over the inequality symbol. Several variations of the combination remain today. The two most common being a inequlity superimposed over an equal sign, and over a single bar .



Kilograms, kilogram mass, slugs, and pounds

Students who learn physics using the SI system are often confused when they encounter kilograms as weight or what the Marks Standard Handbook for Mechanical Engineers calls kilogram force. This term comes as a surprise to students who have been taught that mass in the metric system was kilograms and force was newtons. They are further bewildered when presented with a spring scale calibrated in kilograms (a force/weight), use a Rockwell hardness tester with loads(force/weight) measured in kilograms or find the term kilogram force in a bearing catalog. They are then more totally confused when they take a dynamics course and they are told to divide the weight of an object in pounds by g=32 ft/sec2 to obtain its mass in slugs but that the weight in kilograms is not divided by the metric g=9.8 m/sec2 to get kilogram mass.

The Kilogram is the base unit of mass in the SI and MKS versions of the metric system. The kilogram is defined as the mass of the standard kilogram, a platinum-iridium bar in the custody of the International Bureau of Weights and Measures (BIPM) near Paris, France. Copies of this bar are kept by the standards agencies of all the major industrial nations, including the U.S. National Institute of Standards and Technology (NIST). It is unique because it is the only physical object used to define an international standard.

The weight of one kilogram of mass near the surface of the earth is called a Kilogram Force, and is about 2.204,622 pounds. This is NOT the same as a Newton, in fact it is about 9.8 Newtons. A Newton is the amount of force(weight) created by a mass of 1 Kg under an acceleration of 1 meter/sec2. A Kgf (kilogram force) is the force/weight created by a mass of 1 Kg under the acceleration of gravity on the earth, about 9.8 meters/sec2.

So what the heck is a slug? Well in the British and US system of weights in which a pound is the unit of weight/force under the acceleration of gravity, which in English Units is 32 feet/sec2, the unit of mass is the slug. The word slug seems to have been created around the beginning of the 20th century by A. M. Worthington [from the OED]"1902, A. M. WORTHINGTON Dynamics of Rotation (ed.4) p. viii, I have ventured to give the name of a ‘slug’ to the British Engineer's Unit of Mass, i.e., to the mass in which an acceleration of one foot-per-sec.-per-sec. is produced by a force of one pound." The OED also gives clues to why the term is so little known to students; "1936, F. W. LANCHESTER Theory of Dimensions & its Application for Engineers v. 37 Even amongst the advocates of Perry's system.., the slug has never taken shape except on paper; it has, and has had no real material existence." And if it ever did live, it may have been short lived, [OED]"1973, Nature 20 July 184/3 The statement that the unit of mass in the British system is the slug is several years out of date."

The origin of the word pound is explained on its own page. The word Kilogram combines the Latin kilo, for one thousand, and gramme from the earlier Greek term gramma which meant a small weight.



The Petersburg Paradox

The end of the 20th century may be marked by historians as the age of the lottery. In lotteries people bet small amounts, often less than five dollars, in the hopes of winning millions, and a few do. What would you pay, then, for a chance to win unlimited wealth? For most people, it seems, not very much. The contradiction is the root of what has come to be called the Petersburg Paradox

Suppose I offered to give you money based on how many times you could flip a coin before the first "tails" came up. If it occurs on the first flip, you win $1. If the first tail comes up on the second flip, you win $2, on the third $4, on the fourth $8 etc with the award doubling for each additional heads before the first tail.

According to the rules of probability your chance of winning $1 is 1/2. The chances of $2 is 1/4, $4 is 1/8, ... and $2n-1 has a chance of 1/2n. Adding all the possible winnings together you get 1/2 + 12 + 1/2 + ..... but that is an infinite amount of money; would you pay $1000 to play??.. even $100? Most people would not, and with good reason. When Georges Leclerc, the Comte de Buffon simulated the game for a long period, he found that he would have only won $10,057 in 2084 games [This is in the age before computers, so he had a child flip a coin to simulate 2048 games, something in excess of 4000 flips of a coin]. That would be an average of about $5 per game if he worked in dollars. In his report of the 2084 games, no run of heads exceeded nine, for a payoff of $256

The name for the problem comes from the fact that it was first discussed and written on by Nicholas and Daniel Bernoulli while they were professors at the Academy in St. Petersburg. You can find a translation of some extracts from this correspondance. The problem is also called the St. Petersburg Game for the same reason.

If you think you are willing to pay $20 a game, here is a java scripted simulation that will show you how well you would do.... Good luck, if it had been real money I would be totally broke now.



Pie Chart
The Pie chart is usually credited to William Playfair, in hisThe Statistical Breviary(1801) which includes a pie chart showing the division of the land area of the Turkish Empire. A nice paper describing the history of the pie graph is given by Ian Spence of the University of Toronto. The introductory abstract reads, "William Playfair’s pie chart is more than 200 years old and yet its intellectual origins remain obscure. The inspiration likely derived from the logic diagrams of Llull, Bruno, Leibniz, and Euler, which were familiar to William because of the instruction of his mathematician brother John. The pie chart is broadly popular but—despite its common appeal—most experts have not been seduced, and the academy has advised avoidance; nonetheless, the masses have chosen to ignore this advice. This commentary discusses the origins of the pie chart and the appropriate uses of the form. "

Power of a Point to a Circle

When a line is drawn from a point through a circle, the product of the distance from the point to the two intesections will be the same no matter where the line cuts the circle. In the figure, the Product of the lengths PY and PX is the same as the product of the lengths PH and PG. This product is called the Power of P in relation to the circle. If the line is drawn from P through a point of tangency to the circle, there are not two intersections, but the Power is the square of the distance to the point of tangency. If the point is inside the circle, the power is referred to as negative. According to HSM Coxeter, the use of the word power for this relation was originated by Jacob Steiner in 1826.

If two circles are not concentric then there must be a point that has the same power in respect to both circles. In fact, there must be an infinite number of such points, and they all lie along a straight line, called the radical line or radical axis. If the two circles intersect in two points, the radical line is the line passing between their two points of intersection. If they do not intersect, the radical line can be found by a simple construction. Draw a circle that intersects both the original circles in two places each. Construct a secant through the intersections of both circles and mark the point where the secants intersect. Repeat this process a second time. The line connecting the two points of intersection of the secants is the radical line. In the figure the line through K and R is the radical axis. The radical axis is always perpendicular to the line joining the center of the two circles. The distance from the center of the circles to the radical axis along the line joining the two centers can be found by the equation below. Let r1 be the radius of the first circle and r2 be the radius of the second, and d be the distance between the centers. The distance from the center of circle one to the radical axis is given by .



Term

The word term is used in reference to things which are somehow bounded, limited, or set apart from other things. Colleges often divide the school year into three or more terms. In a similar way the specialized language of a discipline, like math or science, are often referred to as terms, and the words in the index of this page may well be referred to as a list of mathematical terms. You can see this relation to a boundary in words like terminal (where it stops or is bound) and terminate (to put a stop or boundary to something) as in terminating decimal; a decimal which has a finite number of digits.

In mathematics the simplest operations are addition, and its opposite, subtraction. When two or more quantities are connected by addition or subtraction, we refer to these quantities as terms. In the expression 3xy + 7z2 the 3xy forms one term, and the 7z2 forms another. Occasionaly we also refer to the two parts of a ratio or a fraction as terms. In the ratio 2 : 3 = 4 : 6 (or 2/3 = 4/6) the two and six are called the extreme terms, and the three and four are called the medial or median terms.

The origin of the word dates back at least to the early Latin language, and Terminus was the name of the God of boundaries. I'm not sure how long the term has been used in a mathematical sense.




Tetragon

The most common term in the USA for the polygon formed by sides connecting four points in a plane (with no three collinear) is quadrilateral from the Latin roots for "four sided". For many, this seems not in keeping with the common names for the other polygons; pentagon, hexagon, octogon... all of which combine Greek names for number with the Greek root gon for angle. For this reason many mathematicians extend this list down to "tetragon", for four-angled in Greek. This conforms to the use of tetrahedron for the solid with four faces. Although triangle is also not Greek, that name seems too well imbedded in language at such a low level that it does not seem wise to try to alter it.

John Conway posted a message to a discussion list that said the use of quadrilateral is fairly recent. "By the way, the term "quadrilateral" only came into routine use for a general four-sided polygon about a century ago, although there are some isolated early instance of its use in specialized senses. The older term was "trapezoid"; but (as I said here some time ago) this got confused with "trapezium", necessitating the introduction of a new word."

Many mathematicians also prefer the term quadrangle from the Latin for four-angled, which fits well with triangle. According to Jeff Miller's web sight on the first use of some math words, quadrangle dates back in English to at least the fifteenth century, and was used by Shakespeare



Ton, tun

In the very old English, German, and Danish languages there seems to have been a word something like "tunde", "toone" or "tun" which expressed the idea of a great amount. The current residue of these words or their singular ancestor, is a wide range of different units that share the name ton in English.

Among the many units called a ton is the most commonly known in the US, a weight of 2000 pounds, or 20 hundredweights. Just as there are several different standards for what a hundred is, there are also variations of what a ton is. A British ton is 2240 pounds, 20 times 112 pounds in a hundredweight. In the US the British ton is often called a Long ton and the smaller US standard is sometimes called a short ton. The metric ton was established as 1000 kg, so it weighs in at just a little less than 2205 pounds.

A large English shipping casks for wine was called once called the tun. The names of many English towns ending in -tun or ton represent the presence of a brewery in the village. The tun was not standardized, but generally ran about 250 gallons. Merchant ships were rated according to the number of tuns they could carry, and over time this became the merchant ton which today is a measure of volume equal to 100 cubic feet. Because this value was on the ship's registry, it was also called the Registry Ton, and abbreviated RT. Unfortunately this abbreviation was also used for the refrigeration ton, which is neither a weight nor a volume, but a measure of power. The ton in refrigeration is the amount of power required to turn one ton (US) of water at 0 degrees into ice in a period of 24 hours. For large explosions the amount of energy can be expressed in tons also. In this case a ton of energy is the energy equivalent to a ton (US) of dynamite exploding.

The tonde is a measure in Denmark that is used for wine measure and is the equivalent word for barrel. But just to keep things confusing, the tonde is also a measure of land equal to about 1 1/3 acres.