Mathwords, pg 10

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Acre

The root of the word acre is the Greek agros for field. The same root gives us many modern words related to agriculture. Although the original root described any open area, by the time of the Anglo-Saxons it was used more specifically as the amount of land a yoke of oxen could plow in one day. Sometime in the middle ages it was set equal to its current size, a unit of land equal to 4840 square yards, or 43,560 square feet. In the times of plowing with oxen or horses, an acre was not visualized as a square unit of land, but as a rectangle that was a furlong (furrow long) by a chain (four rods) wide.

Antares

The brightest star in the Southern Hemisphere is actually a double star system approximately 424 light years from the earth in the constellation of Scorpio. The name antares tells what it is not, literally meaning "not Mars" drawn from anti, for not, and Ares which was the Greek name for the Planet Mars. Ares was the Greek God of War. Because both the planet and the double star have a reddish color, they may have often been confused, leading to the name of the star.

Bernoulli Numbers
Student's often learn to find the sum of the counting numbers, 1+2+3+...+n by using the formula $$\frac{n(n+1)}{2}(n)$$. Later they may even discover a formula for the sums of squares, $$1^2 + 2^2 + 3^2 +\dotsb + n^2 = \frac{1}{3}( n^3 + \frac{3}{2} n^2 + \frac{1}{2}n)$$ . Around 1631, a mathematician named Johann Faulhaber (1580-1635) derived a formula for sums of any power. His method used an expansion that looked very similar to a binomial expansion with a sequence of special constants. These constants, 1, 1/2, 1/6, 0, -1/30, 0, 1/42, ... could be continued indefinitly to find sums of $$1^n + 2^n+ 3^n + \dotsb + k^n$$for any n and k of interst. 82 years later the Ars Conjectandi a masterful document written by Jacob Bernoulli (1654-1705) and published posthumously discribed this method in great detail giving full credit to Faulhaber. Bernoulli claimed that he was able to add the tenth powers of the integers up to 1000 in 7.5 minutes. And in the strange but common method of mathematical crediting, these numbers are known today as Bernoulli Numbers. Ivo Schneider, in a note to the Hisotria Matematica newsgroup, stated that de Moivre and Euler began calling them "Bernoulli Numbers." Between Faulhaber and Bernoulli, it seems that a Japanese mathematician named Takakazu Seki (?1642? - 1708)also may have discovered the sequence independently. You can find more of the Bernoulli number sequence by expanding (B-1)^n = B^n where B^k is the kth Bernouli Number.

In note G of Ada Lovelace's notes on the analytical engine from 1842, Lovelace describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.

Bode's Law
Bode's Law is a sequence which predicts the distances of the planets from the sun in astronomomical units (AU). The distance from the Earth to the Sun is defined as one AU. The law was published in 1772 by German astronomer Johann Elert Bode, and was based on the work of German mathematician Johann David Titius. The sequence is often referred to by both names, as Bode-Titius or Titius Bode. Titius name is also frequently mis-written as Titus.

When the law was first published there were only six discovered planets; Mercury, Venus, Earth, Mars, Jupiter, and Saturn. The formula states that the distances from the sun should correspond to the terms of the sequence dis(n)=$$\frac{3*2^n + 4}{10}$$ except for an initial term of .4 for Mercury. The first few terms of the sequence, .4, .7, 1, 1.6, 2.8, 5.2, 10, 19.6 predicted the known planets very well, except that there was no planet at distance 2.8 between Mars (1.6) and Jupiter (5.2). In 1781 the planet Uranus was discovered and its distance of 19.2 AU seemed close to the predicted 19.6. Bode encouraged astronomers search for a planet at the missing 2.8 distance. In 1801 Ceres, the first of the many asteroids at the 2.8 AU distance was found, and thought to be the missing planet. Hundreds more asteroids were quickly found and it was thought that the asteroids might either be the remains of a planet which had broken up, or the unassembled clutter of a planet that had not formed.

I recently read that the French philosopher Hagel had begun to circulate his thesis to "prove" that there could only be seven planets, when the news of Ceres discovery forced a quick change of the document. The discovery of Neptune in 1846 at 30.1 AU instead of the predicted 38.8, and then of Pluto in 1930 which was at 77.2 AU instead of 39.6 led to a discrediting of Bode's Law, although many astronomers still believe it may have some physical foundation.

Brachystochrone
In the last part of the 15th century one of the science questions of great interest was the question of what path a particle should take to move from point A to point B under the influence of gravity in the shortest time. The curve was called the brachystochrone problem. The word joins two Greek roots, brakhus for short, and the more common khronos for time. The first root only exists in a few scientific terms such as brachypterous for short winged insect. Chron appears as a much more common root in words like chronic (at all times), and chronology (in time order).
In 1696 John Bernoulli challenged the world to find the path, and Newton returned a correct solution anonymously. It is said that upon receiving the solution, Bernoulli replied, "We recognize the Lion by his claw." Newton's solution showed that the path was along part of a cycloid

For more detail about the history of the problem visit this link at the St.Andrews Math History site.

Carry

Although the word is now less popular with teachers, the "carry" was once a formal part of every American school child's vocabulary, with its opposite action of "borrowing". Today the "carry" is more often labeled as grouping or composing. The name carry most likely came from a similar act in accounting when amounts are "carried" from one column, account, or ledger to another. The early Indo-European root is from kers for run. The root emerged in Latin with cursus, to run, and carrus for a two wheeled wagon. Can you see the origin of our present word car? Words like current, cargo, caricature (exaggerated, or carried to an extreme???), and that new computer word, cursor (runs before the type) all come from the same source.

Although the topic of "carrying" may seem trivial to some, David Wells points out in the Penguin Book of Curious and Interesting Mathematics that carrys play an important part in the solution of Hilbert's Tenth Problem. If we let N be the number of ways that a objects can be chosen from a+b items, and p be any prime number, Ernst Kummer (1810-1893) showed that the largest power of p that would divide N is px where x is the number of carries needed when a is added to b in base p. As an example if we let a=7 and b=15 then N is the number of ways of choosing 7 things from 22 things, or 170544. Now if we wish to know the greatest power of 2 that divides the number 170544, we simply convert 7 and 15 to base 2, then add them and count the number of carrys. Seven in base 2 is 111, and 15 is 1111, so we need to add 111 + 1111 in base two

01111
00111
10110
which is the base two number for 22 and required 4 carries, that means 24 or 16 should be the largest power of two that will divide 170544, and indeed 16*10659 = 170544.

Dice / Die

At least as far back as 2000 BC, the rich and the mystical have had dice to play with. Very early dice were often in the shape of a tetrahedron with the vertices marked on one or more faces. The modern cube shape came later, but was common by the birth of Christ. The first dice-like objects to be used for games were probably made from the astralagus of deer, cow or oxen. F. N. David in Games, Gods, and Gambling described them thus:

The words astragalus, talus, huckle-bone, and knuckle-bone appear to be used indiscriminately in both ancient and medieval literature. The astralgalus is a bone in the heel lying above the talus, which later, in strict anatomical sense, is the heel bone. The huckle bone is astragalus while the knuckle-bone, strictly the bone in the hand, has been used to mean huckle-bone also since the sixteenth century.
An image of four astragali from the book can be found here. These astragali usually had two sides that were rounded and therefore they could only come to rest on four of the edges. They have been found in large quantities in Egyptian finds dating to around 3500 BC. The tombs of Ancient Egyptian Kings have also produced square based cylinders (think of a square pencil) that were apparently used much like dice, for games and predicting the future.

Sophocles stated that dice were invented by Palamedes during the Tojan War. Heroditus, on the other hand, gives credit for the creation to the Lydians, a kingdom that ruled Western Anatolia in the 7th Century BC.

The Greeks called their dice tessera from the Greek word for four. This could be related to the four resting faces of the astragali, or because it had four corners on a face. Dice were so common by the Roman period that when Julius Ceaser ordered his troops across the Rubicon River to begin his invasion of Italy, he declared, "Iacta alea est.", "The die is cast."

The alea of the phrase was the Latin word for a die. The only word I know in modern English with the same root is aleatory, depending on chance. Our word die (singular of dice) comes from the Latin word datum which is also the root of data. There is some indication that the word was used in the common Latin to mean a playing piece.

The prefix eo came to symbolize not only the dawn of a day, but the dawn, or beginning, of anything. For this reason we have scientific terms like eohippus, "dawn horse", for the ancient ancestor of our modern horse. The Cenozoic (literally "new animals") period in history after the demise of the dinosaurs was rich with the emergence of new forms of mammels and birds, and its early part is called the Eocene (dawn of new)

It was known from the work of Gregory of St. Vincent and others that the logarithms were somehow linked to the area under the hyperbola f(x)=1/x because the area under the curve matched the logarithmic property Log(AB)= Log(A)+Log(B). The Area under the curve from 1 to x=ab is equal to the areas from 1 to x=a plus the area from 1 to x=b. The value of e is such that the area under the hyperbola from 1 to e is 1 square unit. It has been conjectured that Euler may have used e as an abbreviation of the word Eins, the German word for one.

One of the most remarkable properties about e turned out to be the simple infinite sequence Euler used to approximate its value so accurately. He showed that e is equal to the sum of the reciprocals of the factorials. $$e = \sum_{n=0}^{\inf} \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!}+ \frac{1}{2!} + \dotsb \approx 2.718281828459045...$$

One oddity that students and teachers may use to remember the first 15 digits of e, given above, is to recognize their relationship with Andrew Jackson's presidency and an isosceles right triangle. Confusing? Just wait, all will be clear. We begin with 2, because Jackson was president for two terms. The 7 tells us he was the seventh president of the US. 1828 is the year he was elected, and we repeat this because of the two terms. Then we give the three angles of an isosceles right triangle, 45, 90, 45, and we have completed 15 digits of the base of the natural logarithms. I am almost 100% sure I picked that up from one of Martin Gardner's Scientific American columns.

Euler was one of the most influential mathematicians of the period and his prestige was sufficient that his use of a variable often marked it for posterity, but there were other symbols that were suggested occasionally. D'Alembert used c for the same constant in 1747, and Benjamin Peirce suggested a symbol that looked like a paper clip, or the @ symbol now used for e-mail addresses instead of pi, and the same symbol reflected in a vertical line for e.

Exponent is the union of the Latin roots exo(out of) + ponere (place). The literal interpretation is to make something visible or obvious. That seems to be what happens when the index is raised "out of" the line. The English word expound, from the same source, means to make clear. An exponent is also used in English to describe a person who explains or interprets. Exponent, as a math term, was introduced by Michael Stifel (1487-1567) in his book, Arithmetica Integra, in 1544.

One of the "math Doctors at the Math Forum, Dr. Peterson, took mild exception to my descriptive use of raising the exponent out of the line in one of his answers. He wrote, "I'm not convinced by the idea that "exponent" refers to the raising of the number above the line ("out of place", or "placed out of line") because a raised exponent notation, though used by Chuquet in the 1400's, was not popularized until the 1600's. Stifel did not use such a notation, but a complicated set of symbols for different powers. Literally, Stifel's "exponens" would be "setting out": "3 is the setting out of 8" might mean "3 is the number of times you write 2 as a factor, in order to get 8 by multiplying". Other authors of that period would call 2 (I think he means 3 here) the "index" of 8, meaning its place in the list of powers, as "indicated" by the top row of numbers. That word is still used, especially in England, for what we call the exponent. " Ok, he makes a good point.

The earliest known symbols to indicate the power of a number appear in ancient Babylon and Egypt. It seems the Babylonians had ideographs for both squaring and cubing a value. In an Egyptian papyrus now in Moscow, a symbol, which looks like a drawing of a pair of walking legs, is used to indicate multiplying a value by itself. The same symbol is used in the Ahmes Papyrus for addition when facing left and subtraction when facing right. Then, as now, there was inconsistency in the use of symbols.

The Hindu and Arabic writers often used different words for x and x2. Al-Khowarizmi used mal for the value of x2 and thought of this as the principal unknown. The square root of mal, x, he called jidr which translates to base or lowest part.

The earliest idea of an exponential index was developed by Nicole Oresme (1323-1382) but none of his contemporaries followed his lead. Even more remarkably, he conceived the idea of fractional powers and a symbol system for them. A century later Nicolas Chuquet independently developed a system of exponential notation. Because at the time almost all work was with a single variable, it was not necessary to use a symbol for the variable, so when Chuquet writes 43 he meant 4x3, not 64.

The first record of notation like we use now sprang from the pen of Descartes in his Geometrie. Descartes wrote 3x5 just as we would now, although for squares he preferred xx to x2, and he only used positive integral exponents. Later (1656) Wallis raised the idea of fractional and negative "indices", but never used them in symbolic form. Finally, Newton (1676) explained the idea of fractional and negative exponents in a letter to the Royal Society of London. When complicated variable quantities started to appear in exponents, De Morgan suggested (1845 _Calculus of Functions_) the use of the carat symbol, "^" as in x^(a+bi).

After I had written the above, I found the following post from Sam Kutler:

From A HISTORY OF MATHEMATICAL NOTATIONS by Florian Cajori, we learn that:

In 1636 James Hume brought out an edition of the algebra of Vieta, in which he introduced a superior notation, writing down the base and elevating the exponent to a position above the regular line and a little to the right. The exponent was expressed in Roman numerals... Except for the use of Roman numerals, one has here our modern notation. Thus, this Scotsman, residing in Paris, had almost hit upon the exponential symbolism which has bexome universal through the writings of Descartes.

Toward the end of 2004, a series of posts on the Historia Matematica discussion list talked about the earliest use of negative and fractional exponents that indicate that the attribution to Nicole Oresme above may be in error. Here are some snips from those posts:

Wallis (1655) showed x^0 = 1 and establish relations of x^{-1} = 1/x ...
See History of mathematics by D.E. Smith, Volume II, page 414
Sounny Slitine
--------------
There seem to be two issues here:
Who first thought of using 0 and negative numbers as exponents; and who actually used x^-1 etc.
Chuquet indicated the use of negative numbers as exponents; as did Wallis. However, on June 13, 1676 Newton wrote Oldenburg:
" Since, algebraists write a^2, a^3, a^4, etc. for aa, aaa, aaaa, etc, so I write a^1/2, a^ 3/2 ... {for roots of numbers} and I write a^ -1, a^-2, a^-3, etc. for 1/a, 1/aa, 1/aaa, etc."
In a later letter, Newton even used irrational exponents. Newton was undoubtedly influenced by Wallis. See Cajori, History of Mathematical Notations.
Sanford Segal
---------------------
Roshdi Rashed gives a much earlier reference than any mentioned so far, in "Histoire des sciences arabes 2: Mathématiques et physique". On p.38 he attributes to al-Samaw'al (died 1174) a definition of general integer powers together with the rule whose modern equivalent is (for every m, n in Z) ((x^m)(x^n) = x^(m+n)).
He gives a French translation of this rule as al-Samaw'al stated it, apparently thinking of the powers in a horizontal list:
"Si les deux puissances sont de part et d'autre de l'unité, à partir de l'une d'elles nous comptons en direction de l'unité le nombre des éléments du tableau qui séparent l'autre puissance de l'unité, et le nombre est du côté de l'unité. Si les deux puissances sont du même côté de l'unité, nous comptons en direction opposée à l'unité."
Ken Pledger.

Gamma Function
The factorial function only applies to non-negative integers. The gamma function is an extension of the factorial function to cover the rest of the reals and the complex plane. Euler described the function, but not the name in his letters to Goldbach around 1730. The name, and the symbol, $$\Gamma(n)$$, were first used by Legendre in 1808. The gamma function is given by the equation

By calculus methods it is possible to show that $$\Gamma(n+1)= (n) \Gamma(n)$$, a recursive definition identical to the method of generating factorials, and in general, when n is a positive integer $$\Gamma(n+1) = n!$$. One of the well known values of the funciton is $$\Gamma (\frac{3}{2}) = \frac{1}{2}! = \frac{\sqrt{\pi}}{2}$$

The $$\Gamma(n)$$ function is related to the Harmonic Series and the relationship can be seen in an alternate earlier version of the equation

Gram

The French revolution led to a rush to redesign all the measurements and the proposal for a unit of weight (actually mass) was first based on the proposed meter even before the meter had been established. Originally in early 1793 Lavoisier had suggested the standard unit of weight, first called a grave would be the weight of a cubic decimeter of water weighed in a vacuum at freezing temperature. Later the final measure presented to the assembly and passed into law had adjusted the temperature to the maximum density of water (about 4 degrees C). The name grave was changed to Kilogram in 1795 and the gram was established as 1/1000 of a kilogram.

The Latin prefixes, kilo, hecto, etc.; first appeared in a report of 1793. Until then the multiples and fractions of a unit used common French names such as the perch {also called the rod in English) for ten meters, the stade, a Greek term picked up and transferred thorugh the Romans was used for 100 meters, and palme for .1 meter.

Ludophh's Number
Ludolph von Ceulen devoted much of his life to finding ever more accurate approximations to
pi. During his lifetime he extend the known accuracy of pi out to 35 decimal places. He was so proud of his achievements that he directed that his estimation of pi be inscribed on his tombstone. Because of his work in this area, Pi is sometimes referred to as Ludolph's number in some places in Europe. Today, with the use of computers and advanced mathematical methods, we can calculate pi to millions of decimal places.

Mode

The early Indo-European root from which mode was derived had more of an "e" sound, and is written as med in dictionaires today. Its meaning was related to measuring and more often, to an exact or appropriate measure. When the term made its way into Latin it became modus and was generalized to apply to the appropriate method. The French truncated it to its present form, and used it for that which was fashionable or popular. In 1895 when Karl Pearson wrote, "I have found it convenient to use the term mode for the abscissa corresponding to the ordinate of maximum frequency." Every since then, Mathematicians have referred to the most "popular" value in a set of discreet data as the mode.

Many words you might not expect are from this same Indo-European root. Medical and meditate are examples as are modern and modest. Other math words from the same root are median, modulus, and module.

Meter

The Greek word for measure was metron and the meaning still persists in the many "ometer" suffixed words, such as speedometer. When the French people decided to throw out all the old reminders of the Royalists after the French Revolution, they set about creating a new system of measurements. They named the unit of length a meter and set its length to be one ten-millionth (1/10,000,000) of the distance from the North Pole to the equator.

There were several serious challenges to the idea of a decimal system based on a measurement of a part of the earth. As early as 1620 the French mathematician Father Marin Mersenne had suggested a standard for length based on the length of a pendulum with a one-second swing. In 1775 it was just this unit that Turgot had in mind when he had asked Condorcet to draw up a system of measurement based on the unit. By this time however, the scientists of the world knew that the period of a pendulum varied somewhat with latitude, and so there must be a location associated with the choice of length. The French proposed a site on the 45th parallel near Bordeaux, although some suggested the equator as a more natural spot, and Thomas Jefferson, as secretary of State in the US, thought the 38th parallel appropriate as it was the median latitude for the US (and also very near his home in Monticello). There were also concerns by many that the meter of one ten-millionth of a quarter meridian would be too long, and suggestions for a unit of one one-hundred-millionth of the full meridian, which would be closer to the pied (about a foot) then in use in Paris were frequent. It was Borda, then the chairman of the Commission of Weights and Measures who finally pushed through the choice of a unit based on the measure of the earth when he presented the report in 1791. Between then and the creation of a provisional meter (the need for a unit was considered too dire to wait for the final measure of the meridian from Dunkerquue to Barcelona) in 1793, there were even calls for the meter to be divided in duodecimal units to facilitate division. Conversly, the deimal supporters were looking at the creation of a day with 10 hours of 100 minutes each lasting 100 seconds. [Here is a picture of a clock face designed with both 12 hour and ten hour movements.] Finally in 1793 the provisional meter was created at a length of 443.44 lignes. The ligne was a unit equal to 1/12 of a pouce {inch) or 1/144 of a pied(foot). It is interesting to note that the grave, predicessor of the kilogram, was proposed as a measure based on the meter, even before the meter had been created.

Today the length is determined by a more complex approach. Here is the explanation as given by Russ Rowlett from the University of North Carolina at Chapel Hill on his Dictionary of Units of Measure

"Because the Earth is difficult to measure (not to mention the many irregularities in its shape!), this is not a practical definition. For a long time, the meter was the length of an actual object, a bar kept at the International Bureau of Weights and Measures in Paris. In recent years, however, the SI fundamental units (with one exception) have been redefined in abstract terms which can (in principle, at least) be reproduced to any desired level of accuracy in a well-equipped laboratory. For the meter, the 17th General Conference on Weights and Measures in 1983 adopted this "simple" definition: the meter is defined as that distance which makes the speed of light in a vacuum equal to exactly 299,792,458 meters per second. The speed of light in a vacuum is one of the fundamental constants of nature. Experiments previously made to measure the speed of light are now reinterpreted as measurements of the meter instead. The meter is equal to approximately 1.0936133 yards, 3.280840 feet, or 39.370079 inches."

Pons Asinorum

When Geometry was first taught in schools, it was taught straight from the translations of Euclid's Elements. In book one, proposition five, Euclid proved that the base angles of an isosceles triangle were congruent by using a figure like this one. . The illustration, and the proof are often referred to as the Pons Asinorum, Latin for bridge of asses. Although some claim the name comes from the fact that it looked somehow like a steep bridge, too steep for a horse but not for an ass, it is more likely that it is named because it was a challenging obstacle to the students who were "too stubborn" to learn their proofs. Because Euclid chose to prove this theorem before he had developed many of the congruence relationships, it was made much more difficult than it might have been. It is said that when given the problem in 1959 a computer solved the problem by simply showing that triangle ABC was congruent to ACB and thus the base angles were congruent. The same solution was found much earlier by Proclus in the fifth century.

You can read the fifth proposition in Greek at the Persius Project. For a nice English translation of the proposition, see the page by D.E. Joyce at Clark University.

The basic coin of the Roman Empire for many years was the As, and when a coin was created that had a value of 1/4 of an As, it was called a Quadrans. Later the word was generalized to quadrant and its meaning broadened to apply to many things that were shaped like 1/4 of a circle or of a circular arc. As a circular arc is drawn, each quater circle marks off one of the four quadrants of the plane.

A common mathematicians instrument of the Middle Ages is called a quadrant because of its shape. A beautiful example of a hand sized "universal" quadrant, "capable of carrying out mathematical, astronomical, astrological and military calculations" is one of several that can be seen on the web page of the Oxford Museum of Scientific History. For greater accuracy, very large quadrants were built and mounted on Walls aligned with the meridian. One of the most famous was the 2 meter radius "great quadrant" at Uraniborg completed by Tycho Brahe in 1582. An image of the Brahe Quadrant is found here as shown in an engraving from Tycho's Astronomiae instaurata mechanica, published in 1598.

I have been searching for when we began to number the quadrants on a coordinate graph as we do (I, II, III, IV in anti-clockwise order). The earliest use I have found so far is the quote below from Augustus De Morgan in his 1849 Trigonometry and Double Algebra.

De Morgan was at the time the Secretary of the Royal Astronomical Society, and a professor of University College, London.

A later copy of the same appears in the 1895, Plane and Spherical Trigonometry, Surveying, and Tables of George A Wentworth.

Scalene

Most students associate the word scalene with triangles, but there are also scalene cylinders and cones. In each case the word scalene has the same meaning, uneven. A scalene triangle is a triangle in which no two of the sides (or angles) are congruent. Scalene comes from an early Indo-European root that related to chopping. Such chopped up edges were often uneven. According to Rick Miller's site on the Earliest Known Uses of Some of the Words of Mathematics in English, "The earliest use of scalene as an adjective to describe a triangle is in 1734 in The Builder's Dictionary."

Soliton

Soliton is a shortened word to represent the solution of a differential equation representing a stationary wave. The term seems to date to about the 1965 and was invented by Martin Kruskal of Rutgers Univ.

Here is a quote from the site of Kanehisa Takasaki about the nature of solitons

....the scientific research of solitons had started in the 19th century when John Scott Russell observed a large solitary wave in a canal near Edinburgh. In the days of Scott Russell, there was much debate concerning the very existence of this kind of solitary waves. Nowadays, many model equations of nonlinear phenomena are known to possess soliton solutions. Solitons are very stable solitary waves in a solution of those equations. As the term "soliton" suggests, these solitary waves behave like "particles". When they are located mutually far apart, each of them is approximately a traveling wave with constant shape and velocity. As two such solitary waves get closer, they gradually deform and finally merge into a single wave packet; this wave packet, however, soon splits into two solitary waves with the same shape and velocity before "collision".

Totient of a number/ Euler's totients/ Euler's Phi Function

The totient of an integer, N, is the number of integers Less than N which are relatively prime to N, that is, they share no common factors. The symbol for the Totient (by the way, the word is pronounced to rhyme with "quotient"... I didn't know that for a long time) of N is usually the greek letter Φ. We would write Φ(10)=4, to indicate there are four numbers less than 10 which have no factor in common with the number ten. The four numbers 1,3,7, and 9 are called the totitives of 10. Euler showed that if a number n had prime factors p, q, and r; then the totient would equal N(1-1/p)(1-1/q)(1-1/r). It doesn't matter if it has some prime factors more than once, for instance 20 factors to 2x2x5, but the number of prime factors is found by 20 (1-1/2)(1-1/5) = 8; with totatives of 1, 3, 7, 9, 11, 13, 17, and 19. The word totient is drawn from the Latin root tot for "so much", and probably to the Greek tosos, "so great". The word tot is still used occasionally for a mark made beside a list to acknowledge receipt, and is sometimes called a "tot mark". Euler defined the rule for the number of totients around 1761 in proving what we now call the "Euler-Fermat" theorem. Euler didn't use the word totient. That term was introduced by J J Sylvester, (According to Graham, Knuth and Patashnik, "Concrete Mathematics") who had a habit of making up new words for math terms. "The reference they give is to a paper from 1883: "On the number of fractions contained in any 'Farey series' of which the limiting number is given".

The name "PHI" for the function was created by Gauss in his Disquitiones arithmeticae .

Here is a list of the first few integers, their totients, and their totitives...
N_____f(N)______ totitives
1_____ 1 ____ 1
2_____ 1 ____ 1
3_____ 2 _____ 1,2
4_____ 2 _____ 1,3
5_____ 4 _____ 1,2,3,4
6_____ 2 ______1, 5
7_____ 6 ______1.2.3.4.5.6
8_____ 4 ______ 1,3,5,7

There is actually an interesting unproven math theorem about totients. It seems that any number that appears in the sequence of f(N) must appear at least twice. It has been proven that if there is an exception, it must have more than 10,000 digits. This is sometimes called the Carmichael Conjecture after R. D. Carmichael.

Wilson's Theorem

Wilson's Thm states that any number p is prime if, and only if, the value of (p-1)!+1 is a multiple of P.  This is often written in congruence format as (p-1)! = -1 mod(p).  The theorem is named for Sir John Wilson (1741-1793), who discovered the theorem while he was still a student at Peterhouse College, Cambridge, although it seems he did not have a formal proof.  Wilson went on to be a Judge and seems to have done little else in mathematics.  The theorem was first published by Edward Waring around 1770 and it was he who attached Wilson's name to the theorem, although it is now known that the result was known to Liebniz, and perhaps was known to Ibn al-Haytham (965-1040) much earlier, as noted in this quote from St Andrews Web site.

Al-Haytham, is also the first person that we know to state Wilson's theorem, namely that if p is prime then 1+(p-1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after al-Haytham before number theory surpasses this achievement of Arabic mathematics.

The quantity n! + 1 also appears in Brocard's Problem, which asks for what values of n does n! + 1 equal a perfect square. The only known solutions are n=4, 5, and 7, but no proof exists that there is not another but if there is, it must be very large. The great Indian mathematician Ramanujan studied the same problem in 1913 not knowing of Brocard's work. Brocard is more famous for his work in Geometry, and Brocard's Point.

Xenon

Xenon, atomic number 54, is a colorless, oderless, and very non-reactive gas. It was first isolated in 1898 by M. W. Travers and William Ramsey. It is the only non-radioactive noble gas which forms stable chemical compounds at room temperature. The name xenon was given by Ramsey and comes from the Greek word xenos for strange. The gas has been used in TV tubes and in bacteriacidal lamps.