Ratio, Rational and Irrational ... History and Etymology of Math Terms

  Ratio

The ancient root of ratio comes from the same early Indo-European root that gave us arithmetic. It is sometimes given as ar and sometimes ree. In its earliest incarnations the word may have related to "fitting together", but quickly took on a meaning related to counting (putting all the items together into one group, perhaps). By the Latin reri it had taken on the ideas of "reason", from which comes rational, and ratio for a comparison of two magnitudes. Rate is a synonym for ratio and comes from the same source. The word rational is used in common language to mean a method of thinking based on logic and reason, and in mathematics to describe a comparison of two magnitudes. A rational number is a number that may be expressed as a ratio of two integers. The letter Q is generally used to represent the set of rational numbers. At Jeff Miller's web site on the earliest use of some math symbols, I found the statement, "Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. (N. Bourbaki was a group of mostly French mathematicians which began meeting in the 1930s, aiming to write a thorough unified account of all mathematics.) The letters stand for the German Quotient and Zahlen. These notations occur in Bourbaki's Algébre, Chapter 1. "

The real numbers may be divided into two sets by separating numbers into the rational numbers, and the irrational numbers. Rational numbers are numbers that may be expressed as the ratio of two integers. All common fractions would be in this category, 2/3 or 5/4, as well as the integers themselves since 3 can be expressed as the ratio of 3/1. Any decimal expression that terminates after some time can be expressed as a rational number also. As an example, .35 can be written as 35/100 or 7/20. Decimal numbers that repeat the same string of digits forever are also rational numbers. Expressions like .444….. can also be represented as 4/9, and in general it is easy to express any decimal fraction that repeats right from the decimal point by writing the repeating string in the numerator and as many nines as there are digits in the repeat string in the denominator. For example the three digit repeat sequence .453453453…. can be written as 453/999. A little algebra allows us to show that if the number repeats after some initial non-repeating sequence, it can still be rewritten as a rational. For example .23453453…. can be written as a rational with the numerator equal to 23453-23= 23430; and the denominator equal to 99900 (note that three digits repeated, hence three nines, and two did not, hence two zeros).

Irrational numbers are real numbers that can NOT be expressed as a ratio of two rational numbers. The story of the irrationals probably starts with the Pythagorean discovery that the diagonal of a square could not be expressed as a ratio of the sides in any way. If the sides of the square are 1 unit in length, the diagonal will have a length that is the square root of two, so  is irrational. The  is approximately equal to 1.41423156… but the decimal expansion never reaches a point where some cycle repeats itself forever. In fact all square roots of integers that are not perfect squares (numbers like 1, 4, 9, 16, etc) are irrational. Other famous numbers that are irrational include Pi, which is appx 3.14159265… and e, which is appx 2.7182818284590… and the golden ratio which is appx 1.6180339… .

A recent discussion on the Historia Matematica list explains the origin and development of irrational. I have clipped parts of several documents.

In the eminent website Earliest Known Uses of Some of the Words of Mathematics I read about the history of the word irrational:
Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius (6th C)was the first writer to use the terms 'rational' and 'irrational' in the sense now current in arithmetic and algebra."
Irrational is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."
Heath mentions (Vol 1 p. 92) that Magnus Aurelius Cassiodorius presented Greek geometry in his encyclopaedia "De artibus ac disciplinis liberalium literarum" (about 475 A.D.). I suppose it is there Cajori has found the terms 'rational' and 'irrational'. Perhaps in his writing about proposition 47, book I? But did Cassiodorius really mention irrational numbers? Does 'now current' mean that he in some sense had a numberline?
I know that Jacques Peletier in 1563 uses the word irrational number and he means that the sum of a rational and a irrational number is always irrational according to, what he calls a philosophical axiom. [Staffan Rodhe]

Let me intervene in this learned discussion with the following remarkable observation made by Johan Kepler in the first pages of his Harmoniae Mundi. The Greek words translated by the Latin "rational" and "irrational" (segments, i.e., numbers) are "logos" and "alogos", resp. When in Greek mathematics one mentions "logoi" or "alogoi" in connection with segments (such as the side and diagonal of a quadrilateral) it means "expressible" or "un-expressible", resp., Hence the Latin translation "rational" and "irrational" is a mis-translation, and it should better be translated as "expressible" or "inexpressible", resp., when appearing in the mathematical context. But now is too late for such a reformation of terminology. Yaakov S. Kupitz

There are actually three Greek words having similar meaning. In Plato one finds occasionally "arrhetos" (unspeakable, inexpressible, related to "rhetoric") and "alogos" (irrational, "illogical"). The word in Euclid is "a-sym-metra" (plural) referring to two in-com-mensurables (a piece-for-piece translation of "asymmetra", and somewhat distinct from the English cognate "asymmetric"). I find it interesting that the English word "unspeakable" carries a heavy emotional connotation of being "too horrible for words," but that connotation is not in the Greek "arrhetos". [Roger Cooke]

---

R

Dissection Puzzles, a Little History

  It is known that Archimedes created a game/puzzle with the dissection of a square into 14 pieces.  The object of the puzzle is to put the pieces back together to form a square. A more difficult question, unknown for over 2000 years, is how many unique ways are there of putting the pieces together to form a square. Bill Cutler used a computer program to show that there are 536 unique ways to assemble the pieces not counting similar rotations and reflections. All 536 solutions are visible in this article from Ed Pegg's web site.

The Archimedes Palimpsest is a parchment codex palimpsest, which originally was a 10th-century Byzantine Greek copy of an otherwise unknown work of Archimedes of Syracuse and other authors. It was overwritten with a Christian religious text by 13th-century monks. The erasure was incomplete, and Archimedes' work is now readable after scientific and scholarly work from 1998 to 2008 using digital processing of images produced by ultraviolet, infrared, visible and raking light, and X-ray.

The Palimpsest is the only known copy of "Stomachion". The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the ancient Greek word for throat or gullet, stomachos (στόμαχος). Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots ofὀστέον (osteon, bone) and μάχη (machē – fight). The puzzle is also known as the Loculus of Archimedes or Archimedes'Box. Loculus seems to be a word related to the division of a tomb area into small chambers for different bodies and is related to the diminutive of locus for a point or place, thus "a little place". (I don't yet get the connection between the puzzle and stomach. Bob Mrotek pointed out to me in a comment that the Stomahion may be a later condesation of the Greek Ostomachion, Ὀστομάχιον, "a word meaning a fight (μάχη, mákhion) with bones (ὀστέον, ostéon) in reference to the pieces which were often made out of ivory." Which now appears as well on Wikipedia.)

The puzzle is sold by Kadon as Archimedes Square:

Amazingly, the next "put-together" puzzle of geometric shapes didn't appear until 1742.  The wisdom plates of Sei Shonagon is a seven piece puzzle.  They appeared in China 71 years before the more famous Tangram puzzle.  The second edition of this puzzle, published in 1743, is the oldest known surviving puzzle of this type.

*http://www.indiana.edu

Tangram is a name of a Chinese puzzle of seven pieces that became popular in Europe around the middle of the 19th century. It seems to have been brought back to England by Sailors returning from Hong Kong. The origin of the name is not definite. One theory is that it comes from the Cantonese word for chin. A second is that it is related to a mispronunciation of a Chinese term that the sailors used for the ladies of the evening from whom they learned the game. [Concubines on the floating brothels of Canton, Hong Kong, and many other ports belonged to an ethnic group called the Tanka whose ancestors came from the interior of the country to become fishermen and pearl divers. They were considered as non-Chinese by the governments of China until 1731. They were unique among Chinese women in refusing to have their feet bound. ] A third suggestion is that it is from the archaic Chinese root for the number seven, which still persists in the Tanabata festival on July seventh in Japan which celebrates the reunion of the weaver (Vega) and the herdsman (Altair). David Singmaster, below, suggests the name was made up by puzzle master Sam Loyd, but I favor Harvard President Thomas Hill (below) . Whatever the origin of the name, the use of the seven shapes as a game in China were supposed to date back to the origin of the Chou dynasty over one thousand years before the common era. The Chinese name is Ch'i ch'iao t'u which translates, so I am told, as "ingenious plan of seven".

It appears however, that the game and the name are both much more modern than believed. From the MathPuzzle.com website, I found that " The Tangram was invented between 1796 and 1802 in China by Yang-cho-chu-shih. He published the book Ch'i ch'iao t'u (Pictures using seven clever pieces). The first European publication of Tangrams was in 1817. The word Tangram itself was coined by Dr. Thomas Hill in 1848 for his book Geometrical Puzzles for the Young. He became the president of Harvard in 1862, and also invented the game Halma.
When Tangrams hit Europe they were an immediate success.  A puzzle museum on-line boasts a collection of a dozen books which were all written within a year of 1817 when Tangrams were supposedly introduced into Europe:

Here is some history of the game by David Singmaster, one of the world's foremost authorities on recreational mathematics,

TANGRAMS. These are traditionally associated with China of several thousand years ago, but the earliest books are from the early 19C and appear in the west and in China at about the same time.
(although the image below with tangram problems to create was printed in Japan in 1795, and  Utamaro’s “Tagasode” is a famous 1804 Japanese blockprint that shows Tagasode and her servent trying to solve Tangrams), Indeed the word 'tangram' appears to be a 19C American invention (probably by Sam Loyd). A slightly different form of the game appears in Japan in a booklet by Ganreiken in 1742. Takagi says the author's real name is unknown, but Slocum & Botermans say it was probably Fan Chu Sen. There is an Utamaro woodcut of 1780 showing some form of the game (not yet seen by me). I have seen a 1786 print - Interior of an Edo House, from The Edo Sparrows or Chattering Guide - that may show the game. Needham says there are some early Chinese books, and van der Waals' historical chapter in Elffers' book Tangram cites a number with the following titles.
Ch'i Ch'iao ch'u pien ho-pi. >1820.
Ch'i Ch'iao hsin p'u. 1815 and later.
Ch'i Ch'iao pan. c1820.
Ch'i Ch'iao t'u ho-pi. Introduction by Sang-Hsi Ko. 1813 and later. remarks inserted from a description of the book in Tangram by Joost Elffers, {Located in the Leiden Library #6891: This book, with an introduction by Sang-Hsi Ko, is, as far as is known, the oldest example of a Chinese game-book. ] I would like to see some of these or photocopies of them. I would also be interested in seeing antique versions of the game itself. The only historical antecedent is the 'Loculus of Archimedes', a 14 piece puzzle known from about -3C to 6C in the Greek world. Could it have traveled to China? I found a plastic version of the Loculus on sale in Xian, made in Liaoning province. I wrote to the manufacturer to get more, but have had no reply.
For the 10th International Puzzle Party, Naoki Takashima sent a reproduction of a 1881 Japanese edition of an 1803 Chinese book on Tangrams which he says is the earliest known Tangram book.
Jean-Claude Martzloff found some some drawings of tangram-like puzzles from a 1727 booklet Wakoku Chie-kurabe, reproduced in Akira Hirayama's T“zai S–gaku Monogatari Heibonsha of 1973. Takagi has kindly sent his reprint of this booklet, but I am unsure as to the author, etc.

*Utamaro’s “Tagasode” http://www.indiana.edu
The "Ganriken" mentioned in Dr. Singmaster's post was a pseudonym, and it seems unclear who the actual author was.  The book was titled "Sei-Shonagon Chie-no-Ita", which translates to "the ingenious pieces of Sei Shonagon". From Wikipedia, "Sei Shonagon was a lady-in-waiting at the Japanese Imperial Court in the beginning of the 11th Century. She kept a personal diary of sorts in which she wrote down her experiences but mainly her feelings. Such diaries were common at the time and were called pillow books because these books were often kept next to people's pillows in which they would write their experiences and observations. The Pillow Book of Sei Shonagon gives an invaluable insight into the world of the Imperial Court of Kyoto a thousand years ago. Sei Shonagon's observations are witty, wry, poignant, and at times condescending."

Tangrams received another boost in popularity when Charles Dodgson, writing as Lewis Carroll, used them to create illustrations of the Characters in the "Alice" books. In the Penguin Books translation of Tangram by Joost Elffers he states that English Puzzle writer H. E. Dudney purchased a copy of a play-book called The Fashionable Chinese Puzzle from Dodgeson's estate. This book seems to be the most common source of the assertion that Napoleon was an avid Tangram player,

And recently I found this beautiful set of 19th Century Tangram dishes made in China,which are in the Hikimi Town Puzzle Museum in Japan.

#23 Trapezoid and Trapezium ...History and Etymology of Math Terms

  Trapezoid and Trapezium Both words come originally from the Greek word for table. Today, in the USA, the term trapezoid refers to a quadrilateral with one pair of sides parallel and a trapezium to one with NO parallel sides. Actually, the term for the case with no parallel sides is almost never used, so trapezium is an archaic term at best in the US. This is exactly the reverse of the original meanings and the meanings in some countries, particularly England, today. Here is a short comment on how this came about from Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida, who maintains an excellent page on the first use of some common mathematical terms:

"TRAPEZIUM and TRAPEZOID. The early editions of Euclid 1482-1516 have the Arabic helmariphe; trapezium is in the Basle edition of 1546. Both trapezium and trapezoid were used by Proclus (c. 410-485). From the time of Proclus until the end of the 18th century, a trapezium was a quadrilateral with two sides parallel and a trapezoid was a quadrilateral with no sides parallel. However, in 1795 a Mathematical and Philosophical Dictionary by Charles Hutton (1737-1823) appeared with the definitions of the two terms reversed: Trapezium...a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid. No previous use of the words with Hutton's definitions is known. Nevertheless, the newer meanings of the two words now prevail in U. S. but not necessarily in Great Britain (OED2).

John Conway recently pointed out in a post on the use of the terms that;

What is true now is that the thing with two parallel sides is called a "trapezium" in England and a trapezoid in America, and that neither term is used in either country for the thing with no parallel sides. (The latest date for which I'vbe seen either of them so used was in a geometry book of 1912, which however, was a reprint of a 19th-century one.) Instead, to avoid confusion, the term "quadrilateral" is now standard for the general case. (This has a few earlier uses, dating back to about 1500, but was decidedly uncommon - like "trilateral" - before 1900.)

I also have an English textbook that uses trapezion (note the n ending) for the shape we more commonly call a kite. In A Junior Geometry by Noel S. Lydon published in 1903 the definition on page 55 states A trapezion is a four-sided figure having two pairs of adjacent equal sides. It goes on to show the method of construction.

Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid. Euclid did not define the shape we now call a trapezoid, and the "trapezia" is defined by default.... "let quadrilaterals other than these be called trapezia" [from the Heath translation]. Heath's translation states that the language used implies that Euclid may have been creating a new word, or using an existing one in a new way. Proclus seperated out trapeziums and trapezoids (backwards to what we now do as explained in the quote above from Jeff Miller's page) but it seems clear he meant that a trapezium had exactly one pair of sides parallel (an exclusive definition) rather than at least one pair of sides parallel(an inclusive defintion). Many mathematicians today prefer the inclusive definition so that a parallelogram is a special case of trapezoid. Apparently this has been a question in geometry for a while as I recently read a note from Neal Silverman which suggests that the inclusive definition has appeared in some books back at least to 1900.

" I recently acquired a copy of "New Plane & Solid Geometry" by Wooster Woodruff Beman and David Eugene Smith (Ginn 1900), a revision of their earlier 1895 work. ... But as to quadrilaterals, consider their definition of trapezoid: "A quadrilateral that has one pair of opposite sides parallel is called a trapezoid." They go on to state that "[b]y the definition of trapezoid here given it will be seen that the parallelogram may be considered a special form of the trapezoid. (section 97 at p. 59).

It seems that the inclusive definition was not well received as in the same post Neal adds, "D. E. Smith went on to write many more books on geometry, some of which were revisions of the old Wentworth books. I have never seen this statement in any of his later books."

John Conway recently posted a note about the etymology of trapezoid;

The etymology of "trapezoid" is quite interesting. It's a corruption of "tetra-pes-oid", whose three parts mean "four-leg-shaped", or perhaps more familiarly "table-shaped", since "tetra-pes" was a familiar Greek name for a small table.

A Non-calculus Explanation for the Volume of a Bead

 

After I wrote about the "surprising constant" that a bead formed by drilling a hole through the center of a sphere had a volume that was dependent only on the height of the remaining bead. At the time I admitted I could not think of a good "non-calculus" explanation for why the volume for a given bead height was a constant. Afterward Arjen Dijksman,(see his blog) who regularly provides helpful insights, suggested that the volume could be explained using the ideas of Mamikon Mnatsakanian that I discussed here. While I was not able to see Arjen's solution, it did lead me to realize that a "non-calculus" proof existed (with some allowances). Thinking of Mamikon's annular sweep led me to realize that every cross-section through the bead perpendicular to the axis of the drilled hole would be an annulus. In fact, using R as the original sphere's radius, r as the radius of the hole, and h as the distance above the center of the sphere where the cross section was taken, the area of the annulus would be Pi times (R² - r²-h²), which is just the area of a circle. Now by Cavalieri's Principal, "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal." Several candidates exist with circular cross-sections; cylinders, cones, and spheres are examples. The cross-sections of the bead went from zero at the top, to a maximum at the center, so the cylinder is excluded. We can exclude the cone since their radii decrease linearly from top to bottom and our bead does not. And since we are already tempted with the idea that the volume of the bead is the same as a bead with a radius equal to half the total height of the bead, why not try to compare those? To make the notation easier, I decided to call the height of the bead 2H (letting H² equal the value (R² - r²). So the cross sectional area of the sphere with raidus H at any height, h, above the center will be Pi(H² - h²). As we have shown above, the cross-sectional area of the bead at the same height h is Pi (R² - r²-h ²). and since H² is equal to (R² - r²) these two are equal. So we have proven that the volume of the bead is the same as the volume of a sphere with the same height;... but did we use calculus? I fear that the proof of Cavalieri's Principal is founded in the calculus, and thus we still have a "calculus-based" explanation...but it's the best I have to offer to date.

Over the last few days I came upon a way to explain the volume of a bead with an (almost) non-calculus approach. My solution was inspired by a comment posted by Arjen Dijksman. I admitted I didn't understand his suggestion to use Mamikon's visual calculus, so he did what good teachers do. He posted a very clear explanation that even I could not miss. It's a beautiful blog, pretty math... check it out.. and Thanks loads Arjen.