Wednesday 13 May 2020

Volume I-M


Icosahedron The regular icoshedron has the most faces of any
Platonic solid, 20.  It has five equilateral triangles meeting at 12 vertices, with 30 edges joining the faces.  The roots are from the Greek roots eikosi for twenty, and hedra for seat.  The hedra root is common today in mathematical polyhedra (many seats, or faces) and cathedral (where the Bishop's seat is). (there is actually a concave icosahedron that could be called regular, called the great icosahedron. It's vertex figure is  a pentagram instead of pentagon.  

Identity (equations) The term identity, in relation to an equation came somewhat late to Algebra.  Identities were probably recognized as a special kind of equation well before the use of variables, but the name seems to have appeared as late as 1830,  when R. Blakelock wrote in his Complete Course of Pure Math, "We may here assume for the arbitrary quantity h any number we think proper, and the identity will still subsist." 

A good example of the meaning appears in B. Smith's Arithmetic and Algebra in 1859, he wrote, "Such and expression as (x+1)^2= x^2 + 2x + 1 \ ...."  There were (still are?) occasional uses of the word as we would say equal.  The word came from a cluster of terms, particularly in French, for sameness or oneness.  Florian Cajori wrote that, the sign now more commonly used for equivalence, (like the equal sign but with three parallel bars) ; was used by Riemann for identity.  

Identity Element The use of the term identity element seems first to have occurred in 1894 when the term was used as if it were one already clear to the reader, in the conditions for a proof, "Given an abstract Group... with identity s(1)".   The general definition is an element of a set which, if combined with any element by a (specified) binary operation, leaves the latter unchanged.  In the Group of real numbers under the binary operation of addition, zero is the identity, since adding it to any number leaves the number unchanged.  In the real number group using multiplication as the operation, one produces the similar result and is thus called the "multiplicative identity."  

Ides (of March) Every month had its ides, but a line from Shakespeare made the "Ides of March" linger in our language long after the ides of the month were still used.  The Ides were the early Roman name for the day of the full moon.  Since months began on the new moon, the ides normally fell near the middle of the month, and the term came to mean the 15th of the month.  The Kalends (see Calendar) were the days that debts come due.  This practice of bills and rents being due on the first of  the month persisted into the 20th century in many parts of the U S.  The nones occurred around the sixth or seventh of the month, but the term comes from the Roman name for 9, counting how many days it was before the Ides.  

Idiot  The Latin use of the Greek root for idiot was for a layman, and unskilled person; not a writer, or soldier, or politician, but the literal meaning of the Greek word, idiotes, was a private person (one who didn't take part in public affairs, most particularly voting. (so don't be an idiot, VOTE).  The idios root (ones own) is the same as the use in idiom.  

iff The iff symbol that teachers love to write for " if , and only if" is a relative newcomer to the symbology of mathematics.  It seem it was introduced into the language in 1955 by J L Kelley in a book on topology, "F is equicontinous at x, ifff (if and only if) there is a neighborhood of x.....".  He credited the creation of the term to Paul Halmos, who introduced several symbols into mathematics.  

Imaginary Numbers The word imaginary was first applied to the square root of a negative number by Rene Descartes around 1635.  He wrote that although one can imagine every Nth degree equation had N roots, there were no real numbers for some of those imagined roots.  Around 1685 the English mathematician John Wallis wrote, "We have had occasion to make mention of Negative squares and Imaginary roots".  Some meathematicians have suggested the name be changed to avoid the stigma that it seems to create in young students, "Why do we have to learn them if they aren't even real."  (To be honest, in thirty years as an educator, I never heard the question.)  Perhaps the weight of history is too much to support the change.  

The first person ever to write about employing the square roots of a negative number was Jerome Cardin (1501-1576).  In his Ars Magna (great arts) he posed the problem of dividing ten into two parts whose product if forty.  After pointing out that there could be no solution, he proceeded to solve the two equations, x+y = 10, and xy=40 to get the two solutions, \( 5 \pm \sqrt{15} \)  .  He then points out that if you add the two solutions, you get ten, and if you multiply them then the product is indeed forty, and concluded by saying that the process was "as subtle as it was useless."

Cardin also left a seed to inspire future work int he mystery of roots of negative numbers. Cardin had published a method of finding soltuons to certain types of cubic functions of the form \( x^3 + ax + b \) .  His solution required finding the roots of a derived equation.  For functions in which the value was negative, his method would not work, even if one of the three roots was a known real solution.  
About thirty years later Rafael Bombelli found a way to use the approach to find a root to \( x^3 - 15x -4 \) with the known solution of four.  He went on to develop a set of operations for these roots of negative numbers.  By the eighteenth century the imaginary numbers were being used widely in applied mathematics and then in the nineteenth century mathematicians set about formalizing the imaginary number so that they were mathematically "real."

Imaginary Unit The imaginary number with a magnitude of one used to represent \( \sqrt{-1} \), has been the letter i since it was adopted by Euler in 1777 in a memoir to the St Petersburg Academy, but it was not published until 1794 after his death.  It seemed not to have gained much use until Gauss adopted it in 1801, and began to use it regularly.  The term, imaginary unit, was first created, it seems, by William Rowan Hamilton in writing about quaternions in 1843 to the Royal Irish Academy.  For his three-dimensional algebra of quaternions, Hamilton added two more imaginary constants, j, and k, which were both considered perpendicular to the i, and to each other.  

Improper Fractions was used by Robert Recorde in 1542 in his the grounde of arts, for a fraction that was "greater than a unit."  For young students today, they are often taught as fractions in which the numerator is greater than the denominator.    

Included Angle In geometry an included angle is an angle formed by  the intersection of two geometrical objects.  The term seems to have been first used in English in 1657 in J. Newton's Astronomica Britannica, where he illustrates, "First the side FG, Secondly the side FH  and the included angle FGH (sic.?) .  In more modern geometry we might define planer angles as between two Rays with a common endpoint,  or in dihedral angles where two half-planes are joined on a common edge. 

Incenter and Incircle  Every triangle and many closed polygons can be inscribed with a circle tangent to every side.  In all these figures, the general name for that circle is the Incircle.  (There is a circumcircle also for triangles, and regular polygons that connects all the vertices. The center of the incircle, is called the incenter.  The incenter is the point where all three angle bisectors will intersect, which makes it the same distance from each of the three sides.  The Greeks knew four centers, the incenter, centroid, circumcenter, and orthocenter.  The incenter is the only one of these that does not lie on the Euler Line.  The distance from the incenter to the incircle is called the inradius. A rather nice formula ties the inradius, r, and the circumradius, R, to the product of the distance from the incenter to the three vertices;   IA x IB x IC = 4 R r^2

Index The use of a number or variable subscript to identify the terms of a sequence or elements of a matrix serves as a way to "point out" particular terms.  If you are speaking of the terms of the Fibonacci sequence, and want to make mention of what is 5th, we often write F(5) or \( F_5  \).   The 5 is called the index.  The origin of the word comes from that very purpose, to point out.  The Greek root dika is applied to a wide range of terms that relate to making one or mre items distinct from others by showing, saying, or teaching.  To adjudicate is to point out guilt or innocence, and gives us the word judge.  The fingers are called digits, and the finger with which we commonly point is called the index finger, a double use of the root.  Other related words are dictate, predict, valedictorian, dedicate, condition and the ubiquitous ditto.  

Infinity Symbol \( \infty \)  The sypmbol we now use for infinity was first used by John Wallis (1615-1703) in 1655 in his treatise On Conic Serctions.  Why he used it seems lost to history.  The two most popular suggestions are these.  The ancient Romans used a symbol like two hooked together zeros for the number 1000.  Since Wallis was a classical scholar, this may have been one source.  A second theory is that he used a variant of the lowercase symbol for omega, the last letter in the Greek alphabet, to symbolize the "final number" in a sense.   

Inflection point  The Latin root, inflexionem, is for the act of bending, or being bent. The current use in calculus is similarly related to the point on a curve where it stops bending one way, and starts bending the other, switching from concave to convex, or
vice-versa. In the Wikipedia gif above, the inflection points are located where the tangent line changes from blue (concave), to Green(convex).  Fermat used the word "inflexionum" in a work that was published after his death in 1679. It appears in English in 1721 in N. Bailey's Universal Etymology Dictionary.  In 1743 Emerson writing on calculus gives the definition as the changed of concavity stated above.  In calculus students learn that it is a point where the second derivative is zero and changes sign at that point.  

Inner Product The inner product is a term for a vector multiplication that produces a scaler (A pure number with no diminsions) result.  Sometimes called the scaler product, or the dot product.  They are useful in finding the angle between two vectors, and give a value of zero when the vectors are perpendicular. The first use in English was in a 1901 paper in the American Mathematical Monthly about German mathematician Herman Grassman's work with vectors. Grassman is credited with creating the term, as well as the alternate term for a vector product called the outer product, in 1844.  

Inscribed angle An inscribed angle has a vertex on a circle and the sides are either two secants,  or a secant and a tangent.  The
idea dates back to the Elements of Euclid, and came into English as early as Billingsley's translation in 1577.  The Latin roots, the prefix being exactly the same then and now, joins a suffix, scribere, meaning to write or draw.  It is used in many proofs of elementary geometry, including the Theorem of Thales, which says that if the two secants cut the endpoints of a diameter,  then the inscribed angle is a right angle.  As the right angle shows,all the inscribed angles whose sides of the  cut a given chord, have the same measure.  

Integer The base root comes from the proto Indo-European root tag, for touch.  The root evolved into a nasalized tang and became the root for many "touch" words such as tangent and contaminate. 

In integer it means untouched and that was the original Latin base meaning.  It also carried peripheral meanings associated with being untouched, such as virtuous, pure, whole or complete.  The last terms brought it into mathematics.

It appears, according to the OED, that the word was first used as a noun in English by Thomas Digges in a 1557 Pantometria, a book of geometry.  

Integral/ Integration

Intermediate Value Theorem – The theorem says that if f(x) is continuous on [a,b] and \(f(a) < L\) and \(f(b) > L \) then there exists a c in [a,b] such that f(c) = L. When L=0, this was proven in 1817 by Bernard Bolzano (Bolzano’s Theorem) and then in 1821 by Cauchy. Both were trying to generalize Lagrange’s analysis of functions. It is interesting to note that the derivative of any function must have the intermediate value property. Very neat results follow from the IVT, such as the Ham Sandwich theorem and the Borsuk-Ulam theorem. The specific results of the Ham Sandwich theorem state that given two regions in the plane, one can cut both regions with a straight line into two regions of equal area (Pancake theorem). In three dimensions, choose any 3 random objects. The Ham Sandwich theorem says that a single plane can cut all three objects into 2 objects of equal areas. This continues for arbitrary dimensions. The Borsuk-Ulam theorem is similar and says that a continuous map from the n-sphere to R^n will always map some pair of antipodal points to the same place. This means that there are two antipodal points on Earth with the same pressure and temperature.
Intermediate is a redundant use of roots, combining the Latin inter, for between, with medius, for "in the middle". Jeff Miller's web site on the earliest use of words in math says the term "Theorem of intermediate value" is found in English in 1902 in a dictionary.  *Derek Orr

Interpolate See Extrapolate/Interpolate

Inverse The idea of an inverse is tied to the identity element in any operation.  The inverse is the element in a group which when operated upon by the binary operation defining the group any non-identity element of the group gives the identity as it's outcome.  The term comes from the Latin inversus, which meant to turn upside down.  The versus base of the word is the same as for a verse of poetry, and meant a line or row. The inverse then, turns back the line.  It first appeared in English in Isaac Barrows translation of the Elements in 1660 in the sense of an inverse ratio, a proportion in which one quantity increases, or decreases, in proportion to the other.  In the current sense of an inverse element in a group, the use seems to have been in 1813 in the Royal Societies Transactions.  

Iodine An ugly black solid, a beautiful flower, a mythical river nymph, and a colorful flame all come together in the story of iodine.  According to th e Greek myths, when Zeus saw Io, the beautiful daughter of the river god Inachus, he fell instantly in love.  He tried to cloak the world in clouds, but Hera, his wife, found out about his liason from a giant named Argus who had a thousand eyes, and was thus a pretty good lookout.  Zeus then tried to disguise her as a white ox, but Hera was not fooled and sent a pestering fly to torture here.  Tortured, Io traveled the globe trying to elude the pestering fly.  The spot where she supposedly crossed from Europe into Asia, near the present day Istanbul, is called the bosporus which literally means "ox crossing" (Oxford England is named for a less mythical spot where cattle could ford (cross) the river.)  Io finally swam to Italy (across the Ionian sea) where she found rest, but no food.  Zeus was moved to pity when he saw her crying, so he turned each tear into a beautiful violet, which the Greeks called Iode. 

Much, much, much later, around the year 1811, French Chemist Bernard Courtois was studying the ashes from burned seaweed (chemists do stuff like that) and discovered a strange blackish gray solid he could not identify.  Some say it was Gay-Lussac, others say it was Humphrey Davey, but in whichever case, someone thought to heat the solid and it gave off a gas with a beautiful violet blue vapor. The Greek name for the flower was chosen, and the substance was called Iodine.  

Iridium/Iris  Hermes(Mercury) was not the only messenger of the mythical gods, in fact they had a special messenger to run messages to the mortals, her name was Iris.  Since she went back and forth to the heavens so often, she would glide up and down on a rainbow, which was also named for her..  The beauty of the rainbow is in its bright colors, and so other bright colorful things were called the same name. The Danish naturalist who studied the  colored portion of the eye also called it the Iris because of its bright color.  In the U S a colorful plant often called a flag plant in many cultures, is called the Iris from the same name.  The plural of iris is irid, so the rainbow like colors on bubbles or shells is called iridescence.  In 1803 when Tennant found an element which formed many different colors when combined with other substances, he called it iridium, the element of rainbows.  

Isogon/Isogonic Lines A curve along which a function of two or more variables has a constant value. There is also a much earlier use as the name for an equiangular polygon, with the gon root related to angle, and knee from the Greek.  The modern use appears on weather maps where lines of equal pressure may appear on your TV weather forecast almost nightly.  Isogonic lines of land altitude are more often called contour lines, and the term is used more than isogon in computer graphing.  Isogon is a little used term for an equiangular polygon.

Isometry is from the Greek roots iso, for same or equal, and metros for measure, and literally means same measure. It seems that Archimedes used the term isometria. Two systems are isometries if they preserve measures.  In geography two points on the Earth are isometric if they are the same distance above sea level. In geometry, and isometry is a transformation in which the image and the preimage are congruent.  A glide translation would be an example students might well know.  The OED cites the earliest print use of the term to Birkhoff and McLane's Survey of Modern Algebra in 1941.  

Isomorphic joins the iso, same,  root to the root morph for body, or shape.  Two systems are isomorphic if the image of A operate B is the same as the image of A operating on the image of B. A simple example would be 2a +2b = 2(a+b)  .  It doesn't matter if you add the numbers and then double, or double the numbers and then add them.  The term seemed to appear in geography before it appears in math.  J J Sylvester used the term as early as 1864 writing in the Philosophical Transactions.

Isosceles is the union of the Greek iso, same or equal, and skelos, legs and refers to the two sides of an object as being the same length, as in isosceles triangle and isosceles trapezoid.  The root iso shows up in many scientific and mathematical words, such as isometry (same measure) and isomorphic (same (body) shape).  Isobar is used in chemistry (two atoms with equal atomic weight) and in meterology (lines connecting points of equal barometric pressure).  

Jacobian - See Matrix

January  The first month of the year was originally a period of festival between the end of one month and the beginning of another in honor of the Roman God Janus.  Janus was the God of beginnings and endings, and is portrayed with two faces, one looking forward and one back.  The months of January and February were added to the Roman calendar around 700 BC

Jerk The origin of the common English word jerk is unknown (at least to me). The mathematical origin seems to have been in 1955 by J. S. Beggs.  The mathematical meaning of jerk is the instantaneous rate of change of the acceleration \$ \frac{da}{dt}\$ or the third derivative of position with respect to time.  Imagine in a car as the driver is accelerating and suddenly slams on the breaks.  

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

Josephus problem – Created by Flavius Josephus back in the 1st century. The story goes that he and 40 soldiers were trapped in a Roman cave and they chose to commit suicide in a peculiar way. The problem goes as follows: Consider n people in a circle. Start with one person and “eliminate” them from the circle. Then skip a certain number of people and eliminate the next person. Then skip the same number of people and eliminate the next person again. Continue this method “indefinitely”. As an example, say there are 20 people in a circle and we start with eliminating person 1. Say we skip 3 people so person 2,3,4 are safe and next we eliminate person 5. Then person 9, person 13, person 17, then person 21…but because there are only 20 in the circle, person 21 would be the person after person 20 which is person 2 (because we already eliminated person 1). This pattern continues. The problem is this: Given the number of people in the circle, the starting point, the order of people and the number skipped, which position will be the last one remaining? Since the problem was created, there have been variations such as different groups of people (historically Turks and Chrisitians on a boat, how should everyone be ordered so only the Turks/Christians get eliminated?). *Derek Orr

Julia Set  When a point on the complex plane is iterated repeatedly on some function, the magnitude (distance from the origin) of the function output will either grow so large as to be unbounded, or it will continue to forever move inside some bounded region of the plane.  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 Julia set lies inside the closed region of the Mandelbrot set, the Julia set will be connected.  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 famous work, Memoire sur Piteration 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 in an artillery blast.  He wrote between the several painful operations necessitated by his wounds.  Fatou dust is named after Pierre Fatou, (1878-1929) . He had entered the same French Prize competition, and published before Julia, who had sealed his work and sent it to the Prize committee.  When he asked the Prize committee to open the sealed envelope he had sent them, and that work was published, Fatou, who had done very similar work, withdrew from the competition. 

People often describe the Mandelbrot set as a catalog of Julia Sets.  The Julia set animation below from Wikipedia shows Julia sets for \( Z^2 +.7855 e^{ai} \} \) with  a ranging from 0 to 2 Pi


Kaprekar Numbers Take a number (I'll use 45 as an example), square it (45^2 = 2025) and add the left half (20) to the right half (25) and if you get the number you started with, the number is a Kaprekar number.  If the squared number has an odd number of digits, keep the middle number with the right hand half of the digits and do the same process.  The first few Kaprekar numbers are 1, 9, 45, 55, 297, 703, 2223, 2728.... It seems that for every Kaprekar number K, of n digits, 10^n - K is also a Kaprekar number; for example the two digit number 45 is a Kaprekar number, and 10^2 - 45 == 55 is as well. I have decided to call these associated Kaprekar numbers "ten-pals" as a play on pen-pals.

The numbers are named for D R Kaprekar who came up with the idea of searching them. .  He is also known for discovering that if you take a four digit number, write the digits in ascending and descending order and subtract, and then repeat the process with this new result, eventually you end up with 6174.  This process is called the Kaprekar process.  It is an interesting exercise for elementary students to try to find the length of the Kaprekar sequence (how many times you have to iterate the process to get 6174) for three and four digit numbers (and they never realize they are practicing subtraction.)

Keith Numbers/ Keith Clusters  Keith numbers were created by American mathematician Mike Keith, while working for Sarnoff in 1987.  A Keith number is a number that will repeat itself under an itertion process that begins with adding the digits of the original number, for example, the smallest base ten Keith Number is 14, so we add the two digits 1+4 = 5.  Repeatedly delete the first value and replace with the most recent sum, in a Fibonacci like way.  4+5 = 9, 5+9 = 14, and we have the original number back.  Non-Keith numbers will eventually jump the target and go on to biger and bigger values.  
Keith numbers are searched for in any base.  The first few Keith numbers in base ten are 14, 19, 28, 47, 61, 75, 197... 

Keith Clusters Some Keith numbers also have multiples that are Keith numbers, such as 14 and 28.  These are called Keith Clusters, There are only three known clusters, another cluster of two at 1104 and 2208, and a single cluster of three 31331, 62662, 93993.  

Kissing Circles Theorem  See The Kiss Precise

Kite The name of a quadrilateral with two pairs of  adjacent congruent sides is probably drawn from the name of the flying toy, which it resembles (although kites of many shapes are found in other cultures).  The toy itself, and perhaps the shape, probably drew its name from the bird commonly called a kite, or kyte.  It is not clear if it is the shape of the bird, its flight behavior, or the long scissored tail that was responsible for its name.  A non-convex kite is often called a dart, which I credit to Roger Penrose who used the name in a proof of his non-periodic tiling of the plane.   
In a discussion group in the late 20th century, John Conway responded to the question , "Is there a name (other than kite) for a quadrilateral that looks like a kite---with no parallel sides, but with two pairs of unequal sides"  (I took this to mean something with the two equal sides of a kite a little unequal, but Mr Conway may not have seen it that way.  His response was to describe a suggestion for a new word, (or an old word new to English) Strombus.  Here is Mr Conway's  own words. 

I was trying to coin an acceptable word for htis for a long time, without success until after being prompted by some considerable discussion on the net about a year ago, I eventually came up with "Strombus", which is derived from the Greek word for a spinning top.  I think it is the best of the terms that were suggested.  It is interesting that the  word rhombus is ultimately derived from the same source, a fact that lends the new term some respectability.

Kurtosis The root of kurtosis dates back to the ancient Greek word kurtos, which meant bent.  The related Greek kirkos for a ring or circle is also the Latin antecedent of our word circus, and circle.  The Greek root also influences such present day words as convex, curly, and crisp.  On a not very pleasant historical note, it seems that one of  the three K's in the name of racist social societies formed after the American Civil War was from the same Greek root for circle.  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.  The word represents a measure of how sharp the peak of a frequency distribution is.  Leptokertic distributions have sharper peaks, and platykurtic are broad and dull pointed.  Leptos is the Greek root for thin or fine,  and platys is the Greek root for broad or wide, leading to a trivia fact, Plato was so called because of his broad shoulders.  


Lame's Theorem is about the number of division needed using the Euclidean algorithm to find the greatest common divisor.  Lame proved that the number of divisions is less than or equal to the five times the number of digits in the smallest number, so if you want to find the greatest common divisor of 873, and 34,506, Lame's theorem shows it will never take more than 5*3= 15, since the smaller number has three digits.
Another mathematical/scientific law called Lame's Theorem describes the tension of a stationary point when three coplaner forces are acting on it.  Incredibly, the theorem is essentially the same as the law of sines for sides and angles of triangles, only here the side lengths are replaced by the magnitude of the forces in the three directions : F(a) /(sin(180-A) =F(b)/(Sin (180-B)=F(c)(Sin (180-C)) . In this application, the angle A is the angle between the directions of forces b and c.  

Lame Curves were first discussed by Gabriel Lame in 1818 and this work is the reason they have his name.  The formula in Cartesian form is a variation of the general formula for the ellipse, given by (x/a)^n +  (b/y)^n = 1 .  The ellipse has n=2.  The exponents Lame considered included all real numbers. The curves where n is even become closer to a rectangle as n increases, and for smaller and smaller fractions of n, the curves begin to approach the perpendicular axes. 

 For odd n, the curves behave like the even powers in the firs and third quadrant, but diverge to infinity in the 2nd and 4th quadrant, for example, when n=3 the curve is the Witch of Agnesi.  

When the exponent is a fraction, the branches in each quadrant turn inward, so that for n= 2/3, you get an asteroid.  

Soma Cube *Quadrant Dan Wordpress
The special case when n= 2.5 picked up a special name, Super Ellipse, in 1959 when Danish poet and architect Piet Hein designed a  roundabout with a and b in a 6/5 ratio.  He continued to use the shape in furniture and many other things, and a super-egg based on the three dimensional rotation of the Super Ellipse which would stand erect on its ends.  

Many children remember Piet Hein for a three dimensional cube assembly problem he marketed as Soma Cube. It is said that he thought up the idea in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Its name is alleged to be derived from the fictitious drug soma consumed as a pastime by the establishment in Aldous Huxley's dystopic novel Brave New World.

Latus Rectum The latus rectum is a line segment in conic sections parallel to the axis and passing through the vertex from one side of the curve to the other.  The term comes direct from the Latin words for side, latus, and straight, rectus.  Students may know latus from the large muscle group in the side of the back often called "lats", the Latisimus Dorsi.  Rectus is the root that gives us words like rectangle, erect, and through the German, right. The first use in English seems to have been in 1702 in J. Raphson's Math Dictionary, "In a rectangle the rectangle of the diameter, and latus rectum.  

Lb, Pound The roman name for their standardized weight was the Libra pondo. Pound came from the second part of the word, but the symbol came from the abbreviation of the first, Lb.  Libra is representing the balance scales which were used to determine weight.  

Law of Cosines  See Cosine

Law of Large Numbers (Probability theory)  The law of large numbers in its simplest form says that with lots of experimental trials, the results will approach the statistical expected value.  The first statement of the idea of the rule was around 1502 when Giralmo Cardano said that empirical statistics tend to improve with more trials.  This is the idea that with five consecutive flips of a fair coin, you may get all heads or something, but if you keep on going, the number will get closer and closer to a ratio of 1/2.  By 1713 Jacob Bernoulli gave a proof for a specific example, the Binomial Distribution, which he published in his classic Ars Conjectandi.  He called it his "golden theorem."  It may have been French mathematician Simeon Poisson who fist used the name Law of Large Numbers.  Actually he used "grand" numbers, but Large was the term that stuck. The earliest citation in the OED for "law of large numbers" is 1837 in the London Medical Gazette, "The universal Law of Large Numnbers is...."

Law of Sines, law of sines gives a relationship between the sine of an inscribed angle and the chord subtended by that angle in a circle. Since a triangle can always be circumscribed by a circle, the rule states that the ratio is constant for any angle of the triangle.  Unfortunately (in my humble opinion) this is often taught with no reference to the circular relationship.  Most students learn in geometry that angles that subtend the same arc, and chord, are equal in measure.  If an angle C inscribed in a circle intersects chord AB, then we can draw a diameter from A to another point D, and then connect D to B and we have another inscribed angle that cuts the same chord.  But ADB is a right triangle because it has a diameter for it's longest side, so Sin (D) = AB/{diameter}.   Since the angle at D is the same as the angle at C, their sines must be equal. This means that Sin(C) =AB/diameter  as well.  Manipulating this equation  gives us diameter = 2r=AB/Sin C . This works for any inscribed angle anywhere around the circle, so it applies to all three vertices of a circumscribed triangle, and remember that every triangle is circumscribable.  So we can  write the complete relationship,a/Sin (A) = b/Sin (B) = c/Sin (C) = 2r .

The law of sines is credited to 15th century German mathematician Regiomontanus, but the name didn't appear until around 1840.  A letter from English polymath William Whewell includes the term in relation to a writing of Fermat, and Snell's Law.  

Lemma Mathematicians use the word lemma to describe a proof which is a preliminary step to a more important or more complete result.  The word is directly from the Greek and descended from lambanein which means to receive, take, grasp, or seize.  The roots is the source of other words like syllable, to take together, and epilepsy, to seize upon.  

Lemniscate The word lemniscate comes from the Greek word lemniskus for ribbon.  The mathematical curve, a sort of figure eitht, does look somewhat like a bow for a package from a twisted ribbon. The word seems to be disappearing from textbooks, and is completely missing in my high schools edition of the American Heritage Dictionary (which dismisses the heritage of many mathematical Americans.).  The only related term I could find was lemniscus, a nerve bundle in the brain (also known as Reil's Ribbon).  

The curve  is related to the rectangular hyperbola through the following relationship.  If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn through the origin, the point where the perpendicular meets the tangent is on the lemniscate.  

Jacob Bernoulli used lemniscus in 1694.  OED's earliest citation is 1n 1781 in Chamber's Cylopedia.  The name for an algebraic function describing a lemniscate was first applied by Cayley and appears in his Collected Papers in 1879.  

Lemoine Point in Triangle   See  Symmedian Point

Leyland Numbers are a number theory idea created by Paul Leyland, a British professor of mathematics at Oxford University.  The Leyland numbers are numbers of the form x^y + y^ x where x is greater than or equal to y and both are greater than 1.  The sequence of Leyland numbers is  8173254571001451773203685125939451124 (sequence A076980 in the OEIS).

L'Hopital's Rule  (pronounced like lopital) is a calculus tool to evaluate limits of indeterminate forms, It is name for French mathematician Guillaume L'Hopital (sometimes spelled L'Hospital) .  It is used to find the limit of the ratio of two functions at some point c, which might be a discontinuity, when the limit of each function goes to infinity or both go to zero. 

The rule in simple language says such cases may be resolved if we simply apply the ratio of the limit of their slopes (derivative) as they approach the point. The general form is \$ \lim_{x \to c} \frac {f(x)}{g(x)} = \lim_{x \to c} \frac {f'(x)}{g'(x)} \$

L'Hopital published the rule in his 1696 text, Analysis of the Infinitely Small for the Understanding of Curved Lines,  which was the first Calculus textbook. 

Limacon (of Pascal) is a graph or function of the curve (roulette)

of a point fixed to a rolling circle when it rolls around another circle of equal radius. The point may be fixed at any length along the radius of the rolling circle.  

The shape can also be defined by a fixed point P, and a given circle C, and the limacon will be the shape formed by all the circles that can be drawn with a center on the given circle, and passing through P.  

The polar form of the curve is given by r = b + a cos(t).  

The shape seems first to have been studied by the Renaissance artist Albrecht Durer in his Instruction in Measurement written in 1525. His work included specific instructions for how to produce the curves.   Later the father of Blaise Pascal, Etiennine, would investigate the curve also. His work was known to Giles de Roberval, who also studied the curves, and gave them that name.  He credited Pacal in the new name in honor of his work on them.   

The word came into English and is first cited by the OED in George Salmon's 1852 Treatise on Higher Planes.  

Limit The Latin word for the fence or path between two pieces of land was limes.  The idea of a point or boundary beyond which something does not extend is the basis for both the common English meaning of limit, and the mathematical meaning.  Isaac Newton first used limit in a mathematical sense around 1725, although Gregory St. Vincent had earlier used "terminus" for a similar meaning in relation to progressions.  The idea extends back at least to Archimedes efforts to find the Area of a circle, or a section of a hyperbola.   

Line The roots of line are hidden in the genus of the flax plant, linum and the fibers of the plant which were spun together to create a thread called linen.  The idea of stretching the thread between two points to mark a straight line leads easily to the name of the imaginary one dimensional object passing through the two points.  

Linear Algebra A simple definition was given by Benjamin Pierce in The American Journal of Math in 1870, "An algebra in which every expression is reducible to the form of an algebraic sum of terms, each of which consists of a single letter  with a quantitative coefficient, is called a linear algebra."  That is also the earliest mention of the term in English in the OED.  I have an old note that Bombelli used the term "algebra linera" in is Algebra discussing relations between Algebra and Geometry. 

The first known use of solving systems of linear equations occurred over the period from the 10th to the 2nd century BC in China in The Nine Chapters on the mathematical art.  The method is preserved in one of the oldest surviving texts of Chinese mathematics. The method shown is essentially the same as what
we now call Gaussian elimination, and teach in advanced high school classes.  In a modest way, the earliest solution to the intersection of two lines in Cartesian equations is an example. The introduction of vectors into mathematics (they had appeared earlier as a term in astronomy) around 1850 led to a much greater variety of working with linear equations.  In 1844 Grassman's Theory of Extension introduced new topics into what would come to be called linear algebra, and in 1846 William R. Hamilton introduced the term vector in the Philosophical Magazine of the Royal Irish Academy on a paper on Quaternions., Two years later J J Sylvester introduced the term matrix, another way of dealing with systems of equations, and working with transforming one system to another. In 1856 Cayley introduced matrix multiplication which allowed these transformations to be done more efficiently.  By 1876 James C. Maxwell was using the terms in his work on Electricity and Magnetism.   By the first decade of the 20th century, linear algebra materialized into a rigorously defined system and the transformation of finite dimensional spaces had arrived.  By the middle of that century it had evolved into an important element in modeling and simulations, and by the close of the century, it was deeply involved in film animation. 

Two of my posts from 2009 that may be of interest to high school students and their teachers are "Why Bother with Vectors,"  and "More on Vectors in the High School Curriculum".  

Linear programming (LP) – Finding the optimal value of a linear equation given linear constraints. The history dates back to Fourier in 1827 (see Fourier-Motzkin elimination). However, real movement began in the 1900s. In 1939, Soviet economist Leonid Kantorovich developed a formulation for solving linear programming problems, hoping it would help with the war at the time. At the same time, T.C. Koopman developed some economic problems involving linear programming. These two later shared the 1975 Nobel Prize in Economics. Real change came when George Dantzig independently discovered linear programs and used these problems for the Air Force in 1946 and 1947. He discussed this with John von Neumann and Neumann realized his work was very similar to Dantzig’s, this is where the theory of duality arose. After the war, industries used a lot of their work in their daily planning. Later on in 1979, it was proven that linear programming problems are solvable in polynomial time (Leonid Khachiyan). *Derek Orr

Lissajous Figures/Bowditch Curves A figure or a 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 lenmiscates and other complex figures.   They can be created mechanically with pendulums or harmonographs, and electronically by tuned oscillators. 

They are named for Jules Anton Lissajous, who wrote about them in 1887.  They had been written about earlier by American Nathanial Bowditch, and some people suggest they more correctly should be named for Bowditch.  

I knew them only as Lissajous figures until in the first decade of the 21st Century 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 1852, so I was surprised to read 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 of Bowditch, his book, The New American Practical Navigator, that Bowditch, and the Marine society had published in that year.  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. 

Students may make the curves on graphing calculators or software with parametric graphing capability. 

Locus The set of all points are "places" that meet a set of geometric conditions is called the Locus of htese conditions.  For example, a geometer might say, the locus of all points equidistant from another point is a circle."  The ancient Greek geometers would have used the word topos, the Greek word for place that shows up in words like topology and topological.  When these Greek works were translated into Latin, they were replaced with the Latin word. The word has persisted in mathematics into the present.  The OED cites the word appearing in a non-mathematical sense in 1648, as we might say location (from the same root).  The mathematical use in English seems to first appear in print in the first half of the 18th century. 

Logarithm is the combinations of two Greek roots, logos, reason or ratio, and arithmus, number. The ration refers to the original method of constructing logarithms by geometric sequences.  The name, and the original method were created by John Napier, although Joost Burgi had discovered logarithms at about the same time.  Napier first used the term in the Latin form, then subsequently into English in Correspondence with Henry Briggs, who would add a table of logarithms in base ten, with log(1)=0 and log(10) = 1.

Logic – via Old French logique and late Latin logica from Greek logikē (tekhnē) ‘(art) of reason’, from logos ‘word, reason’. Logic really started a long time ago in India, China, and Greece. In India (BCE), Medhatithi Gautama founded anviksiki, the school of logic. Panini also developed Sanskrit grammar which contained logic similar to Boolean logic. One of the Hindu schools of logic and philosophy was Vaisheshika. This school accepted two sources of knowledge: perception and inference. The school also taught a form of “atomism”, that everything is created from five substances and each of these substances are one of two types: indivisible (paramanu) or composite. Anything perceptible by a human was composite and could be split. The other school of logic, Nyaya, accepted four sources of knowledge: perception, inference, comparison/analogy, and word/testimonies. The school also developed five-step guide to inference: initial premise, reason, example, application and conclusion. They were a school of realism as well, saying anything that really exists is humanly knowable. The most important contribution to Hindu thought has been the Nyaya works on epistemology and the system of logic. Further, the Nyaya school heavily influenced Buddhism and Buddhist philosophies. In China, Mozi “Master Mo” founded the Mohist school which was primarily focused on valid inferencing and how to arrive at correct conclusions. The school also favored rhetorical analogies over mathematical reasoning, however during the Qin Dynasty, much of the Mohist school was forced to stop studying logic in general. Logic wouldn’t return again until Buddhism logic, which was so highly influenced by Indian logic, Chinese Buddhists misunderstood a lot of the culture because they lacked the background from Indian logic. In the West, it is believed that the discovery of geometry from ancient Egyptians was the catalyst that sparked the discussion of philosophy and logic. People in ancient Babylon also had a sense of logic as they predicted planetary systems. Looking towards ancient Greece, their biggest contribution was to use the empirical proofs discovered by Egyptians and replace them with demonstrative proof. They had three ground rules: there must be propositions that are accepted as true, without needing proof (axioms), any proposition that is not accepted true must be demonstrated true using these axioms, and the demonstration must be formal, meaning it can be shown independent of the particular subject matter in question. Thales is generally accepted as the first philosopher in the Greek community and he was the first to use deductive reasoning in geometry. The study of proof first began in the school of Pythagoreas, whose students were the first to study “form” rather than just “matter”. Later, Heraclitus was the first to write “Logos” when describing philosophy. He stood by the belief that everything is changing and the only thing that unifies everything is “Logos”. Parmenides however, claims at “all is one and nothing changes”. He says that something that exists cannot possibly not exist. Saying the phrase “X is not” would be false or meaningless. His interpretation of “logos” was it represented the Truth. Plato looked into more philosophical logic and had 3 questions: What is it that can properly called true or false? What is the nature of the connection between the assumptions of a valid argument and its conclusion? What is the nature of definition? Later, Aristotle had a huge influence on logic by being the first logician to write variables and symbolically show a logical argument. His works, called the Organon, gave great study guides to his reasoning and provided overwhelming importance in the history of logic. The Stoics (still ~3 BCE) gave three main contributions to the theory of logic: Modality, conditional statements, and their take on “meaning and truth”. *Derek Orr

Loxodrome  The shortest path between two points on the surface of the Earth, or any true sphere, is along a great circle arc, but this path is often not possible for ships, at least in early navigation.  One reason is that a great circle arc takes constant changes of the compass heading. Because it is not much longer  in the middle latitudes, ships often sail a path of constant compass heading called a "loxodrome"(and sometimes a rhumb line) .  The word comes from a Greek root loxos, for slanted, and drome, which means path or course.   Until the middle of the 18th century, finding ships longitude at sea was nearly impossible.  This forced seamen to navigate along a steady compass heading and use their estimated speed to "dead reckon" their position.  A straight line drawn on a Mercator projection map is a loxodrome. 

The word is also used for a logarithmic spiral because it always cuts a line through the origin with the same angle.  A true complete loxodrome spiral on a sphere will endlessly circle the poles without reaching them.   Earliest citation in OED is for J. Morse's New System of Math.  

Lozenge is used mathematically to describe a rhombus with a "diamond" shape, but what exactly defines that varies with the user.  The word in French is used as the name of the shape we call a rhombus, but in English speaking country's, it is generally applied to a rhombus oriented with the opposite vertices in horizontal and vertical fashion.  Many will only use the term if there are two acute, and two obtuse angles, and some require that the acute angles are 45 degrees.  The shape is common in antique art, heraldry, and tiling.  The word may have originated from the Latin lapis, for stone, and seemed to work its way through the Portugase, Catalonian, and into France.  The earliest use in English was by Robert Recorde in his Pathway to Knowledge in 1551, "The thyrd kind is called lasenges or diamondes whose sides bee all equal, but it hath neuer a square corner.  

Lucas' Sequence The Lucas sequence is similar to the Fibonacci sequence, but much newer. The Lucas sequence begins 1, 3, 4, 7 11... and follows the same rule as the Fibonacci numbers after the first two numbers.  The limit of the the ratio of consecutive terms in the Lucas sequence, like the Fibonacci sequence, approaches the Golden Mean, There is also a equaton that ties the Lucas numbers to the Fibonacci numbers, L(n) = F(n-1) + F(n+1).  Thus the fifth lucas number, 11, is the sum of the fourth and sixth Fibonacci numbers, 3 + 8.  

The sequence is named for Edouard Lucas, a French mathematician of the last half of the 19th Century.  He used his sequence, and the Fibonacci sequence for testing if large numbers of the form 2^p - 1, were prime.  

Lucas is also remembered for his unusual death, caused when a waiter dropped a plate.  The shattered plate sent a piece into Lucas' neck.  Lucas died several days later from a deadly inflammation of the skin and subcutaneous tissue caused by streptococcus.  The disease, officially listed as erysipelas (from the Greek for "red skin" was more commonly called St. Anthony's Fire.   

Lucky Numbers were introduced to the public in 1956 by Gardner, Lazurus, Metropolis and Ulam. They suggested naming the sieve that defines it as a Josephus Flavius sieve, because it resembled the counting out sieve in the Josephus problem from the 1st century. The sieve begins by counting out every second number and eliminating them (thus eliminating all the evens). Then counting again from the start, eliminate every nth number where n is the next number in the list after the first survivor. It should proceed something like this:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21...
1, 3, x, 7, 9, x, 13, 15, x, 19, 21, x
1, 3, 7, 9, 13, 15 x 21
Lucky searching to you all. Seems like a really good computer project for young programming students.

Ludolph's Number Ludolph von Ceulen devoted much of his life to finding ever more accurate approximations to pi.  During his lifeime he extended the known accuracy for 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.  

Lune The word lune is from Luna, the Roman Goddess of the Moon.  The more ancient Indo-European root is leuk, which refers to light.  Lunatic is from the same root, perhaps becuase the ancients (and some moderns) believed it was the effects of the moon that accounted for the crazy behavior.

Lune is used in mathematics to describe two different figures (see below).  The most common is the area on a sphere between two semicircles with common endpoints at polar opposites of the sphere.  Imagine two lines of longitude running from the North pole to the South pole.  The (usually smaller) surface of the sphere between the two longitudes is called a lune.  There is also a two dimensional shape called a lune, the area between two circles, that plays an important part in the mathematics of the ancient Greeks.  This lune was studied in depth by Greek mathematician Hippocrates of Chios (around 400 BC) in an attempt to find a square with the same area as a circle using the classic tools of a compass and straightedge.  This problem of "squaring the circle" was one of hte three great unsolved problems of Greek mathematics.  Much later it was proved using algebra tools, that the task was impossible, but the Greeks discovered lots of important mathematical ideas in their search to prove the impossible.  



"m" for Slope   see Why m for slope

Magic Square A square grid, or array, of numbers, usually required to be all different numbers, in which the sum of each row, column, or diagonal has the same sum.  This sum is called the sum of the magic square, or sometimes, the magic constant.  The order of a magic square describes the number of rows or columns in the square.  

The order 3 magic square was known to the Chinese before 200 BC. The Lo Shu Square ( literally: Luo (River) Book/Scroll) is the unique normal magic square of order three. Except for rotations or reflections it is the only order three magic square that can be formed with the digits 1-9. 

Chinese legends concerning the pre-historic Emperor Yu tell of the Lo Shu: In ancient China there was a huge deluge: the people offered sacrifices to the god of one of the flooding rivers, the Luo river, to try to calm his anger. A magical turtle emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell: circular dots giving unary (base 1) representations of the integers one through nine are arranged in a three-by-three grid.

The odd and even numbers alternate in the periphery of the Lo Shu pattern; the 4 even numbers are at the four corners, and the 5 odd numbers (outnumbering the even numbers by one) form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (the number of days in each of the 24 cycles of the Chinese solar year. 

 Beyond the basics of the magic square, O'Shea points out several other interesting relations. First, the sum of the  squares of the numbers in the top and bottom row are equal. 42 + 92 + 22 = 82 + 12 + 62 = 101. You can do the same thing with the two outside columns, 42 + 32 + 82 = 22 + 72 + 62 = 89. Notice that both sums are prime numbers.

The Lo Shu square is also a guide to creating magic squares of any odd order, put the one at the bottom center, draw a diagonal down to the right (or left) going off the page, but as if the page was instead a torus (doughnut) and the next square is the top one row to the right (or left).  Continuing down on the diagonal until you go off the page again on the right side and rejoin the motion on the left side of the row one below.   All the above, and more about the Lo Shu diagram is on my post on "The 3x3 Magic Square, More Magical than you thought."

  One famous magic square was in the work of Albrecht Durer in his famous 1514 Melencholia painting.  It is an order four magic square with a magic constant of 34.  

Another is on the Passion Facade of Gaudi's famous Barcelona Cathedral,  the Sagrada Famalia.  It differs with numbers repeated to give a constant of 33, supposedly the age at which Christ died.   

My post on a way to raise matrices of magic squares to produce more magic squares is here.

And here is a post on a unique way of filling in odd order magic squares.   An on my daily On This Day in Math blog you can find the first prime 3x3 magic square on the 111th Day, which is usually April 21, but as I'm writing this in a leap year, it will appear on April 20. 

Magnitude The Roman Goddess Maia (She who is great) from whom we get the name of the month of May, is also the source of the word magnitude, and many other words that related to greatness of size. The greatest value in a set, the maximum is drawn from the same root.  The suffix is from the Latin tudo, and was indicative of a condition or state, and appears in words like altitude, amplitude, and fortitude.    The main root actually runs back to origins in Sanskrit and has relation to words like omega, maharajah, and Almagast, which was the name of Ptolemy's great work about 150 AD which chronicled the knowledge of the time about astronomy and geography.  Current English words related to the same root includes magnificent, magistrate, mayor, matador, and master.  

Mandelbrot Set is the set of all points c=m+ni in the complex plane the remain bounded when iterated through the function F(z) =
z^2 + c.  Each result becomes the new z, and the c in each iteration stays the same.  If that iteration settles down to a single point, or eventually begins to orbits on each of some finite number of points, then the point c is part of the Mandelbrot set.  The points which diverge to infinity are the  black(generally, but white is also common) space you see in the images, frequently the colors change to indicate how quickly the points diverge.  It can be categorized as a catalog of Julia sets, since the points in the Mandelbrot set are just the points that are a closed Julia Set. It is said that Adrian Douady was the first to name them after Benoit B. Mandelbrot, perhaps because the border of the Mandelbrot set is a fractal, a word created by Mandelbrot. The actual definition of the object, and the first drawings of it, were actually by Robert Brooks and Peter Metelski in 1978. And the origin of Complex Dynamics, as the field is often called, date back to the first quarter of the 20th century and the work of Gaston Julia, and Pierre Fatou (see Fatou Dust)  

The French mathematician, Douady, has a fractal object named for himself,  Douady's Rabbit, a class of Julia sets that cluster around a three-bulb on the Mandelbrot set, that somewhat resembles the ears of a rabbit. The numbers on the bulbs describe the length of the orbit cycle for the points in that area.  

Mantissa/Charastic  This is probably one of those math words that calculators will render obsolete.  Every logarithm has a characteristic (think orders of magnitude) which is the whole number part of the logarithm. In base ten, all numbers between 10 and 100 have the same characteristic, 1. Those from 1 to 10 have a characteristic of 0.  The fractional part is called the Mantissa.  The mantissa of log(2.3), log(23) and log (230000) are all the same, thus tables of base ten logarithms needed only to have the value of numbers from 1 though 10 since the logs of 10 through 100 would all be the same except for the whole number characteristic. It was assumed a capable mathematician at this level would be able to figure out the power of the base to determine the characteristic.  

The word seems to have come to us from the ancient Etruscans through the Latin where it meant some small addition or something of minor value.  I remember reading somewhere that the Etruscan meaning was related to "a little tail" but can not find the old notes to confirm the source or accuracy.  Over time it has come to mean an appendix.  John Wallis used the word "appendage" in his English works, but then in the Latin translation applied the word "mantissa".  As with many other words, it was not accepted widely until it was used by Euler. At one time Gauss suggested using the word for the decimal part of any mixed number.  

Markov Chain is named for Russian mathematician Aundrey Markov(there are so many variations of spelling in both the first and last name, I will not try to give them)  and is a probabilistic method of determining states over time, where the probability in a state is constant, and does not depend on the path that happened in the past. Markov published his first paper on the method in 1906 work on extending the law of large numbers.  

They are often calculated by matrices raised to a power based on the transition graph of the set of states and their probability. A transition graph is a graph that shows each state with the probability of transition to another state.
An example of a use of Markov chains to predict the probability of a modification of the  birthday problem illustrates a transition graph and examples of how the power of a matrix can produce the probability of being in any state after n trials.  

Mathematics The origin of the word mathematics is from the Greek word manthanein, to learn.  The meaning is preserved today in words like polymath, for a person of great learning.  The word seems more seldom today. The root of Education, I like to point out, is from the root educare, to draw out.  So the teacher works to draw out knowledge, and the mathematician (or student) is challenged to acquire it.  It's a team thing.  

Matrix/Matirces The earliest use of the word matrix in English had nothing to do with math, and in fact the first use about the term in math came from this same word in Latin, matric, for  "a supporting or enclosing structure" or the womb of a mammal.  The first mathematically related use was from J J Sylvester in 1850 in which he uses the term as "...a matrix out of which we may form various systems of determinants..."

A matrix in mathematics is now a rectangular array of numbers, or expressions, or symbols arranged in rows and columns.    The dimension is written as r x c (read "a by c")  where the r and c represent the number of rows and columns. Elements of the same order are called conformal, and they can be added or subtracted element by element in corresponding positions.  For oblong matrices to be multiplied, the number of columns of the multiplicand must equal the number of rows of the multiplier, so a matrix with order a x 4 can be multiplied by a matrix with order 4 x b to produce a product with order a x b.  

For square matrices of order n x n,    they may be added, subtracted or multiplied, and for every non-singular square matrix [A], there is an Identity matrix, usually labeled [I],  such that for any matrix [A], [A] [I] = [I] [A] = [A], and an inverse so that \$ [A]^{-1} [A] = [A] [A]^{-1} = [I] \$  In general the product of two matrices is not commutative, so the order of multiplication may produce different results.  They do, however obey the associative and distributive laws.  

The earliest matrices, in much the same form, existed in early Chinese writing as far back as at least the 1st century. The method was called fang cheng which best translates into English as rectangular arrays.  The system is a way of solving n equations in n unkowns, very similar to the method Gauss created in the early 19th century.
For high school students the most common application of matrices is to solve systems of equations, both by elimination and Cramer's Rule for determinants, and for transformation of objects in the plane.  They may also be introduced to some uses in modeling, or in Markov chains for certain probability problems.  

Jacobian The Jacobian Matrix,  named after Carl Gustav Jacob Jacobi who created them in 1841, and its determinant are both referred to as "the Jacobian" in many instances.  When a Matrix  with  vector valued functions for each row, the Jacobian of the Matrix is the set of all first order partial derivatives of  the vectors.  If the number of partial derivatives in each row is equal to the number of functions, that is, the Jacobian matrix is a square matrix, then the determinant of the matrix is a well defined funciton.

Trace of a Matrix the trace, often abbreviated tr or (tr) is the sum of the elements in the main diagonal of a square matrix, the diagonal starting at the top left and proceeding down to the bottom right entry.  The trace is important because it is the sum of the complex eigenvectors of the matrix, and does not vary with a change of base.  The trace is related to the derivative of a determinant, using "Jacobi's Formula", the same Jacobi who created the Jacobian above.  


Mean/  Median  If you sometimes get mean and median confused, it is no wonder.  Both words come from the same origin, and originally meant about the same thing.  Both words, as well at the more common word middle, come from the Indo-European root medhyo for pertaining to the middle.  When most students think of the mean they think of it as the same as the average, but this is only one of the many means in mathematics, called the arithmetic mean.  Here are the definitions of three of the most common:

Arithmetic Mean--- The sum of the values divided by the number of values, also called the arithmetic average.  The average of 3, 8 and 10 is (3 + 8 + 10) /3 = 7.  If you subtract each number from 7, the total comes out to zero.   

Geometric Mean ---- The nth root of the Product of the numbers where n is the number of values multiplied.  The Geometric mean of 3, 8, and 10 is (3*8*10)1/3 = 2(30)1/3 ; which is approximately 6.214.  The Geometric mean gives the length of an edge of an regular n dimensional hypercube, which is the general term for extensions of the idea of a square beyond the square and cube. A box with sides 3, 8, and 10, has the same volume as a cube with sides equal to their geometric mean.  

Harmonic Mean--- The reciprocal of the arithmetic mean of the reciprocals of the values.  The harmonic mean of 3, 8, and 10 is  found by taking adding 1/3 + 1/8 + 1/10 , then dividing by three to get the mean of the reciprocals of the values, \$ \frac{67}{360} \$  .  The reciprocal of that value, |$ \frac{360}{67}\$ is the Harmonic mean of the three numbers. See here more on the history of the Harmonic mean, and some applications. 

The Means and extremes ratio, which today many people call the Golden Ratio, or the Golden Mean, was known well be fore the 5th century BC when Hippasus discovered that it was not a rational number. The first known written definition occurs in Euclid's Elements, "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser." This was not just a mathematical idea but represented a model of a way of life, a balance between excess and deficit. For a long time Martin Ohm was created for first using the term (goldener schnitt) in 1835. Recently (2019) two earlier uses have been discovered and reported. One in 1789 by Johann Samuel Traugott Gehler, and another in 1717 reported by Holger Becker. The full descriptions were published in Historia Mathematica, but I have not had access to the journals yet to see where they were found, or by whom. It still seems that the first English usage was by James Sully in 1875.

For much more on the "golden mean", and other metallic means See "All That Glitters is not Golden".

The Median is generally the center term when a group of numbers are ordered by size.  If there are an even number of values, then the arithmetic mean of the middle two is the median.  It is said that the ancient Greeks studied ten different means.  Other words drawn from the same root includes medium, mitten, meridian, and Mediterranean  (the middle of the world, at least to the people who lived around it.)  

Mean Value Theorem This theorem is an incredibly important calculus theorem that, when explained to students can seem almost obvious.  It says that for smooth unbroken curves, If you draw a secant line joining two points on the curve (call them a and b) then in between there exists some point c, at which the tangent of the curve has the same slope as the secant.  It is actually a generalization of Rolle's theorem which says if there are two points on such a curve where the values of A and B are the same, then the slope at some point C, must be zero.  The earliest idea of the rule, or at least a part of it, seems to date back to Indian mathematician Parameshvara before 1460.  This rule and Rolle's Thm did not use calculus.  The origin of the present theorem is credited to Augustine Cauchy in 1823. 

Measure  Measurement, exact or approximate, stretches back to man's earliest mathematical efforts, and the Indo-European root of measure, me, shows up today in words that, at first, often seem unrelated to measuring.  The Greeks extended the word to metron, from which we get meter and metric.  The Latin variants of this took the forms of med or men, and gave us words like moon, month, menopause, and semester (semi-meter).  The old German variation had a 'L" ending and gives us meal, which originally meant an appointed time.  The time meaning of meal still shows up in the word piecemeal, literally, a piece at a time.  

Medial Triangle This is the triangle connecting the three points where the three medians of a triangle intersect the opposite sides, thus, the three midpoints, and I admit to often using "midpoint triangle" as often as this term.  This means that each side of the triangle is parallel to a side of the original triangle, and 1/2 its length, making the area of the medial triangle 1/4 the area of the original.  The Orthocenter (intersection of the altitudes) of the medial triangle is the Circumcenter of the original triangle. 

Median See Mean

Medians of a triangle/tetrahedron A line that joins a vertex of
 the triangle to the midpoint of the opposite side, it is an area bisector of the triangle. The three medians of a triangle intersect in a single point, the centroid or center of gravity of the triangle.  Theoretically if you made a triangle out of uniform material, and place that point on a pin tip, it would balance.  The intersection point of the three medians divides each median into a 2/1 ratio, with the longer portion toward the vertex.  

A similar relationship can be constructed in a tetrahedron.  If you connect a segment from the vertex of the tetrahedron to the centroid of the opposite triangle, then these four segments also intersect in a point, but with the relationship divided into a 3/1 ratio.  The point represents the center of mass of the tetrahedron.

MEDIAN (of a triangle) is found in 1876 in Lessons in elementary mechanics. Introductory to the study of physical science by Sir Philip Magnus, with emendations and introduction by Prof. DeVolson Wood: "In the same way it may be shown that the centre of gravity of the triangle is in the median CE (fig. 109). Hence the centre of gravity of the triangle is at G, where the two medians intersect" [University of Michigan Digital Library]. *

Menelaus' Theorem   is a rule about triangles in a plane that are
cut by a transverse line cutting all three sides (some or all extended to intersect out side the triangle).   The rule states that \$ \frac{AF}{FB} x \frac{BD}{DC} x \frac{CE}{EA} = -1 \$ Each ratio is treated as positive if the cutting point is between the vertices, and negative if it is not.  The theorem is an iff since the converse is also true, any three points whose distances meet the product relation are colinear. 

The theorem is named for Menelaus of Alexandra who wrote about it around 100 AD.  He is known to be one of the earliest to recognize the great circle arcs on a sphere were an analog of straight lines on a plane. Ptolomy used the theorem above in such a manner on several proofs about spherical geometry in his famous almagest (the greatest composition). It is important to note that Menelaus' Theorem was used by Persian and Arabic mathematicians of the tenth century to extract the Law of Sines.

Meter The Greek word for measure was metron and the meaning still persist in 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 lenght a meter, and set its length to be one ten-millionth of the distance from teh North Pole to the equator (which strangly, they didn't know very accurately at all at that time).  

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 Mathematical Units:

Because the Earth is difficult to measure (not ot mention the many irregularities in its shape!) this is not a practical definition.  For a long time, the meter was the lenght 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 principal, at least) be reproduced to any desired level of accuracy in a well-equipped laboratory.  For the meter, the 17th General Confederation 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 299,742, 458 meters per second.  The speed of light 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 equall to approximately 1.0936133 yards, or 3.280840 feet. 

Mile The name for the distance we now call a mile comes from the Latin phrase mille passes which means one thousand paces.  Since the paces(one step with each foot) at which the Roman army marched were supposed to be two steps of 2.5 feet long, one thousand paces is very close to the statute mile, 5,280 feet.  A nautical mile was developed to be a distance equal to one minute of arc (1/60 of a degree) distance along a great circle.  The length of a nuatical mile is usually given as 6076 feet.  The OED gives mile first appearing in English in 1225 as milen.

Million The root of million is the same Latin mille for a thousand, that gives us mile and millennium.  The ion suffix implies large or great, and so a million is just a "large thousand".  The ancient Greek and Roman life had little demands for numbers of this size, and the largest named number of the age was the myriad (see), which stood for ten thousand.  Although the word million seems to have come into use in the 13th Century, and into England 70 to 100 years later.  Most writers would use the phrase "a thousand thousands" to avoid confusion.  Although it did not gain common use in mahtematics until the 1700s, it was in print well before that time.  The word was sometimes used to describe an exact amount, and sometimes as we might say uncountable.  The word appears in Shakespeare in Hamlet, act 2 Scene 2, "for the play, I remember, pleased not the million."   

Minor  The word minor, for smaller or lesser, comes to us from the Greek root meton for small. This is the same root which gives us minus, minimum, and minister (a servant, or one of low standing, a magistar, think magistrate, was a master or person of high standing) 

The word was introduced into mathematics by J J Sylvester in 1850 to describe a smaller matrix made by eliminating one row and one column from a larger matrix.    

Here is an excellent explanation of the process from  Wikipedia:

And if you use Wikipedia alot, like I do, send them a donation once in a while. We would miss them if they were gone.

Minuend In a subtraction problem, like 12 - 5, the number which is to be diminished, or subtracted from (12 in this example) is called the minuend. The Latin root, minuendus, is similar to the roots of minute and minus, and means the same as the English adaptation, that which is to be reduced.  The earliest citation in the OED is in William Jones Synopsis Palmariorum Mathesos, "The greatest of the given numbers is called the minuend. "

Minus See Minute and Plus

Minute When early sailors from the Eastern Mediterranean chose to cut an arc into parts, they chose fractions in the sexagesimal (base 60) system that was common throughout much of the Mediterranean at the time.  Later when writers described these small parts of an arc, they used the Latin phrase, pars minuta, Latin for small parts.  Out unit of time for 1/60 of an hour adopted and contracted this phrase into minute.  The conjugate word with the same smelling, by pronouced mi nyoot', continues to refer to something very small.  

The word minus for subtraction is drawn from the same root and refers to making something smaller.  The verbal use of the words plus and minus date back to the Romans when the terms were used the words just as we now use the words more and less.  Other related words are minor (the smaller of two), minced (cut into small pieces), miniature (on a small scale), and menu (in a small list).  

Miquel's Point & Theorem The theorem is about a relationship on planer triangles.  It is named for August Miquel, a little known high school teacher in France. His theorem was published in Liouville's Journal in 1838.  The rule is sometimes called the Pivot theorem.  

Miquel observed that if three points are selected,  one one each side of the triangle, and then three circles are created that each pass through a vertex and the points on the sides meeting at that vertex, all three circles will share a common point of intersection.  That Point is called the Miquel point.  Miquel went on to find various other similar theorems about quadrilaterals and pentagons. 

There is a 3D analogy of the theorem for a tetrahedron with points selected on each of its six edges.  

Mobius' Strip Mobius Strip – Discovered in 1858, it is the simplest non-orientable surface. Non-orientable means if you stand on the Mobius strip and point your hand outward, walk a full revolution and your hand will now be pointing inward. In other words, your body rotates as you walk a full circle. Two German mathematicians Johann Listing and August Mobius independently discovered this object those it is said that Roman mosaics share similar patterns. A fun fact about Listing is he was the first to coin the word “topology” in a 1847 article. The Mobius strip is also famously said to only have one side (because if you keep walking more than one rotation, you will have walked on “both sides” of the strip). Why did the chicken cross the Mobius strip? To get to the same side! *Derek Orr

Mode another word introduced to young students as a measure of center is mode.  The mode of a set of values is the value that occurs the most often. 

The early Indo-European root from which mode was derived had more of an "e"sound, and is written in dictionaries as med 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.  Th eFrench 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 fo the abscissa corresponding to the ordinate of maximum frequency"  (the highest point on the graph).  Every since then mathematicians have referred to the most "popular" value in a set of discrete 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.  

Modular Arithmetic

Monte Carlo Method During WWII the scientists and mathematicians working on the Manhattan Project to develop the atomic bomb were forging into untested waters in both science and math.  To answer some of the scientific questions, they would repeatedly sample from their best estimates of the partial results, then apply the math they knew to the interactions and study the range of results.  This process, which they named after the famous Monaco gaming casino town of Monte Carlo, was created by Stan Ulam and John von Neumann.  The term and a description of the method seems not to have been published until some time after the war. In his autobiography Adventures of a Mathematician Stanislaw M. Ulam (1976, pp. 196-200) wrote that such a method came to him while playing solitaire during an illness in 1946. Ulam described the method to John von Neumann and they “developed the mathematics together.” In an unpublished manuscript, “The Origin of the Monte Carlo Method,” dated Apr. 12, 1983, Ulam adds that what seems to be the first written account of the method was given by von Neumann in a letter to Robert Richtmyer of Los Alamos in early 1947. *Jeff Miller's web site.  The OED gives 1949 as the earliest print citation, for an article appearing in the American Journal of Statistics. 


Month's and their Names

Morley's Theorem/Triangle 
Just down the road about an hours drive  from the  Lakenheath RAF station, where I taught for a decade, in the East Anglia region of England is the pretty little village of Woodbridge.  Situated on the banks of the River Deben, Woodbridge is renowned for its sailing, riverside walks, pubs, restaurants, independent shops and theater.  It has a high-street, the British version of a "main street" in the USA, with a number of pretty little shops.  In 1860, one of the shops along the high street was a china shop owned and operated by Joseph Roberts Morley.  Joseph's son Frank was a bright young man who would have gone on to be a life long and productive resident perhaps, except by chance he developed a love of chess, and found himself to be very good at it.  This interest led him to meet another chess aficionado, who had spent some of his formative years not far away on his uncle's farm in Suffolk.  George Biddell Airy, the Astronomer Royal, recognized young Frank's chess skills and his potential in mathematics, and had an influence on getting him to compete for a scholarship to Kings College in Cambridge, which Frank won.  

Morley went on to graduate from Cambridge in 1884 and seemed to have not overly impressed anyone there with his mathematical power.  Like thousands of mediocre graduates before and after, he went into teaching.  As a teacher Morley would prove to be exceptionally capable.  After he moved to the US, first at Haverford College , then to St. Johns, where he supervised 48 graduate students.  His research seems not to have been on the mainstream of mathematics, but he is remembered for his ability to "have on hand a sufficient variety of thesis problems to accommodate particular tastes and capacities."  (as another connection to my interest in Morley,  one of my ex-calculus students graduated from Haverford

some decade ago and one another from St. Johns the year before)

But Morley is remembered most today for a singular theorem which bears his name in recreational literature.  Simply stated, Morley's Theorem says that if the angles at the vertices of any triangle (A, B, and C in the figure) are trisected, then the points where the trisectors from adjacent vertices intersect (D, E, and F) will form an equilateral triangle.
   In 1899 he observed the relationship described above, but could find no proof. It spread from discussions with his friends to become an item of mathematical gossip. Finally in 1909 a trigonometric solution was discovered by M. Satyanarayana. Later an elementary proof was developed. Today the preferred proof is to begin with the result and work backward. Start with an equilateral triangle and show that the vertices are the intersection of the trisectors of a triangle with any given set of angles. For those interested in seeing the proof, check Coxeter's Introduction to Geometry, Vol 2, pages 24-25.

About 1796, the nineteen year old Karl Gauss proved that certain regular polygons could be created with only the classic construction tools of straight edge and compass. Some forty years later, Wantzel completed this work by showing that only the regular polygons Gauss had described could be so constructed. As a consequence of this, it was finally proven that a general angle could not be trisected with straight edge and compass, thus ending the search for one of the classic problems of antiquity. From then until now, mathematicians and math professors have been beset by "proofs" from zealous amateurs who have triseceted an angle.  H S M Coxeter has suggested that from this time on people felt uneasy about the mention of trisecting an angle. This, he thinks, probably contributed to the reason that Morley's Theorem was not discovered until near the dawning of the 20th century. 

I was impressed that all three of Morley's sons went on to become Rhodes Scholars.  Felix was the editor of the Washington Post, a Pulitzer Prize winner, and then President of Haverford College.  Another, Christopher, became e a novelist, most remembered for " Kitty Foyle" .  A third, Frank V, was a director of the publishing firm Faber and Faber he also was a mathematician and worked with his father. It was Frank who wrote of his father,
. then he would begin to fiddle in his waistcoat pocket for a stub of pencil perhaps two inches long, and there would be a certain amount of scrabbling in a side pocket for an old envelope, and then there would be silence for a long time; until he would get up a little stealthily and make his way toward his study - but the boards in the hall always creaked, and my mother would call out, "Frank, you're not going to work!" - and the answer would always be, "A little, not much!" - and the study door would close.

(It wasn't hard to gather that my father was working at geometry, and I knew pretty well what geometry was, because for a long time I had been drawing triangles and things; but when you examined the envelope he left behind, what was really mysterious was that there was hardly ever a drawing on it, but just a lot of calculations in Greek letters. Geometry without pictures I found it hard to approve; indeed, I prefer it with pictures to this present day.)

 The senior Morley continued to be an impressive chess player and once defeated the algebraist and world chess champion,  Emmanuel Lasker.

Multiply comes from the combined Latin roots multi, for many, and plico, for folds, as in a number folded upon itself many times.   The first use I have found of the word as a verb in English, as in "multiply two by three",  is credited to Chaucer in his 1391 work, A Treatise on the Astrolabe.

Myriad The early Greeks had little use for numbers larger than a thousand, and sometimes referred to larger numbers as murious, uncountable. When the need, or desire, to create a number word for 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 the original meaning, uncoutable.  A general term for many legged anthropods such as millipedes and centipedes (which never have 100 legs) is myriapods.  

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