03/02/10

1.  GRAVITY  A lady asked the postmaster to weigh a package. When told it was too heavy and needed one more stamp, she said: ”I don’t understand.  Will adding a stamp make it lighter?”

2.  TIME MACHINE  An Amish boy and his father went to a large department store for the first time.  They saw an elevator, but did not know what it was.  An old lady with a cane, hobbled up to the elevator, and pushed a button on the wall, and a door opened.  She went into a small room and the door closed.  Then lights flashed above the doorway showing numbers.  The numbers stopped flashing at number five.  After a little while the numbers started flashing again,  and stopped at number one.  The door opened and out walked a beautiful blonde.                     The father shouted to his son:  “Go get your mother!”.

3.  NEGATIVE  & IMAGINARY NUMBERS  Messages to leave on your telephone answering machine are:       a)  “If you received a negative response, hang up, rotate your phone 180 degrees, and try again”.                          b)  “If you think you have reached an imaginary number, rather than a real number, hang up, rotate your phone ninety degrees, and try again.”                                                                                                                                    

4.  BASE 10 & BASE 8    Mathematicians confuse Halloween & Christmas.  25 Dec = 31Oct   That is, 25 in base 10, equals 31 in base 8. 

But, in the Octal System of Base 8   12x12 =144  just like in the Decimal System  of Base 10.  The number 144 in the Octal System is really the number 100 in the Decimal System.

100 in the Decimal System is 144 in the Octal system and in base 6 it is 244.   12 in the Decimal system is 14 in the Octal System and 20 in Base 6.  So  in Base 6, 12x12 = 400.

5.  INTEGRAL CALCULUS Two mathematicians in a restaurant were arguing about the mathematical knowledge of the public.  The cynic said: “I’ll bet you the cost of this dinner that the waitress can’t answer a simple math question." The cynic excused himself to visit the men’s room.  The other called the waitress over and said: "Here is $10. When I ask you a question, say ‘one third  x  cubed’.“  She agreed. 

The cynic  returned, called the waitress over, and said his friend had a question.  The question asked  was:         “What is the integral of x squared?”  After fidgeting and squirming a long time, she said: “One third x cubed.” .        The cynic paid the check. The waitress walked away, and muttered under her breath:  “Plus a constant.". .                                                                                                                                                                                                                                        

7.  INFINITY  Richard Phillips Feynman (1918-1988) “Did you know there are twice as many numbers as numbers?”                                                 

8.  FORCE  If brute force doesn’t work, you’re not using enough of it.

9.  GEOMETRY  EUCLID (325-265 BC)  What is the ratio of the Circumference of a Jack O Lantern to its Diameter?  Pumpkin Pi, of course.  Not Apple Pi.  The ratio of the distance around an igloo to the distance across, is Eskimo Pi.  

10.  NUMBERS  2 raised to the 5th power times 9 raised to the 2nd power = 2592                                                    

111,111,111 x 111,111,111 = 12345678987654321

"A man is a person who will pay two dollars for a one-dollar item he wants. A woman will pay one dollar for a two-dollar item she doesn't want..." -- William Binger

One half of five is four, as one half of five is iv.

Nine is the maximum number of cubes that are needed to sum to any positive integer.  Nineteen is the maximum number of fourth powers needed to sum to any positive integer.

10     10! =6! x 7!  These are the only consecutive integers, 6 and 7, that solve the equation N! = A! x (A+1)!

11  British mathematician J J Sylvester said: "Mathematics is the music of reason."  In 1884 at age 70, he proved that the highest number that cannot be created from using two numbers x and y equals xy - x - 7.  In Rugby, where drop goals are scored as 3 and converted tries are scored as 7, there cannot ever be a score of 11.            That is 3x7 -  3 - 7 = 11. 

19  Every single positive integer can be written as the sum of at most 19 powers.

22, 23, and 24 are the only positive integers (other than 1) for which n! has precisely n digits.  23 is the smallest prime for which the sum of the squares of its digits is also a prime.

Forty is the only number whose letters are in alphabetical order.

88  The number of keys on a piano, 52 white keys and 36 black keys. . There are 7 white keys and 5 black keys to an octave.    88  The number that is called "two fat ladies" in Bingo. 88  The number of feet  per second, when driving 60 miles per hour.

144 is the largest Fibonacci square.

37   (666)/(6 + 6 + 6) = 111/3

11.  NUMBER THEORY

100 =  (1² + 2² + 3² + 4²)

365 = ( 10² + 11² + 12²)  = (13² + 14²)

The formula for the nth square number is n².  And n² equals the sum of the first n odd numbers:

Brocard's Problem:  N factorial + 1 = X squared.  This is true  for X = 5, 11, and 71, but that may be all.                 Pierre René Jean Baptiste Henri Brocard: (1845 - 1922)

Primes, other than 2 or 3 are either of the form 6n + 1 or 6n - 1.     

Lychrel Numbers.  Most numbers become a palindrome by reversing their digits and adding repeatedly.  (349 + 943 = 1292, 1292 + 2921 = 4213, 4213 + 3124 = 7337 a palindrome.  Those that do not convert, are Lychrel Numbers. The name "Lychrel" was coined by Wade Van Landingham – a rough anagram of his girlfriend's name Cheryl.

Catalan's conjecture (occasionally now referred to as Mihăilescu's theorem)  was conjectured by the mathematician Eugene Charles Catalan  in 1844 and proven in 2002 by Preda Mihăilescu.

To understand the conjecture, notice that 2³ and 3² (i.e. 8 and 9) are two powers of natural numbers, whose values 8 and 9 respectively are consecutive. The conjecture states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of

x^a − y^b = 1

for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3.

12.  SQUARE ROOT  The square root of the number 81 is 9.  81 is the only number whose square root is the sum of its digits.

What did Pythagoras say when he was first confronted with the square root of 2?  "There has to be a rational explanation for this."

13.  LIMITS  The limit, as n goes to infinity, of {sin x}/n is 6.  Divide numerator & denominator by n and you get six.

14.  MUSICAL DOW  In 1987, there was an article in the Hartford Courant about Myron Schwager, a cellist,    who teaches at the Hartt School of Music, part of the University of Hartford.  He has a stock market charting system based on scales, and I wrote him a letter. See correspondence.   A month or so later, as I was a        senior officer, Mary and I were invited to a black tie ITT/Hartford Board dinner at the Harford Club.  A small     group played music during the dinner.  I recognized the cellist from his picture in the Hartford Courant article.  After dinner I went up to Myron Schwager and asked him what instrument he was playing.  He said "It's a cello."     I asked if I could look at it and he handed it to me.  I held it up to my ear and exclaimed: "Its playing the Dow!"    He said: "You must be Don Sondergeld."

15.   MATHEMATICIANS and PHYSICISTS  

      Grigori Yakovlevich Perelman (1966-       ):  (from Russia) The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute  in 2000. Currently, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. One of the problems, the Poincare' conjecture, was solved by Perelman in  2002  He was also awarded  the Fields Medal in 2006.  He has not accepted either prize.

     Sir Andrew John Wiles (1953-      ):  (British)  A professor at Princeton University in Number Theory. He published a flawed proof of Fermat's Last Theorem in 1993.  He corrected the error in 1994.

     Edward Witten (1951-      ):  One of the researchers at Princeton working on "string theory" which may help with the clash between the central ideas of general relativity and quantum mechanics when it comes to extremely small scales.   

     Dr. Keith Devlin (1947-     ): This professor from Stamford defines Mathematics as  the Science of Patterns. 

     Persi Warren Diaconis (1945-       ): He is the statistician who demonstrated that it takes the average card player no fewer than seven shuffles to create a random order in a deck of cards.

     Stephen Hawkings (1942-      ):  A physicist from Cambridge wrote "A Brief History of Time".  In it he tells the story of a lady commenting on a statement made in a lecture on astronomy.  She said: "Rubbish, The world is really a flat plate supported on the back of a giant tortoise"  When asked what the tortoise was sitting on, her answer would have made Goedel smile: "You're very clever, young man, very clever.  But its turtles all the way down."    

     Benoit Mandelbrot (1924-      ):  (from France)  The Father of Fractal Geometry.  There are many beautiful pictures to view on the web.  For example:  http://sprott.physics.wisc.edu/fractals.htm  Also there are terrific videos to be found at:  http://www.fractal-animation.net/ufvp.html

    Richard Feynman (1918-1988):  He said:  "Mathematics is looking for patterns" and "Mathematics is only patterns" and "Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry."  Also: "Physics is like sex.  Sure, it may give some practical results, but that's not why we do it".

Murray Gell-Mann commented to the New York Times that, “the Feynman Algorithm to solve a problem is:
1. Write down the problem
2. Think very hard
3. Write down the answer.

    Paul Erdos (1913-1996):  (from Budapest) "A mathematician is a machine for turning coffee into theorems."

    James R Newman (1907-1966):  (Hungarian-American) Von Neumann's first significant contribution to economics was the minimax theorem in1928.  He eventually improved and extended the minimax theorem to include games involving imperfect information and games with more than two players. This work culminated in the 1944 classic Theory of Games and Economic Behavior. Von Neumann was one of the pioneers of computer science making significant contributions to the development of logical design. "The Theory of Groups is a branch of mathematics in which one does something to something  and  then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing."

     Kurt Goedel (1906-1978):  (from Austria) Goedel's Incompleteness Theorem: Any consistent axiom system is necessarily incomplete in that there will be true statements that can't be deduced from the axioms. 

     Andre Weil (1906-1998):  (from France) "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

     Srīnivāsa Aiyangār Rāmānujan (1887-1920):  (Indian) He was a self taught genius, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.

     Hermann Weyl (1885-1955):  "A thing is symmetrical if there is something you can do to it, so that after you have finished doing it, it looks the same as before.

    Frank Albert Benford, Jr. (1883-1948): (American)  Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. Measurements of real world values are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly). This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change. It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

     Albert Einstein (1879-1955):  The speed of light is the same, irrespective of how the source of light or the observer is moving.  Furthermore, space and time cannot be treated as separate entities, rather they are inseparably tethered together by symmetry.  One of the known results of special relativity is that the length of moving bodies, as measured by observers at rest, contracts along their direction of motion.  The contraction is larger,  the higher the speed. Gravity warps and bends spacetime.  One of the key predictions of general relativity was the bending of light rays under the influence of gravity. Guided by principles of symmetry Einstein showed that acceleration and gravity are two sides of the same coin.(If a train is moving very fast to the north and a man in a boxcar drops his keys, they fall to the south.)(If a man in a stationary box car drops his keys, the keys would fall to the south, if gravity was tilted to the south.) 

    Bertrand Russell (1872-1970):  "Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties we can discover." 

    David Hilbert (1862 – 1943): (German) He was recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

    Jules Henri Poincaré (1854 –1912): (French) A mathematician,  theoretical physicist,  and a philosopher of science. philosopher of science. Poincaré is often described  in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.    

     Prime Number Theorem states that if you select a large number N, the probability of  it being prime is about 1/Ln(N) was solved independently in 1896 by Jacques-Solomon Hadamard (1865-1963) and Charles de la Vallee Poisson (1866-1962) by showing that the Riemann Zeta Function has no zeros of the form (1 + ki).            

    Bernhard Riemann (1826-1866):  If the "Riemann Hypothesis" is true,  the exact number of primes less than a given number N, or  Pi(N), can be calculated exactly.   Although thought to be correct, this hypothesis is unproven.    Karl Friedrich Gauss (1777-1855) had an approximation to Pi(N), equal to N/ln(N), where ln is the natural logarithm.   Adrien-Marie Legendre (1752-1833) improved on Gauss's estimate using  Pi(N) = N/{ln(N) - 1.08366}   Gauss then improved upon that estimate using Li(N) , which he called the logarithmic integral. (not shown here)  Leonard Euler (1707-1783) showed that the Riemann Zeta Function: Z(s) =  The sum of 1/n raised to the s power for n = 1 to infinity, is also equal to a product series involving primes.  Z(s) = The product of (1 + 1/p to the s + 1/p to the 2s + 1/p to the 3s  + 1/p to the 4s + 1/p to the 5s +...) over all primes.  It is important  to note: "s" is a "complex number". Riemann then hypothesized that Z(s) = 0 for only complex numbers where the real part = 1/2.  The Riemann Hypothesis has not been proven, but computers have shown the first 6.3 billion zeros all lie on the line s = 1/2 +ki.  If the Riemann Hypothesis is correct, then Riemann has a formula for calculating Pi(N) exactly!    Pi(N) =  R(N) minus an Adjustment.  R(N) is a formula involving the logarithmic integral and the Adjustment is expressed in terms of the zeros of the Zeta Function. The function R(N) was named in honor of Riemann.  

    E'variste Galois (1811-1832):  A symmetry of an object is what you can do to an object to leave it essentially looking like it did before you touched it.  Galois was interested in the collection of all symmetries and seeing what happens if you do one symmetry after another.  He discovered that it is the interactions between the symmetries in a group that encapsulate the essential qualities of the symmetry of an object.

    Niels Henrik Abel (1802-1829):  (Norwegian)  At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had only held for rational numbers. At age 19, he showed there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils."

    Michael Faraday (1791-1867) and James Clerk Maxwell (1831-1879):  They proved that electric and magnetic forces are the same force in different guises.

    Carl Friedrich Gauss (1777-1855): German) Called the Prince of Mathematicians and the greatest mathematician since antiquity. He is ranked as one of history's most influential mathematicians. He referred to mathematics as the Queen of Sciences.  Gauss proved the Fundamental Theorem of Algebra. Gauss claimed to have discovered the possibility of non Euclidean Geometries but never published it.

    Pierre-Simon, marquis de Laplace (1749-1827):  (French) He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France, with a phenomenal natural mathematical faculty superior to any of his contemporaries.

    Joseph-Louis Lagrange (1736-1813): (Italian) Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun, and Moon and the movement of Jupiter’s satellites. In 1772 found the special-case solutions to this problem that are now known as Lagrangian points. He transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called.

    Leonhard Euler (1707-1783):  One of his many contributions was called "Euler's Formula".  The formu states that, for any real number  x,     where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions, with the argument x given in radians. The formula is still valid if x is a complex number.   Richard Feynman called Euler's formula "our jewel" and "one of the most remarkable, almost astounding, formulas in all of mathematics".  

     Gottfried Wilhelm Leibnitz (1646-1716): (German) He invented infinitesimal calculus independently of Newton, and his notation has been in general use since then. He also invented the binary system, the foundation of virtually all modern computer architectures.

     Sir Isaac Newton (1642-1727):  (British) His theory of gravity unified the force  that keeps our feet on the ground, with the force that holds planets in their orbits. His 1687 publication of the Philosophiae Naturalis Principia Mathematica  is considered to be among the most influential books in the history of science.  In this work, Newton described universal gravitation and the three laws of motion. Newton shares the credit with Leibnitz for the development of differential and integral calculus. He also demonstrated the generalized binomial theorem and contributed to the study of power series.

     Blaise Pascal (1623-1662): (French) He helped create two major new areas. He wrote a significant treatise on  projective geometry  at the age of sixteen.   Pascal's development of probability theory was his most influential contribution to mathematics, a subject on which he corresponded with Fermat.  Pascal continued to influence mathematics throughout his life. In 1653 he described a convenient tabular presentation for binomial coefficients, now called Pascal's triagle.

     Pierre de Fermat (1601-1665): (French)  A lawyer and amateur mathematician who contributed to Number Theory and known for "Fermat's Last Theorem".  Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to both Newton and Leibnitz in developing calculus.

    René Descartes (1596-1650): (French)  The inventor of Analytical Geometry.  He was a philosopher, mathematician, physicist and writer. He has been dubbed the "Father of Modern Philosophy".

    Leonardo Pisano Fibonacci (1170?-1250):  (Italian)  Fibonacci is considered to be one of the most talented mathematicians for the Middle Ages. Few people realize that it was Fibonacci that gave us our decimal number system (Hindu-Arabic numbering system) which replaced the Roman Numeral system. When he was studying mathematics, he used the Hindu-Arabic (0-9) symbols instead of Roman symbols which didn't have 0's and lacked place value. In fact, when using the Roman Numeral system, an abacus was usually required. There is no doubt that Fibonacci saw the superiority of using Hindu-Arabic system over the Roman Numerals. He shows how to use our current numbering system in his book Liber abaci. And he gave us the Fibonacci Series. Fibonacci was known as Leonardo of Pisa. He was born in Pisa, home of the famous leaning tower and his statue is located there.

   Archimedes of Syracuse (c.287-c.212 B.C):  (from Sicily) A mathematician and inventor. He determined the exact value of pi, is also known for his strategic role in ancient war and the development of military techniques. "Give me a place to stand and I will move the earth" was his boast when he discovered the laws of levers and pulleys.  His mechanical inventions defeated the Roman fleet of Marcellus.  The word "eureka" comes from the story that when Archimedes figured out a way to determine whether the king (Hiero II of Syracuse), a possible relative, had been duped by measuring the buoyancy of the king's supposedly solid gold crown in water, he became very excited and exclaimed the Greek (Archimedes' native language) for "I have found it": Eureka. Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with the ratio of the cylinder's volume to that of the sphere. Archimedes considered the discovery of this ratio the greatest of all his accomplishments.    

    Euclid of Alexandria (300 BC-       )  (Greek)  He was often referred to as the "Father of Geometry."  His "Elements" is one of the most influential works in mathematics, serving as the main textbook for teaching mathematics, especially geometry, from the time of its publication until the late 19th or early 20th century.   

MATHEMATICS  and  MUSIC

     Heinrich Rudolf Hertz (1857-1894):   A Hertz, or Hz, is named after this German physicist.  It is a measure of the frequency, the number of vibrations of a string per second.  People in different musical traditions have different ideas about which notes they think sound good together.  If you double the frequency, the human ear tends to hear both notes  as the same. This is called "octave equivalency". The doubled frequency is called a higher octave. This "octave"  is two times higher, not eight times higher. In the "diatonic scale", there are 8 notes counting both ends of the octave hence the term "octave".  In the "chromatic scale" there are 13 notes counting both ends, and the "Arab classical scale" has 17, 19, or even 24 notes in its "octave.

The most important scale in the Western tradition is the Diatonic Scale.  Scales can be classified as "just".       (if the ratios between the frequencies are ratios of integers), "tempered" (if the just scale is tempered), and "practice-based" (if it reflects musical practice)   In the Even Tempered Scale going from one semitone to  the next is the 12th root of 2, or 1.05946...  (Pythagoras discovered that frequencies whose ratio is equal to  the ratio of two simple whole numbers yield "harmonious"  and pleasing sounds.  The ratio of 3:2 is 1.5.  A "perfect fifth" corresponds to a separation of seven semitones, as the seventh power of 1.05946 is close to 1.5.  A "perfect fourth" corresponds to a frequency ratio of 4/3 and five semitones.)

The frequency of middle C on a piano is often set at 261.6 Hz.  There are twelve semitones in an octave. octave.  A piano keyboard has 7 white keys and 5 black keys to play notes within any octave. octave. A trumpet has 3 valves that can be all open, two closed, etc. with 7 combinations of fingering, fingering, so a trumpet can only play 7 tones within an octave.