09/28/08

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. 

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

11.  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.

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

13.  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."

14.   MATHEMATICIANS and PHYSICISTS  

      Sir Isaac Newton (1642-1727):  His theory of gravity unified the force  that keeps our feet on the ground, with the force that holds planets in their orbits.

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

      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. 

      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).            

      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.

       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.

       James R Newman (1907-1966):  "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."

       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."   

      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.

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

     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.  

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

     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."

     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."

     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.