Show Posts

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.

Messages - Mohammad Hassan Murad

Pages: 1 [2] 3
Science Discussion Forum / Re: Facts about Newton
« on: June 25, 2012, 01:24:20 PM »
“Hannah went into labor sometime on December 24, 1642. It was the time of the full moon, and the baby was born an hour or two after midnight on Christmas morning. He was so frail that two women who attended Hannah were sent to obtain some medicine from a neighbor. Instead of hurrying, they sat down to rest on the way, thinking the child would already be dead. In the years to come Hannah would tell her son that he was so little when he was born that he would have fit into a quart pot. Moreover, he was too weak to hold his head upright for feeding and breathing. A special collar was fitted to his small neck to support his head; for a long time he remained very much below the size of children his own age.”

Christianson, G. E. Isaac Newton. And the Scientific Revolution Oxford University Press, 1996.
Also republished in
Christianson, G. E. Isaac Newton. Lives and Legacies Oxford University Press, 2005.   

At first I want to thank you for encouraging me to get through this topic.
Yeah, I think there is much more coincidences among ours.

Dear Sir you will enjoy the third part of this topic as you enjoyed the first and second parts. You remind me a very good thing. You can also calculate the chances of sharing birthday of our faculty members.

Dear Asit Sir I hope the other parts of this post will make you more interested on this subject.

We note that the same phenomenon pertains to deaths, for example, the probability that at least two former presidents of the United States share a birth month and day and the probability that at least two former presidents share a death day and month is the same. From the table (See Probability and the Birthday problem Part-I) for n = 44 presidents, the probability is 0.9328. Which shows that, there is about 93.28% chance that the former any two or more US Presidents share their birthday or death day.

James Polk (11th US President)
Warren G Harding (29th US President)
Millard Fillmore (13th US President)
William Howard Taft (27th US President)
John Adams (2nd US President)
Thomas Jefferson (3rd US President)
James Monroe (5th US President)

James Polk and Warren Harding were born on November 2; Milliard Fillmore and William Howard Taft died on March 8. John Adams, Thomas Jefferson, and James Monroe all died on July 4 (Independence Day). Of course, these presidents’ birth and death events do not prove the phenomenon.


Woolfson, M. M. Every day Probability and Statistics, Imperial College Press, 2008.
Deep, R. Probability and Statistics, Academic Press, 2006.
Blinder, S. M. Guide to Essential Math A Review for Physics Chemistry and Engineering Students, Academic Press, 2008.

To know more on the birth and death date/month/year of the US Presidents the following website would be helpful to the readers

To see the previous parts of this topic please visit
Probability and the Birthday problem Part-I,9321.0.html
Probability and the Birthday problem Part-II,9322.0.html

In my previous post Probability and the Birthday problem Part-I I asked-
What is the probability (or chance) that I share my birthday with any two or more of my students (in a section)?
The answer is given in this part.

In the following table I present the chances (in percentage) of sharing my birthday.


Woolfson, M. M. Every day Probability and Statistics, Imperial College Press, 2008.
Deep, R. Probability and Statistics, Academic Press, 2006.
Blinder, S. M. Guide to Essential Math A Review for Physics Chemistry and Engineering Students, Academic Press, 2008.

To see the previous part of this topic please visit
Probability and the Birthday problem Part-I,9321.0.html


In this current semester I take the course Mathematics-I in the department of EEE. There are two sections. In each the number of students are 43 and 42.
Now one day I ask myself what is the probability for sharing my birthday to any other students?
Or, I can ask what is the probability that any two or more students share the same birthday?

It is a possibly surprising fact that if you have 23 students in your class or at a party, there is a better than 50% chance that two people share the same birthday (same month and day, not necessarily in the same year).

Let us solve this problem. If we ignore the complication of leap-days there are 365 days on which a birthday can occur. A moment of thought tells us that if there are 366 people in a room then it is certain (probability = 1) that at least one pair of them must share a birthday (think about it!!). If the number of people were reduced to 365 or 364, then while it is not certain that at least one pair share a birthday, our intuition tells us that the probability of this must be very close to 1 — indeed we should be astonished if no two people shared a birthday with such numbers. If the number of people were reduced further then, clearly, the probability would reduce but would still remain high down to 200 people or so.
Let us now start at the other end with two people in a room. The probability that they have the same birthday is low and is easy to calculate. One way of thinking about it is to consider that one of them tells you his birthday. Then you ask the other one for his birthday. There are 365 possible answers he can give but only one of them will match the birthday given by the first person. Hence the probability of the birthdays being the same is 1/365.
The other possible way to look at this problem is to first ask the question “In how many possible ways can two people have birthdays without any restriction?” Since there are 365 possibilities for each of them then the answer to this is 365×365—every date in the calendar for one combined with every date in the calendar for the other. Now, we are going to ask a question that is not, perhaps, the obvious one to ask — “In how many ways can the two people have different birthdays?”.Well, the first one has 365 possibilities but once his birthday is fixed then, for the birthdays to be different, the second has only 364 possibilities so the answer is 365 × 364. Now, we can find the probability that they have different birthdays as
pdiff = (Number of ways of having different birthdays)/(Total number of ways of having birthdays)
= (365 × 364)/(365 × 365)
= 364/365
= 0.997260
≈ 99.72%
Now, either the two individuals have the same birthday or they do not and these are the only possibilities and they are mutually exclusive so that
pdiff + psame = 1,
or, psame = 1 − pdiff = 1 – 364/365 = 1/365
the answer obtained previously. The reason we have introduced this apparently convoluted way of getting the answer is because it better lends itself to dealing with the problem of when there are three or more people in the room. Now consider three people in a room. The number of ways they can have birthdays, without any restriction, is
365 × 365 × 365.
The number of ways they can have different birthdays is found by noting that the first person can have a birthday in 365 ways, for the second to be different from the first there are 364 ways and for the third to be different from the other two there are 363 ways. Hence the total number of ways they can have different birthdays is
Hence the probability that they have different birthdays is
pdiff = number of ways birthdays can be different number of ways they can have birthdays
= (365 × 364 × 363)/(365 × 365 × 365) = 0.9917958.
If the birthdays are not all different then at least two of them must be the same and the probability of this is
pnot diff = 1 − pdiff = 0.0082042.
We should now be able to see by extending the principle that the probability that at least two people out of five people in a room would share a birthday is
pnot diff = 1 – (365 × 364 × 363 × 362 × 361)/ (365 × 365 × 365 × 365 × 365) = 0.027136.
The advantage of calculating the probability in the way that we do, by finding the probability that all birthdays are different and then subtracting from 1, is now evident. Suppose that, we tried to directly find the probability that two or more people in a set of five shared a birthday. We would have to consider one pair the same and the others all different, three with the same birthday with the other two different, two pairs with the same birthday, and many other combinations. All birthdays being different is a single outcome, the probability of which is comparatively easy to find.
Now, we ask the question that is really the basis of this section—“How many people must there be in a room for there to be a probability greater than 0.5 (or, 50% chance) that at least two of them would share a birthday?” This can be done with a calculator. We work out the following product term-by-term until the answer we get is less than 0.5.
(365/365) × (364/365) × (363/365) × (362/365) × (361/365) × (360/365) × • • •
The value of n for which the answer first falls below 0.5 is when the probability of all birthdays being different is less than 0.5 and hence when there is a probability greater than 0.5 that two or more people have the same birthday. This is the number of people that fulfills our requirement. What is your intuitive assessment of n at this point? In the following table, we give the probability of having at least two people with the same birthday with increasing n. We see that the probability just exceeds 0.5 when 23 individuals are present — a result that most people find surprisingly low.

There is greater than 50% chance that two of these will share a birthday!

Woolfson, M. M. Every day Probability and Statistics, Imperial College Press, 2008.
Deep, R. Probability and Statistics, Academic Press, 2006.
Blinder, S. M. Guide to Essential Math A Review for Physics Chemistry and Engineering Students, Academic Press, 2008.

Science Discussion Forum / The Giants of Calculus
« on: May 31, 2012, 04:01:14 PM »

Calculus was invented as a tool for solving problems. Prior to the development of calculus, there were a variety of different problems that could not be addressed using the mathematics that was available. For example, scientists did not know how to measure the speed of an object when that speed was changing over time. Also, a more effective method was desired for finding the area of a region that did not have straight edges. Geometry, algebra, and trigonometry, which were well understood, did not provide the necessary tools to adequately address these problems.
             At the time in which calculus was developed, automobiles had not been invented. However, automobiles are an example of how calculus may be used to describe motion. When the driver pushes on the accelerator of a car, the speed of that car increases. The rate at which the car is moving, or the velocity, increases with respect to time. When the driver steps on the brakes, the speed of the car decreases. The velocity decreases with respect to time.
             As a driver continues to press on the accelerator of a car, the velocity of that car continues to increase. Acceleration is a concept that is used to describe how velocity changes over time. Velocity and acceleration are measured using a fundamental concept of calculus that is called the derivative.
             Derivatives can be used to describe the motion of many different objects. For example, derivatives have been used to describe the orbits of the planets and the descent of space shuttles. Derivatives are also used in a variety of different fields. Electrical engineers use derivatives to describe the change in current within an electric circuit. Economists use derivatives to describe the profits and losses of a given business.
             Derivatives along with the concept of a tangent line can be used to find the maximum or minimum value for a given situation. For example, a business person may wish to determine how to maximize profit and minimize expense. Astronomers also use derivatives and the concept of a tangent line to find the maximum or minimum distance of Earth from the Sun.
             The derivative is closely related to another important concept in calculus, the integral. The integral, much like the derivative, has many applications. For example, physicists use the integral to describe the compression of a spring. Engineers use the integral to find the center of mass or the point at which an object balances. Mathematicians use the integral to find the areas of surfaces, the lengths of curves, and the volumes of solids.
             The basic concepts that underlie the integral can be described using two other mathematical concepts that are important to the study of calculus are area and limit. Many students know that finding the area of a rectangle requires multiplying the base of the rectangle by the height of the rectangle. Finding the area of a shape that does not have all straight edges is more difficult.

Area under a curve f(x)

Area approximated by 4 rectangles

Area approximated by 10 rectangles

Area approximated by 20 rectangles

            An interesting relationship in calculus is that the derivative and the integral are inverse processes. Much like subtraction reverses addition; differentiation (finding the derivative) reverses integration. The reverse of this statement, integration reverses differentiation, is also true. This relationship between derivatives and integrals is referred to as the "Fundamental Theorem of Calculus". The Fundamental Theorem of Calculus allows integrals to be used in motion problems and derivatives to be used in area problems.


Pinpointing who invented calculus is a difficult task. The current content that comprises calculus has been the result of the efforts of numerous scientists. These scientists have come from a variety of different scientific backgrounds and represent many nations and both genders. History, however, typically recognizes the contributions of two scientists as having laid the foundations for modern calculus: Gottfried Wilhelm Leibniz (1646-1716) and Sir Isaac Newton (1642-1727).
              Leibniz was born in Leipzig, Germany, and had a Ph.D. in law from the University of Altdorf. He had no formal training in mathematics. Leibniz taught himself mathematics by reading papers and journals. Newton was born in Woolsthorpe, England. He received his master's degree in mathematics from the University of Cambridge.
             The question of who invented calculus was debated throughout Leibniz's and Newton's lives. Most scientists on the continent of Europe credited Leibniz as the inventor of calculus, whereas most scientists in England credited Newton as the inventor of calculus. History suggests that both of these men independently discovered the Fundamental Theorem of Calculus, which describes the relationship between derivatives and integrals.
             The contributions of Leibniz and Newton have often been separated based on their area of concentration. Leibniz was primarily interested in examining methods for finding the area beneath a curve and extending these methods to the examination of volumes. This led him to detailed investigations of the integral concept.
Leibniz is also credited for creating a notation for the integral, ∫. The integral symbol looks like an elongated S. Because finding the area under a curve requires summing rectangles, Leibniz used the integral sign to indicate the summing process. Leibniz is also credited for developing d notation for finding a derivative. Both of these symbols are still used in calculus today.
              Newton was interested in the study of fluxions. Fluxions refers to methods that are used to describe how things change over time. As discussed earlier, the motion of an object often changes over time and can be described using derivatives. Today, the study of fluxions is referred to as the study of calculus. Newton is also credited with finding many different applications of calculus to the physical world.
             It is important to note that the ideas of Leibniz and Newton had built upon the ideas of many other scientists, including Kepler, Galileo, Cavalieri, Fermat, Descartes, Torricelli, Barrow, Gregory, and Huygens. Also, calculus continued to advance due to the efforts of the scientists who followed. These individuals included the Bernoulli brothers, L'Hôpital, Euler, Lagrange, Cauchy, Cantor, and Peano. In fact, the current formulation of the limit concept is credited to Louis Cauchy. Cauchy's definition of the limit concept appeared in a textbook in 1821, almost 100 years after the deaths of Leibniz and Newton.

Pierre de Fermat (1601-1665)
Joseph-Louis Lagrange (1736-1813)
Gottfried Wilhelm Leibniz (1646-1716)
Augustine Louis Cauchy (1789-1857)
Leonhard Euler (1707-1783)

Who Uses Calculus Today?

Calculus is used in a broad range of fields for a variety of purposes. Advancements have been made and continue to be made in the fields of medicine, research, and education that are supported by the methods of calculus. Everyone experiences the benefits of calculus in their daily lives. These benefits include the availability of television, the radio, the telephone, and the World Wide Web.
            Calculus is also used in the design and construction of houses, buildings, bridges, and computers. A background in calculus is required in a number of different careers, including physics, chemistry, engineering, computer science, education, and business. Because calculus is important to so many different fields, it is considered to be an important subject of study for both high school and college students.
            It is unlikely the Leibniz or Newton could have predicted the broad impact that their discovery would eventually have on the world around them.

Based on the article by Barbara M. Moskal published in
Brandenberger, B. M. Jr. (editor in chief) Mathematics Volume 1 Macmillan Science Reference, Thomson/Gale, 2002.

Also See
Hall, A. R. Philosophers at War The Quarrel between Newton and Leibniz, Cambridge University Press, 2002.
Edwards, C. H. Jr. The Historical Development of the Calculus, Springer-Verlag, New York, 1979.
Bardi, J. S. Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time,
Thunder's Mouth Press, New York, 2006

Leonhard Euler (1707–1783) is a name well known in many academic fields: philosophy, hydrodynamics (the dynamics of moving fluids), astronomy, optics, and physics. His true fame comes, however, through his prolific work in pure mathematics. He produced more scholarly work in mathematics than have most other mathematicians. His abilities were so great that his contemporaries called him “Analysis Incarnate” (in human form of analysis). A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."
Portrait by Johann Georg Brucker (1756)
                                                                                                                                  A 1753 portrait by Emanuel Handmann. This portrayal suggests
                                                                                                                         problems of the right eyelid, and possible strabismus. The left eye, which here appears healthy, was later affected by a cataract.
             Euler (pronounced “oiler”) was born in Switzerland in 1707. He had originally intended to follow the career path of his father, who was a Calvinist clergyman. Euler studied theology and Hebrew at the University of Basel.
            Johann Bernoulli, however, tutored Euler in mathematics on the side. Euler’s facility in the subject was so great that his father, an amateur mathematician himself, soon favored the decision of his son to pursue mathematics rather than join the clergy.
            Euler first taught at the Academy of Sciences in St. Petersburg, Russia, in 1727. He married and eventually became the father of thirteen children. His children provided him with great joy, and children were often playing
in the room or sitting on his lap while Euler worked. It was in Russia that he lost sight in one eye after working for three days to solve a mathematics problem that Academy members urgently needed but had predicted would take months to solve.
            Euler was a very productive writer, completing five hundred books and papers in his lifetime and having four hundred more published posthumously. The Swiss edition of his complete works is contained in seventy four volumes. He wrote Introductio in Analysin Infinitorum in 1748. This book introduces much of the material that is found in modern algebra and trigonometry textbooks.
            Euler wrote the first treatment of differential calculus in 1755 in Institutiones Calculi Differentialis and in 1770 explored determinate and indeterminate algebra in Anleitung zur Algebra. Three-dimensional surfaces and conic sections were also extensively treated in his writings.
            Euler introduced many of the important mathematical symbols that are now in standard usage, such as Σ (for summation), π (the ratio of the circumference of a circle to its diameter), f (x) (function notation), e (the base of a natural logarithm), and i (square root of negative one). He was the first to develop the calculus of variations. One of the more notable equations that he developed was e = cos θ + i sin θ, which shows that exponential and trigonometric functions are related. Another important equation he developed establishes a relationship among five of the most significant numbers, e +1 = 0.
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. In the middle, it shows his polyhedral formula V-E+F = 2

Old Swiss 10 Franc banknote honoring Euler
                                                                                         1957 stamp of the former Soviet Union commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

            All of Euler’s work was not strictly academic, however. He enjoyed solving practical problems such as the famous “seven bridges of Königsberg” problem that led to Euler circuits and paths. He even performed calculations simply for their own sake.
            Euler later went to Berlin to become the director of Mathematics at the Academy of Science under Frederick the Great and to enjoy a more free political climate. However, Euler was viewed as being rather unsophisticated, and Frederick referred to him as a “mathematical Cyclops.” He returned to Russia when a more liberal leader, Catherine the Great, came to rule.
            By 1766, Euler was completely blind but continued to dictate his work to his secretary and his children. His last words, uttered as he suffered a fatal stroke in 1783, imitated his work in eloquence and simplicity: “I die.”

Brandenberger, Barry M. Jr. (editor in chief) Mathematics The Macmillan Science Library Volume 1-4, Macmillan Reference USA, Thomson/Gale, 2002.
Dunham, W. Euler The Master of Us All, Mathematical Association of America, 1999.
Nahin, P. J. Dr. Euler's Fabulous Formula Cures Many Mathematical Ills with a new preface by the author, Princeton University Press, 2006.
Tent, M. B. Leonhard Euler and the Bernoullis Mathematicians from Basel, A K Peters, 2009.
Suisky, D. Euler as Physicist, Springer, 2008.

Science Discussion Forum / Maria Gaëtana Agnesi and her witch
« on: May 30, 2012, 12:38:12 PM »
Maria Gaëtana Agnesi (1718–1799)
Italian Mathematician and Philosopher

Agnesi (pronounced ahn-YAY-zee or anyesi) was born in Milan, Italy. By the time she was 5 years old, she could speak both Italian and French. Her father was a professor of mathematics in Bologna, and Agnesi enjoyed a childhood of wealth and privilege. Her father provided her with tutors, and she participated in evening seminars, speaking to the guests in languages as varied as Latin, Greek, Hebrew, and Spanish.
            In her teens, Agnesi mastered mathematics. She became a prolific writer and an advocate for the education of women. After her mother died, Agnesi managed the household of eight children, and educated her brothers. Her father remarried and after her stepmother died, Maria became housekeeper for twenty siblings.
            Agnesi continued studying mathematics, mostly at night, and often to the point of exhaustion. In 1748 her mathematical compendium, Instituzioni analitiche ad uso della gioventù italiana (Analytical Institutions), derived from teaching materials she wrote for her brothers, was published in two volumes. In this work, Agnesi had not only written clearly about algebra, precalculus mathematics, differential calculus, and integral calculus, but she had also added her conclusions and her own methods. Analytical Institutions remains the first surviving mathematical work written by a woman.
             So successful was Agnesi’s textbook that it became the standard text on the subject for the next 100 years. Her book was studied and admired not only in Agnesi’s native Italy but also in France and Germany, and was translated into a number of other languages.
             In his 1801 English translation of Agnesi’s work, John Colson, the Cambridge (England) Lucasian Professor of Mathematics, made the mistake of confusing the Italian word for a versed sine curve, aversiera, with another Italian word for witch or wife of the devil, avversiere. Although 200 years have passed, the curve Colson misnamed the “Witch of Agnesi” still bears that name in many calculus texts as an illustration of an “even” function with a “maximum” value. For a = 2, for example, the maximum value of y will be 2. The curve illustrates many basic concepts in calculus.
Witch of Agnesi

Agnesi was recognized during her lifetime with election to the Bologna Academy of Science. The Pope at the time was interested in mathematics and in 1750 made certain that Agnesi was invited to be an honorary lecturer in mathematics at the University of Bologna. However, Agnesi turned down the appointment and instead adopted a religious life, spending much of her time working among the elderly poor and sick women of Milan.
           Although a hospice Agnesi founded has become famous throughout Italy, and Italian streets, a school, and scholarships have been named in her honor, Agnesi is perhaps best known today for her contributions to mathematics.

Maor, E. Trigonometric Delights, Princeton University Press, 1998
Brandenberger, Barry M. Jr. (editor in chief) Mathematics The Macmillan Science Library Volume 1-4, Macmillan Reference USA, Thomson/Gale, 2002.
Bradley, M. J. Pioneers in Mathematics The Age of Genius 1300 to 1800 Vol. 2, Facts on File, 2006.

Science Discussion Forum / Do Mathematicians get Nobel Prize?
« on: May 27, 2012, 06:02:18 PM »
He [Abel] has left mathematicians something to keep them busy for five hundred years.

Do Mathematicians get Nobel Prize? No. They get Abel Prize. It is not a joke. The name comes from a Norwegian mathematician who died at the age of 26.

NIELS HENRIK ABEL, one of the foremost mathematicians of the 19th century, was born in Norway on August 5, 1802. At the age of 16, he began reading the classic mathematical works of Newton, Euler, Lagrange, and Gauss. When Abel was 18 years old, his father died, and the burden of supporting the family fell upon him. He took in private pupils and did odd jobs, while continuing to do mathematical research. At the age of 19, Abel solved a problem that had vexed leading mathematicians for hundreds of years. He proved that, unlike the situation for equations of degree 4 or less, there is no finite (closed) formula for the solution of the general fifth-degree equation. Although Abel died long before the advent of the subjects that now make up abstract algebra, his solution to the quintic (fifth degree equation) problem laid the groundwork for many of these subjects. Just when his work was beginning to receive the attention it deserved, Abel contracted tuberculosis. He died on April 6, 1829, at the age of 26. In recognition of the fact that there is no Nobel Prize for mathematics, in 2002 Norway established the Abel Prize as the Nobel Prize in mathematics in honor of its native son. The amount of money that comes with the prize is usually close to US$ 1 million, similar to the Nobel Prizes (10 million kr. or US$ 1.46 million), which are awarded in Sweden and Norway and do not have a category for mathematics. Norway gave the prize an initial funding of NOK (Norwegian Krone) 200,000,000 (about US$23,000,000) in 2001. The prize is an attempt at creating publicity for mathematics, making the discipline more prestigious, especially for young people. It is a prize of 6 million kronor or approximately (in 2012) 1.06 million US dollars. The Abel Prize is now seen as an award equivalent to the Nobel Prize.

To see the winners of Abel Prize please visit the following webpage

JEAN-PIERRE SERRE of the College de France has been awarded the first Abel Prize of the Norwegian Academy of Science and Letters. Serre is honored “for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.” The prize amount is 6 million Norwegian kroner (approximately US$825,000).

The Norwegian Academy of Science and Letters in Oslo awarded Endre Szemer the one million dollar Abel prize on 21st march for "fundamental contributions to discrete mathematics and theoretical computer science". His specialty was combinatorics, a field that deals with the different ways of counting and rearranging discrete objects, whether they be numbers or playing cards.

Holden, H., Piene, R. (ed.) The Abel Prize 2003-2007 The First Five Years, Springer, 2010.
Jackson, A. Norway Establishes Abel Prize in Mathematics, Notices of the AMS, 49, 39-40,
Jackson, A. Serre Receives Abel Prize, Notices of the AMS, 50, 693, June/July 2003
Atiyah & Singer Receive 2004 Abel Prize, Notices of the AMS, 51, 650-651, June/July 2004.

Further reading

Science Discussion Forum / Life of Pythagoras
« on: May 26, 2012, 06:28:56 PM »
Pythagoras (ca. 569??-475?? B.C.E.) Born in Samos, Ionia, Greek mathematician and mystic Pythagoras is remembered as founder of the Pythagorean School, which claimed to have found universal truth through the study of number. As such, the Pythagoreans are credited as the first to have taken mathematics seriously as a study in its own right without regard to possible application and practical need. What precisely Pythagoras contributed himself is no longer clear, but the school is acknowledged to have discovered the role of ratios of whole numbers in the musical scale, the properties of figurate numbers, amicable numbersperfect numbers, and the existence of irrational numbers. Although Pythagoras's Theorem was known to the Babylonians over 1,000 years earlier, the Pythagoreans are the first to have provided a general proof of the result.
            Very little is known of Pythagoras's life. The sect he founded was half scientific and half religious and followed a code of secrecy that certainly promoted great mystery about the man himself. Many historical writings attribute godlike qualities to Pythagoras and are generally not regarded as accurate portrayals. It is understood that Pythagoras visited with Thales of Miletus (ca. 625-547 B.C.E.) as a young man who, most likely, contributed to Pythagoras' interest in mathematics. Around 535 B.C.E., Pythagoras traveled to Egypt and was certainly influenced by the secret sects of the Egyptian priests. (Many of the practices the Pythagoreans followed were the same as those practiced in Egypt refusal to eat beans or to wear cloth made from animal skins, for instance.)
           In 525 B.C.E., Egypt was invaded by Persian forces, and Pythagoras was captured and taken to Babylon. Eventually, after being released, Pythagoras settled in southern Italy, where he founded his famous school around 518 B.C.E. Both men and women were welcomed as members.
           Pythagoras believed that all physical (and metaphysical) phenomena could be understood through numbers. This belief is said to have stemmed from his observation that two vibrating strings produce a harmonious combination of tones only when the ratios of their lengths can be expressed in terms of whole numbers. (Pythagoras was an accomplished musician and made significant contributions to the theory of music.) Pythagoras, and his followers, studied whole numbers and their ratios, and even went as far as to assign mystical properties to numbers. For instance, they believed that the first natural number, 1, acted as the divine source of all numbers and so was of different stature than an ordinary number. (The number 2 was deemed the first number.) All even numbers were assumed feminine and all odd numbers masculine, and all odd numbers (except 13) represented good luck. The number 4 stood for justice, being the first perfect square, and 5 marriage, as the union (sum) of the first even number (2) and the first odd number (3). The number 6 was perfect since it equals the sum of its factors different from itself.
             In Geometry, the Pythagoreans knew that the interior angles of a triangle always sum to 180° (and, more generally, that the interior angle of an n-sided figure sum to (n - 2)180°), and knew how to construct three of the five Platonic Solids. They could solve equations of the form x2 = a(a-x) using geometrical methods, and they developed a number of techniques for constructing figures of a given area. They attributed great mystical significance to the pentagram because of the geometric ratios it contains. In astronomy, the Pythagoreans were aware that the Earth is round, but believed it lay at the center of the universe. They were aware that the orbit of the Moon is inclined to the equator of the Earth and were the first to realize that the evening star and the morning star were the same heavenly object (namely, the planet Venus).
             The exact date, location, and circumstance of Pythagoras's death are not known. Despite his passing, the factions of the original Pythagorean order endured for more than 200 years. Members of the Pythagorean brotherhood so revered their founder that it was considered impious for any individual to claim a discovery for his own glory without referring back to Pythagoras himself. It is said, for instance, that member Hippasus of Metapontum (ca. 470 B.C.E.) was put to death by drowning for announcing his own discovery of the regular dodecahedroan, the fifth Platonic solid. (Other versions of this popular story submit that he was executed for claiming to have discovered that the square root of 2 is an irrational number.)

To explore more see
Cornelius Lanczos Space Through the Ages Geometrical Ideas From Pythagoras to Hilbert & Einstein, Academic Press, 1970.
James Tanton Encyclopedia Of Mathematics, Facts on File, 2005.
Carl B. Boyer A History of Mathematics, 2nd ed. revised by Uta C. Merzbach and foreword by Isaac Asimov, John Wiley and Sons Inc., 1991.
George F. Simmons Calculus Gems Brief Lives and Memorable Mathematics, McGraw Hill 1992.

Charles Babbage is best remembered for his work on the design and manufacture of a mechanical calculator, the forerunner of a computer. After first constructing a “difference machine,” Babbage devoted the remainder of his life to the construction of a superior “analytic engine” capable of performing all mathematical operations. His work toward this goal laid the foundations of computer design used today. Partly due to lack of funding, however, the machine was never completed.....

To know more on him please see the attached file.

As mathematicians believe most of the mysteries of Nature are hidden in numbers. Now I shall talk about such a mysterious problem in number theory. At first I begin by quoting one of the most greatest mathematician of England,

The famous problem of which I propose to speak tonight is probably as difficult as any of the unsolved problems of mathematics.
G. H. Hardy on Goldbach's conjecture

What is Goldbach's Conjecture?
The Prussian mathematician Christian Goldbach noticed a pattern in the following sums:
4=2+2,                        6=3+3,                  8=3+5
10=3+7,12=5+7,         14=3+11=7+7,
16=3+13=5+11,          18=5+13=7+11,     20=3+17=7+13

(Do you see a pattern here?) Based on his observations, Goldbach, in a letter to Euler in 1742, conjectured that every even integer > 2 can be expressed as the sum of two primes. Euler could not prove it, and his conjecture still remains an unsolved problem.

Is Goldbach'ss Conjecture true for all even integer?
If not, up to when it is true?

To know more on this see the following attached file

If global warming is a reality, as most now agree it is, then its consequences must be considered. The possible consequences of global warming are controversial and difficult to predict. However, most of the scientists who study the atmosphere agree that the following are likely results of increased global warming, although the extent to which each of these phenomena may manifest itself is debatable.

  • Oceans may warm and release more carbon dioxide due to the reduced solubility of gases in warm water. This would increase warming even more and exacerbate the problem. However, warmer oceans might allow more phytoplankton to grow, which could consume more carbon dioxide.
  • Solid, frozen methane deposits at the bottom of the oceans may become gaseous as the ocean warms, and the gaseous methane would be released into air. This would increase warming even more.
  • Ice and snow cover may decrease significantly in parts of the world. This would cut down on reflected light so Earth would absorb even more radiation and warm up at an accelerated rate.

  • Ocean levels may rise as ice and snow at the polar caps melt. The consequences for islands and low-lying areas along coasts could be disastrous. There is already evidence of this occurring. Ice shelves that have been frozen for thousands of years are breaking away. The Arctic Climate Impact Assessment, a study by over 300 scientists released in late 2004, suggested that, among other consequences of such warming, polar bears could become extinct by 2099.

  • More water might evaporate from the oceans as they warm. This could increase global warming because water is a greenhouse gas. However, this could be offset by the reflective effect that additional cloud cover might have on incoming solar radiation.

  • Warming of the oceans could cause major shifts in ocean currents, leading to unpredictable influences on weather, fisheries, etc.

  • Rainfalls would become much heavier in some areas as a result of increased evaporation. This would lead to increased flooding.

  • The tree line would move farther north, causing trees to grow over a broader region. This could consume more carbon dioxide.

  • Deserts and arable regions of the world could shift location and change size as a function of shifts in areas of rainfall. The consequences are difficult to predict.

  • Tropical diseases might become more prevalent and widespread as Earth warms.

  • The intensity of storms and weather systems would probably increase.

  • Increased heat may mean worsened air pollution, damaged crops, and depleted resources in some areas.

  • Some scientists have speculated that when a certain atmospheric concentration of greenhouse gases is reached, it may trigger a relatively rapid and cataclysmic climate system reorganization.

  • Possibly, nothing of consequence, that is, nothing we can’t handle, will happen.
Excerpted from the book
Joesten, M., et al. The World of Chemistry Essentials, 4th edition, Thomson, Brooks/Cole, 2007.

Pages: 1 [2] 3