RAMANUJAM

"An equation for me has no meaning unless it expresses a thought of God."- RAMANUJAM

Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. His most famous work was on the number p(n) of partitions of an integer n into summands.

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on Advanced Trigonometry written by S. L. Loney. He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.

When he was 16, Ramanujan came across the book "A Synopsis of Elementary Results in Pure and Applied Mathematics"  by George S. Carr. This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.

Ramanujan, with the help of Ramaswami Iyer(founder member of the Indian Mathematical Society) , had his work published in the Journal of Indian Mathematical Society.

In January 1913 Ramanujan wrote to G .H. Hardy having seen a copy of his  book Orders of infinity. Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter.

Hardy wrote back to Ramanujan and in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.

On 6 December 1917, he was elected to the London Mathematical Society. 

In 1918, he became a Fellow of the Royal Society , and he was the youngest Fellow in the entire history of the Royal Society.

On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.


TAXICAB NUMBER:
The number derives its name from the following story:
G. H. Hardy told about Ramanujan. I remember once going to see him when he was ill . I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
1729 is the second taxicab number (the first is 2= 1^3 + 1^3). The number was also found in one of Ramanujan's notebooks dated years before the incident.

Every positive integer is one of Ramanujan's personal friends" - John Littlewood, on hearing of the taxicab incident

Ramanujan had problems settling in London. He was an orthodox Brahmin and right from the beginning he had problems with his diet. Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died on April 6, 1920.

EUCLID

FATHER OF MATHEMATICS

The Greek mathematician Euclid’s referred to as “THE FATHER OF GEOMETRY” is well known for his most famous work “ The Elements” which is a collection of geometrical theorems and “Euclidean theorem”.

EUCLID’S FAMOUS QUOTES:

“The laws of nature are but the mathematical thoughts of God”

“ There is no other Royal path which leads to geometry”.

THE ELEMENTS:

The Elements is divided into 13 books.

• The first 6 books deals with plane geometry.
• Books  7to 9 deals with number theory.
• Book 10 deals with the theory of irrational numbers .
• Books 11 to 13 deals with three-dimensional geometry .

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved.

EUCLID'S OTHER WORKS :       

• ON DIVISION deals with plane geometry.
• The book DATA discusses plane geometry and contains propositions.
• PHAENOMENA is a work by what we call today as applied mathematics, concerning the  geometry of spheres for use in astronomy. 
• THE OPTICS, corrects the belief held at the time that the sun and other heavenly bodies are actually the size they appear to be to the eye.
• CONICS was a work on conic sections.

 EUCLID'S APPROACH:

Euclid used an approach called the "synthetic approach" to present his theorems. Using this method, one progresses in a series of logical steps from the known to the unknown.

EUCLID’S CLASSICAL PROOF ON PRIME NUMBERS:

Euclid proved that it is impossible to find the "largest prime number," because if you take the largest known prime number, add 1 to the product of all the primes up to and including it, you will get another prime number. Euclid's proof for this theorem is generally accepted as one of the "classic" proofs because of its conciseness and clarity. Millions of prime numbers are known to exist, and more are being added by mathematicians and computer scientists.

ARCHIMEDES

FATHER OF MATHEMATICS

CONTRIBUTIONS TO MATHEMATICS

Archimedes, a Greek mathematician is considered one of the three great mathematicians along with Isaac Newton and Carl Fredrick Gauss. . His greatest contributions to mathematics were in the area of Geometry. Archimedes was also an accomplished engineer and an inventor.

• Discovered the method to determine the area and volumes of circles, spheres and cones. 
• Discovered the actual value of PI.
• Archimedes‘s investigation on Method of Exhaustion led way to current form of Integral Calculus which is now updated. Though it is outdated it is believed that he invented the method of Integral Calculus 2000 years before Newton and Leibniz.

OTHER CONTRIBUTIONS

• Archimedes performed countless experiments on screws, levers, and pulleys. 
• Archimedes invented the water screw, a machine for raising water to bring it to fields.  
• His work with levers and pulleys led to the inventions of compound pulley systems and cranes.  
• His compound pulleys are highlighted in a story that reports that Archimedes moved a fully-loaded ship single-handedly while seated at a distance.  
• His crane was reportedly used in warfare during the Roman siege of his home, Syracuse. 
• Wartime inventions attributed to Archimedes include rock-throwing catapults, grappling hooks, and lenses or mirrors that could allegedly reflect thesun's rays and cause ships to catch on fire.
• Another invention was a miniature planetarium, a sphere whose motion imitated that of the earth, sun, moon, and the five planets that were then known to exist.

 A FAMOUS STORY 

There are many stories about how Archimedes made his discoveries. A famous one tells how he uncovered an attempt to cheat King Hieron.

The king ordered a golden crown and gave the crown's maker the exact amount of gold needed. The maker delivered a crown of the required weight, but Hieron suspected that some silver had been used instead of gold. He asked Archimedes to think about the matter. One day Archimedes was considering it while he was getting into a bathtub. He noticed that the amount of water overflowing the tub was proportional (related consistently) to the amount of his body that was being immersed (covered by water). This gave him an idea for solving the problem of the crown. He was so thrilled that he ran naked through the streets shouting, "Eureka!" (Greek for "I have discovered it!").

There are several ways Archimedes may have determined the amount of silver in the crown. One likely method relies on an idea that is now called Archimedes's principle. It states that a body immersed in a fluid is buoyed up (pushed up) by a force that is equal to the weight of fluid that is displaced (pushed out of place) by the body. Using this method, he would have first taken two equal weights of gold and silver and compared their weights when immersed in water. Next he would have compared the weight of the crown and an equal weight of pure silver in water in the same way. The difference between these two comparisons would indicate that the crown was not pure gold. 

FLORENCE NIGHTINGALE'S CONTRIBUTION TO MATHEMATICS

The rare photograph of Florence Nightingale was taken by Lizzie Caswall Smith in 1910 .The black and white image of the silver-haired nursing pioneer shows her in the imposing bedroom of her home just off London's Park Lane, before her death in 1910 at the age of 90.


Florence Nightingale is most remembered as a pioneer of nursing and a reformer of hospital sanitation methods. For most of her ninety years, Nightingale pushed for reform of the British military health-care system and with that the profession of nursing started to gain the respect it deserved.

During the American Civil War, Nightingale was a consultant on army health to the United States government. She also responded to a British war office request for advice on army medical care in Canada. Her mathematical activities included ascertaining "the average speed of transport by sledge" and calculating "the time required to transport the sick over the immense distances of Canada."

Unknown to many, Florence Nightingale is credited with developing a form of the pie chart now known as the polar area diagram, or occasionally the Nightingale rose diagram, equivalent to a modern circular histogram to dramatize the needless deaths caused by unsanitary conditions and the need for reform during the Crimean War .

The legend reads:

The Areas of the blue, red, & black wedges are each measured from the
centre as the common vertex.
The blue wedges measured from the centre of the
circle represent area for area the deaths from Preventable or Mitigable
Zymotic diseases, the red wedges measured from the centre the deaths from
wounds, & the black wedges measured from the centre the deaths from all
other causes.
The black line across the red triangle in Nov. 1854 marks the
boundary of the deaths from all other causes during the month.
In October 1854, & April 1855, the black area coincides with the red, in January
& February 1855,(*) the blue coincides with the black.
The entire areas may be compared by following the blue, the red, & the black lines
enclosing them.

With her analysis, Florence Nightingale revolutionized the idea that social phenomena could be objectively measured and subjected to mathematical analysis.

Euler's depiction on SWISS 10 FRANC NOTE

EULER'S POWERS OF MEMORY AND CONCENTRATION

Euler's powers of memory and concentration were legendary :

Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy.

Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as " Cyclops".

Euler later suffered a cataract in his good left eye, rendering him almost totally blind a few weeks after its discovery in 1766.

Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory.

For example: Euler could repeat the Aeneid of virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

JOHANN CARL FRIEDRICH GAUSS

 PRINCE OF MATHEMATICS

Johann Carl Friedrich Gauss , a German mathematician who had a remarkable influence in many fields , including number theory, statistics, analysis, differential geometry, electrostatics, astronomy and optics is ranked as one of history's most influential mathematicians.

CHILD PRODIGY:

At the age of three he amazed his father by correcting an arithmetical error.

In primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add a list of integers. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher. 

Gauss's presumed method, which supposes the list of numbers was from 1 to 100, was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums:

1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050 .

FAMOUS QUOTE:

“Mathematics is the queen of the sciences and number theory is the queen of mathematics”

“Ask her to wait a moment,I am almost done” (he told this while working when he was informed that his wife is dying).

CONTRIBUTIONS:

• In Disquisitiones Arithmeticae, one of the most brilliant achievements in mathematics, Gauss systematized the study of number theory. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
• Gauss proved the Fundamental Theorem of Algebra, which states that every polynomial has a root of the form a+bi.
• He also discovered the Cauchy Integral theorem for analytic functions
• Gauss's work in mathematical physics contributed to potential theory and the development of the Principle of Conservation of Energy.
• Theoria motus corporum celestium (theory of motion of the celestial bodies) is his most significant work on applied mathematics.
• Gauss discovered Ceres, the largest of the asteroids orbiting around the Sun.
• His Theory of Celestial Movement remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant.
• Introduced the Method of Least Squares, a procedure used in all sciences to this day to minimize the impact of measurement error.