SRINIVASA  RAMANUJAN

Contribution to Mathematics

            His chief contribution in mathematics lies mainly in analysis, game theory and infinite series. He made in depth analysis in order to solve various mathematical problems by bringing to light new and novel ideas that gave impetus to progress of game theory. Such was his mathematical genius that he discovered his own theorems. It was because of his keen insight and natural intelligence that he came up with infinite series for π

                     

This series made up the basis of certain algorithms that are used today. One such remarkable instance is when he solved the bivariate problem of his roommate at spur of moment with a novel answer that solved the whole class of problems through continued fraction. Besides that he also led to draw some formerly unknown identities such as by linking coefficients of and providing identities for hyperbolic secant.

      D.R.KAPRELAR DISCOVERIES

Kaprekar constant

In 1949, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have:

4321 − 1234 = 3087, then

8730 − 0378 = 8352, and

8532 − 2358 = 6174.

Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.

A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.

Kaprekar number

Another class of numbers Kaprekar described are the Kaprekar numbers. A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 45²=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 100²=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in the OEIS):

Number	Square	Decomposition
703	703² = 494209	494+209 = 703
2728	2728² = 7441984	744+1984 = 2728
5292	5292² = 28005264	28+005264 = 5292

       P.C. MAHALANOBIS CONTRIBUTIONS

Mahalanobis Distance

                         Mahalanobis Distance is one of the most widely used metric to find how much a point diverges from a distribution, based on measurements in multiple dimensions. It is widely used in the field of cluster analysis and classification. It was first proposed by Mahalanobis in 1930 in context of his study on racial likeness. From a chance meeting with Nelson Annandale, then the director of the Zoological Survey of India, at the 1920 Nagpur session of the Indian Science Congress led to Annandale asking him to analyse anthropometric measurements of Anglo-Indians in Calcutta. Mahalanobis had been influenced by the anthropometric studies published in the journal Biometrika and he chose to ask the questions on what factors influence the formation of European and Indian marriages. He wanted to examine if the Indian side came from any specific castes. He used the data collected by Annandale and the caste-specific measurements made by Herbert Risley to come up with the conclusion that the sample represented a mix of Europeans mainly with people from Bengal and Punjab but not with those from the Northwest Frontier Provinces or from Chhota Nagpur. He also concluded that the intermixture more frequently involved the higher castes than the lower ones. This analysis was described by his first scientific paper in 1922. During the course of these studies he found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted "D²" and now eponymously named Mahalanobis distance, is independent of measurement scale. Mahalanobis also took an interest in physical anthropology and in the accurate measurement of skull measurements for which he developed an instrument that he called the "profiloscope".

        INTERESTING  FACTS  OF  MATHEMATICS-2

1.   The ancient Egyptians did not use fractions.

2.   If you add up all the roulette numbers, you will get a mystical number of 666.

3.   Quadratic equations appeared in India 15 centuries ago.

4.   Euclid left many works on mathematics and geometry, which we still use. Interestingly, no information about Euclid himself was found.

      5. Rene Descartes introduced the concepts of real and imaginary numbers.

       6.Interestingly, the great emperor Napoleon Bonaparte left some mathematical works after his death.

        7.The Indian scholar Budhayana, who lived in the 6th century, is considered the first to use the Pi number.

        8.Johannes Widmann first recorded the classic signs of addition and subtraction. It happened about 500 years ago.

       INTERESTING FUN FACTS OF MATHEMATICS

1. You can cut a cake into 8 pieces by using only 3 cuts.

You just need to make two cuts in a vertical plane and one in a horizontal plane. 

 2. The numbers on opposite sides of a dice always add up to seven

 On a dice the numbers 1,2 and 3 all share a vertex.  If these three numbers run clockwise round this vertex then the dice is called left-handed and if the three numbers run anti-clockwise round the vertex, then it is a right-handed dice.   Chinese dice are normally left-handed and Western dice are normally right-handed.

3. William Shanks calculated pi to 707 decimal places but made a mistake on the 528^th digit.

Amateur mathematicia William Shanks (1812-1882) spent a good part of his life calculating mathematical constants by hand.  Shanks never found out about his mistake as it wasn’t revealed until after his death.

4. The word googol was made up by a 9-year old boy

In the 1930s an American Mathematician named Edward Kasner asked his nine-year old nephew Milton Sirotta to make up a word for him to use. Milton made up the world ‘googol’ which Edward Kasner later used to describe the number  .  The search engine Google was later named after the ‘googol’ meaning that Milton Sirotta had unwittingly helped name one of the world’s most famous companies.

5. The word “hundred” comes from the old Norse term, “hundrath”, which actually means 120 and not 100. In a room of 23 people there’s a 50% chance that two people have the same birthday.

                    DIOPHANTUS

                Diophantus was an Alexandrian Greek mathematician, born somewhere between 200 and 214 BC. Alexandria was the center of Greek culture and knowledge and Diophantus belonged to the ‘Silver Age’ of Alexandria. His life story is not known in detail however we do have some dates acquired from a mathematical puzzle known as ‘Diophantus’s Riddle’. 

Contribution to Mathematics

           ‘Arithmetika’ a major work of Diophantus, is considered to be the most noticeable and influential work done on algebra in Greek history. His style was very different; he never used general methods in working out a problem. A method used for one problem could not be used to solve even another very similar problem. Diophantus wrote many books but unfortunately only a few lasted. He did a lot of work in algebra, solving equations in terms of integers. Some of his equations resulted in more than one answer possibility. There are now called ‘Diophantine’ or ‘Indeterminate’. It was none other than Diophantus who started the use of a symbol to specify the unidentified quantities in his equations. His style of algebra is known as the ‘syncopated’ style of algebraic writing, in which he represented polynomials as one unknown. 

                       HIPPARCHUS

     Hipparchus, better known as Hipparchos was a Greek mathematician born in 190 BC. Not much is known about Hipparchus’s life however it is deduced that his place of birth was Nicaea in Bithynia which is modern day Turkey. Though being one of the most influential mathematicians and astronomers, the details of his work are very scarce the most definite survived piece being his commentary on a poem by Aratus from the 3rd century the ‘Commentary on the Phainomena of Eudoxus and Aratus’. Also in the list of his contributions are his books on optics and arithmetic, writings concerning geography and astrology and a treatise called ‘On Objects Carried Down by their Weight’. 

Contribution to mathematics:

         His contributions to astronomy are believed to be of significant use in modern day applications of the field. Being the first to calculate a heliocentric system he left his work as according to his calculations the orbits were not truly circular as was the belief of science of that time. Hipparchus had observed the stars from a time span of 147 to 127 BC using an instrument called ‘dioptra’. Some historians suggest that he was the inventor of ‘Planispheric Astrolabe’, an astronomical device. It was none other than Hipparchus who raised important questions such as what the length of a year was and what the lunar distances were. Curious to find an answer, Hipparchus extensively studied the solar and lunar motions and their orbits using several calculations and techniques. He also determined the distances and sizes of both the sun and moon.