Great Scientist Albert Einstein

ALBERT EINSTEIN
He was born in Ulm, Germany on March  14, 1879. After education in Germany, Italy, and Switzerland, and professorships in Bern, Zurich, and Prague, he was appointed Director of Kaiser Wilhelm Institute for Phy-sics in Berlin in 1914. He became a professor in the School of Mathematics at the Institute for Advanced Study in Princeton beginning the fall of 1933, became an American citizen in the summer of 1936, and died in Princeton, New Jersey on April 18, 1955. In the Berlin where in 1900 Max Planck discovered the quantum, Einstein fifteen years later explained to us that gravitation is not something foreign and mysterious acting through space, but a manifestation of space geometry itself.
He came to understand that the universe does not go on from everlasting to everlasting, but begins with a big bang. of all the questions with which the great thinkers have occupied themselves in all lands and all centuries, none has ever claimed greater primacy than the origin of the universe, and no contributions to this issue ever made by any man anytime have proved themselves richer in illuminating power than those that Einstein made.

Achievements:
1.His estimate of molecular size based on the change in viscosity of a liquid when  particles are suspended in it.
2.His demonstration, based on the first theory of a stochastic process, that micro-scopic fluctuation phenomena can be observed in Brownian motion.
3.His development of a new kinematics in the special theory of relativity, and the deduction from it of such remarkable features as: The path dependence of proper time intervals (twin-paradox")  The equivalence of mass and energy ("E=mc2").
4.His development of general relativity, still the best theory of gravitation that we have.
5.His proposal of the light quantum hypothesis, which developed into the theory of the photon, the first elementary particle to be given a quantum treatment.
6.His quantum theory of solids, which provided the basis for explaining the anoma-lous low-temperature behavior of crystalline solids.
7.His explanation of Planck’s law based on the introduction of the A&B coefficients, which placed the concept of transition probabilities at the center of atomic physics.
8.The Einstein-Podolsky-Rosen "paradox," which highlighted the nature of the quan-tum entanglement of two or more systems.His work on Bose-Einstein statistics, leading to his prediction of the existence of Bose-Einstein condensates, only recently confirmed.
9.The Einstein-Infeld-Hoffmann derivation of the equations of motion of massive bodies from the field equations of general relativity – the list could go on indefi-nitely.
10.He received Nobel prize in physics for his contribution to the theoretical physics and the photoelectric effect in 1921.

Great Scientist C.V.Raman

C. V. Raman  (Sir Chandrasekhara Raman)
In this Indian name, the name Chandrasekhara is a patronymic, not a family name, and the person should be referred to by the given name, Raman. Sir Chandrasekhara Venkata Raman, (7 November 1888 – 21 November 1970) was an Indian physicist, born in the former Madras Province, whose ground breaking work in the field of light scattering earned him the 1930 Nobel Prize for Physics. He discovered that, when light traverses a transparent material, some of the deflected light changes in wavelength. This phenomenon is now called Raman scattering and is the result of the Raman effect. In 1954, he was honoured with the highest civilian award in India, the Bharat Ratna.
 Family
Raman's maternal grandfather, Saptarshi Sastri, was a Sanskrit scholar who was learned in navya nyaya (modern logic). Raman's father initially taught in a school in Thiruvanaikaval, became a lecturer of mathematics and physics in Mrs. A.V. Narasimha Rao College, Vishakapatnam (then Vizagapatnam) in the Indian state of Andhra Pradesh, and later joined Presidency College in Madras (now Chennai). He was married on 6 May 1907 to Lokasundari Ammal (1892–1980). They had two sons, Chandrasekhar and radio-astronomer Radhakrishnan.
Raman was the paternal uncle of Subrahmanyan Chandrasekhar, who later won the Nobel Prize in Physics (1983) for his discovery of the Chandrasekhar limit in 1931 and for his subsequent work on the nuclear reactions necessary for stellar evolution.
 Early education
At an early age, Raman moved to the city of Visakhapatnam and studied at St. Aloysius Anglo-Indian High School. Raman passed his matriculation examination at the age of 11 and he passed his F.A. examination (equivalent to today's Intermediate exam 10+ 2) with a scholarship at the age of 13.
In 1902, Raman joined Presidency College in Madras where his father was a lecturer in mathematics and physics. In 1904 he passed his Bachelor of Arts (B.A.) examination: He stood first and won the gold medal in physics. In 1907 he gained his Master of Arts (M.A.) degree with the highest distinctions.

Achievements
In 1917, Raman resigned from his government service after he was appointed the first Palit Professor of Physics at the University of Calcutta. At the same time, he continued doing research at the Indian Association for the Cultivation of Science (IACS), Calcutta, where he became the Honorary Secretary. Raman used to refer to this period as the golden era of his career. Many students gathered around him at the IACS and the University of Calcutta.
During a voyage to Europe in 1921, Raman noticed the blue colour of glaciers and the Mediterranean sea. He was motivated to discover the reason for the blue colour. Raman carried out experiments regarding the scattering of light by water and transparent blocks of ice which explained the phenomenon.
There is an event that served as the inspiration of the Raman effect. On a December evening in 1927, Raman's student K. S. Krishnan (who later became the Director of the National Physical Laboratory) gave him the news that Professor Compton had won the Nobel Prize for his studies of the scattering of X-rays. This led Raman to theorize that if the Compton effect is applicable for X-rays, then it may be for light also, and to devise some experiments.
Raman employed monochromatic light from a mercury arc lamp which penetrated transparent material and was allowed to fall on a spectrograph to record its spectrum. He detected lines in the spectrum which he later called Raman lines. He presented his theory at a meeting of scientists in Bangalore on 16 March 1928.
 On 28 February 1928, Raman led experiments at the IACS with collaborators, including K. S. Krishnan, on the scattering of light, when he discovered what now is called the Raman effect and he won the Nobel Prize in Physics in 1930.
Books
For compact work, see: Scientific Papers of C. V. Raman, S. Ramaseshan (ed.).
Vol. 1 – Scattering of Light (Ed. S. Ramaseshan)
Vol. 2 – Acoustics
Vol. 3 – Optics
Vol. 4 – Optics of Minerals and Diamond
Vol. 5 – Physics of Crystals
Vol. 6 – Floral Colours and Visual Perception


Honours and awards
He was elected a Fellow of the Royal Society early in his career (1924) and knighted in 1929.
He won the Nobel Prize in Physics in 1930
 He was awarded the Franklin Medal in 1941
He was awarded the Bharat Ratna. in 1954
He was awarded the Lenin Peace Prize in 1957.
 The American Chemical Society and Indian Association for the Cultivation of Science recognised Raman's discovery as an International Historic Chemical Landmark in 1998
India celebrates National Science Day on 28 February of every year to commemorate the discovery of the Raman effect in 1928.
Achieve  of   Raman Research Papers
The Raman Research Institute, founded by Raman after his tenure at IISc, curates a collection of Raman's research papers, and articles on the web.

Great Mathematician Pythagoras

PYTHAGORUS
Pythagoras  (570 BC – 495 BC)  was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.  He born at island of Samos. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a seal engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.
Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus. He attributed various numbers and forms to physical elements for example 5 is the cause of colour,  6 is the cause of cold, 7 is the cause of health & 8 is the cause of love.
It was said that he was the first man to call himself a philosopher ("lover of wisdom")  and that he was the first to divide the globe into five climatic zones. Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy.



Contribution of Pythagorus:
In the field of  mathematics:

 The Pythagorean theorem: Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is  a2+ b2= c2
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Although Pythagoras is most famous today for his alleged mathematical discoveries, classical historians dispute whether he himself ever actually made any significant contributions to the field. Many mathematical and scientific discoveries were attributed to Pythagoras, including his famous theorem, as well as discoveries in the fields of music, astronomy,  and medicine.
The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, but it is possible that he may have been the first one to introduce it to the Greeks. Some historians of mathematics have even suggested that he—or his students—may have constructed the first proof.  Burkert rejects this suggestion as implausible,  noting that Pythagoras was never credited with having proved any theorem in antiquity.  Furthermore, the manner in which the Babylonians employed Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources.  Pythagoras's biographers state that he also was the first to identify the five regular solids  and that he was the first to discover the Theory of Proportions.

In the field of  music:
Pythagorean tuning and Pythagorean hammers:
According to legend, Pythagoras discovered that musical notes could be translated into mathematical equations when he passed blacksmiths at work one day and heard the sound of their hammers clanging against the anvils. Thinking that the sounds of the hammers were beautiful and harmonious, except for one, he rushed into the blacksmith shop and began testing the hammers.  He then realized that the tune played when the hammer struck was directly proportional to the size of the hammer and therefore concluded that music was mathematical.  However, this legend is demonstrably false, as these ratios are only relevant to string length (such as the string of a monochord), and not to hammer weight.
In the field of  astronomy
In ancient times, Pythagoras and his contemporary Parmenides of  Elea were both credited with having been the first to teach that the Earth was spherical, the first to divide the globe into five climactic zones,  and the first to identify the morning star and the evening star as the same celestial object. of the two philosophers, Parmenides has a much stronger claim to having been the first and the attribution of these discoveries to Pythagoras seems to have possibly originated from a pseudepigraphal poem.  Empedocles, who lived in Magna Graecia shortly after Pythagoras and Parmenides, knew that the earth was spherical.  By the end of the fifth century BC, this fact was universally accepted among Greek intellectuals.
Pythagorus had great interest in the properties of numbers and that he devoted considerable attention to the study of areas , volumes, properties and five regular solids. Under the influence of mysticism the pythagorus belived the unoversed to be composed of five elements and foe each of the five regular solids. Thus the earth arose from  the cube , fire came from the pyramid , air from the octahedron, water from the icosahedrons  and sphere  of the universe from the dodecahedron.

Great Mathematician : Aryabhatta

                                                                            ARYABHATA
 Aryabhata  (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya (499 CE, when he was 23 years old) and the Arya-siddhanta.
Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.
Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.
The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.

Contribution of Aryabhata in the field of  Mathematics:
Place value system and zero:
The place-value system, first seen in the 3rd-century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients.
However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.
Approximation of π
Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational "Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."
This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

Trigonometry:
In Ganitapada 6, Aryabhata gives the area of a triangle the result of a perpendicular with the half-side is the area." Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means "half-chord".
Indeterminate equations: 
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to Diophantine equations that have the form ax + by = c.

Algebra: 
In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes: 

 1²+2²+3²+----------+n² =  [n(n+1) (2n+1) ] ÷ 6

1³+2³+3³+----------+n³ =  [n(n+1)  ÷ 2]²

Astronomy:   Motions of the solar system
Aryabhata seems to ascribe the apparent motions of the heavens to the Earth's rotation. He may have believed that the planet's orbits as elliptical rather than circular.
Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view, that the sky rotated.  In the same way that someone in a boat going forward sees an unmoving (object) going backward, so (someone) on the equator sees the unmoving stars going uniformly westward.
Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles. They in turn revolve around the Earth. The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac.

Eclipses
Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by Rahu and Ketu (identified as the pseudo-planetary lunar nodes), he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth's shadow.
Honour  to Aryabhata:
ISRO built India’s 1st  satellite,  Aryabhat  which was  launched  by Soviet Union on 19th April 1975.It was named after the Mathematician Aryabhata.

Correlation of Mathematics with other School subjects:

Relation of Mathematics with other School subjects:

   Mathematics in Economics

The level of mathematical literacy required for personal and social activities is continually increasing.  Mastery of the fundamental processes is necessary for clear thinking.  The social sciences are also beginning to draw heavily upon mathematics.

Mathematical language and methods are used frequently in describing economic phenomena. According to Marshall – “The direct application of mathematical reasoning to the discovery of economic truths has recently rendered great services in the hand of master mathematicians.” Statistical methods are used in economic forecast different issues of economics can be represented statistically such as ‘Trade Cycles’, Volume trade, trend of exports and imports, population trends, industrial trends, thrift, expenditure of public money etc.,

In economic theory and econometrics, a great deal of mathematical work is being done all over the world. In econometrics, tools of matrices, probability and statistics are used. A great deal of mathematical thinking goes in the task of national economic planning, and a number of mathematical models for planning have been developed. 

  Mathematics in Political Science

In Mathematical Political Science, we analyze past election results to see changes in voting patterns and the influence of various factors on voting behavior, on switching of votes among political parties and mathematical models for Conflict Resolution. Here we make use of Game Theory. 

Mathematics in Geography

            Geography is nothing but a scientific and mathematical description of our earth in its universe.  The dimension and magnitude of earth, its situation and position in the universe the formation of days and nights, lunar and solar eclipses, latitude and longitude, maximum and minimum rainfall, etc are some of the numerous learning areas of geography which need the application of mathematics.  The surveying instruments in geography have to be mathematically accurate.  There are changes in the fertility of the soil, changes in the distribution of forests, changes in ecology etc., which have to be mathematically determined, in order to exercise desirable control over them.

Mathematics in History:

Ø  To know the time period

Ø  To know the birth and death of historical persons

Mathematics helps in Calculation of Dates like duration of Britishers ruled in India? When Gandhi ji was born? Celebrate National Days and festivals, Cost in building of Taj-Mahal. Tenure of President in India. This gives us new information of the historical world. When the First and second world wars were fought? On account of economic considerations industrial revolution in Europe.

MATHEMATICS AND LANGUAGE:

• Math and Reading:- Students read about the discoveries or work of great mathematicians, and they can make poem on numbers.
• Math and Writing (numbers are converted into writing):- A student makes the pie chart and interprets in his own words.

e.g. Counting of alphabet, vowel, Read About The Life History of Mathematicians. Student can draw make a bar graph of time spent in school and home the whole week and can interpret. (Interpretation of Non-Verbal Data)

Nature & characteristic of Science

                                                        Nature & characteristic of science

Science is a systematic method of continuing investigation that uses observation, hypothesis, testing, measurement, experimentation, logical argument and theory building to lead to more adequate explanation of natural phenomena.  

Ø Methodological activity, discipline, or study. An activity that appears to require study and method.

Ø Knowledge, especially that gained through experience.

Ø Is the ongoing process of exploring and discovering the way nature works.  We may never know everything there is to know about life and the universe, but we can continue to expand our understanding by making observations, asking questions, & seeking answers.

Science is our attempt to understand the objects and events we experience in nature and the means by which we do so.  People develop understanding about things they experience by asking question and finding answers.  What is life?  Why does my heartbeat?  Why does it pump blood?  What causes diseases such as Herpes, Diabetes, or AIDS?  Why won’t my car start?  In attempting to find answers to questions such as these, one is actually doing science.  In biology most of the questions involve living things.  Finding answers to them involves you in doing biology… the science of life.

Steps to make Theory:

Observation → Analysis the questions → Hypothesis → Prediction → Testing Hypothesis / Experimentation if it is success then becomes Theory if not the again, we have to make new Hypothesis.

Nature of Science

1.     Science is a particular way  of looking  at nature

2.     Science is  an accumulated and systematized body of knowledge

3.     Science is an interdisciplinary area of learning.

Characteristics of Science: 

1.  It is guided by natural law

–   The pursuit of scientific knowledge must be guided by the physical and chemical laws that govern the universe (state of existence).

2.  It has to be explained by referencing these natural laws.

–   Scientific knowledge must explain what is observed by reference in nature.  We cannot invoke the explanations based on supernatural deities (ghosts, angels, gremlins, fairies, etc.) miracles, or magic.

–   Science must only rely on observable, testable evidence which must either support or not support hypotheses.  Extraordinary claims require extraordinary evidence.

3.  Science is testable against the observable world. 

–   We must be able to make observations in the real world, directly or indirectly, ask questions, or form and test hypotheses = a tentative, causal explanation/answer for an observation or phenomenon.

_   We use observations and/or tests to answer questions about the natural world.

–   Science relies on observable, testable evidence, which must either support or not support hypotheses.

4.  Its conclusions are tentative, that is, are not necessarily the final word.

_   If we draw a conclusion based on some observation or test on some event, we must be ready always to discard or to modify our conclusion, if further observations falsify it.

_   Can’t be scientific if you start with a conclusion and refuse to change it regardless of the evidence developed during the course of the investigation.

5.  It is falsifiable.

-       You must be able to disprove any statement.  If there is no possibility that the statement cannot be correct, then it isn’t science.  What this means is that science will seek out errors and correct them.  Unlike other philosophies, it’s a self-correcting system.  We add to and take away information on a daily basis depending on new discoveries and new evidence. 

6.  It relies on evidence that is testable (from observations and experimentations).  If we cannot make repeated observations or experiments to gather information, then it is outside the realm of science (e.g. UFO’s, haunted houses, etc.). 

7.     Science is logical & rational

8.     Science makes well-defined claims

9.     Scientific experiments are repeatable

10.   Science insists that extraordinary claims require extraordinary evidence

Audio-Visual Aids in teaching mathematics

Audio-Visual Aids in teaching mathematics

“Beauty attract to the person”

Popular saying on Audio- visual Aids:

The thing which I hear, I may forgot

          The thing which I see, I may remembered

      The thing which I do, I cannot  forget

 The use of Audio-Visual aids in teaching is not a fashion but is a matter belief and actual practice. That to say of ordinary visual aids e.g. charts. Graphs, map, models, etc. they are using films film-strips, epidiascope, tape-recorder, radio and television to make education valuable and worthwhile.

 Audio-Visual Aids an Fit Well in:

 a) Traditional system (from primary stage to higher secondary)

 b) Basic system of education

c) Project method kindergarten Montessori etc.

Features of Audio-Visual Aids:

 (i) Arouses interest,  (ii) Modifies attitude,  (iii) Claries concepts,  (iv) Stimulates thinking,  (v) Summarizes contents,  (vi) Demonstrates knowledge and  (vii) Concretizes knowledge.

 We  should  use to  Audio-visual packages to teach abstract and difficult oriented mathematical concepts, to enhance easy retention and high academic performance of our students. It will be  effective and efficient in the  classroom teaching-learning process.