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maths

Mathematics (colloquially, maths, or math in American English) is the body of knowledge centered on concepts such as quantity, structure, space, and change, and the academic discipline which studies them; Benjamin Peirce called it "the science that draws necessary conclusions".[1] It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.[2] Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in ancient mathematical texts originating in ancient Egypt, Mesopotamia, Ancient India, and Ancient China with increased rigour later introduced by the ancient Greeks. From this point on, the development continued in short bursts until the Renaissance period of the 16th century where mathematical innovations interacted with new scientific discoveries leading to an acceleration in understanding that continues to the present day.[3] Today, mathematics is used throughout the world in many fields, including science, engineering, medicine and economics. The application of mathematics to such fields, often dubbed applied mathematics, inspires and makes use of new mathematical discoveries and has sometimes led to the development of entirely new disciplines. Mathematicians also engage in pure mathematics for its own sake without having any practical application in mind, although applications for what begins as pure mathematics are often discovered later.[4]
EtymologyThe word "mathematics" (Greek: μαθηματικά) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art. The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical".[5] Despite the form and etymology, the word mathematics, like the names of arts and sciences in general, is used as a singular mass noun in English today. The colloquial English-language shortened forms perpetuate this singular/plural idiosyncrasy, as the word is shortened to math in North American English, while it is maths elsewhere (including Britain, Ireland, Australia and other Commonwealth countries).
History

A quipu, a counting device used by the Inca.
Main article: History of mathematicsThe evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges have something in common was a breakthrough in human thought. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like timedays, seasons, years. Arithmetic (addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[6]
Inspiration, pure and applied mathematics, and aesthetics

Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.
Main article: Mathematical beautyMathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton was one of the infinitesimal calculus inventors, Feynman invented the Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.
Notation, language, and rigor

In modern notation, simple expressions can describe complex concepts. This image is generated by a single equation.
Main article: Mathematical notationMost of the mathematical notation we use today was not invented until the 16th century.[7] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way. Mathematical language also is hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. It was said that Henri Poincaré was only elected to the Académie française so that he could tell them how to define automorphe in their dictionary.[citation needed] But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor". Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[8] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous. Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Mathematics as science

Carl Friedrich Gauss, while known as the "prince of mathematicians", did not believe that mathematics was worthy of study in its own right[citation needed].Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[9] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[10] Many philosophers believe that mathematics is not experimentally falsifiable,[citation needed] and thus not a science according to the definition of Karl Popper. However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[11] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.[12] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right. The opinions of mathematicians on this matter are varied. While some in applied mathematics feel that they are scientists, those in pure mathematics often feel that they are working in an area more akin to logic and that they are, hence, fundamentally philosophers. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics. Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[13][14] established in 1936 and now awarded every 4 years. It is usually considered the equivalent of science's Nobel prize. Another major international award, the Abel Prize, was introduced in 2003. Both of these are awarded for a particular body of work, either innovation in a new area of mathematics or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.
Fields of mathematics

Early mathematics was entirely concerned with the need to perform practical calculations, as reflected in this Chinese abacus.As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.
QuantityThe study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Number theory also holds two widely-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture. As the number system is further developed, the integers are recognised as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalised to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalise the concept of counting to infinite. Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
Natural numbers
Integers
Rational numbers
Real numbers
Complex numbers
StructureMany mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change.




Number theory
Abstract algebra
Group theory
Order theory
SpaceThe study of space originates with geometry - in particular, Euclidean geometry. Trigonometry combines space and number, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.




Geometry
Trigonometry
Differential geometry
Topology
Fractal geometry
ChangeUnderstanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.




Calculus
Vector calculus
Differential equations
Dynamical systems
Chaos theory
Foundations and philosophyIn order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studying the results of such a framework. As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that there are always true theorems which cannot be proven. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.

Mathematical logic
Set theory
Category theory
Discrete mathematicsDiscrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. This includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model - the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically soluble by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy. As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems. [15] It is widely believed that the answer to this problem is no. [16]



Combinatorics
Theory of computation
Cryptography
Graph theory
Applied mathematicsApplied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation.
Mathematical physicsAnalytical mechanicsMathematical fluid dynamicsNumerical analysisOptimizationProbabilityStatisticsMathematical economics • Financial mathematics • Game theoryMathematical biologyCryptographyOperations research
Common misconceptionsMathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
misunderstanding of the implications of mathematical rigor;
attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.
Relationship between mathematics and physical realityMathematical concepts and theorems need not correspond to anything in the physical world। Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while most systems of axioms are derived from our perceptions and experiments, they are not dependent on them. Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led Eugene Wigner to write an essay,
piyushdadriwala

मथ्स formula

Mensuration
Area of a Triangle, (sides a,b,c).. Area = ( b . c sin A )/ 2
Area of a Triangle , s = (a + b + c)/ 2.. Area = Sqrt (s .(s - a). (s - b). (s - c))
Area of a Circle (r = radius).. Area = π . r 2
Area and Volume of a Cylinder
Area and Volume of a Cone
Area and Volume of a Frustrum of a Cone
Area and Volume of a Sphere
Area and Volume of a Pyramid
Trigonometry...
Definitions...Sin A = Opposite / Hypotenuse = a / c Cosine A = Adjacent / Hypotenuse = b / c Tangent A = Opposite / Adjacent = a / bCosecant A = 1 / sin = c / a Secant A = 1/cosine = b / c Cotangent A = 1/tangent. = b / a Trigonometric Relations...Sin ( - A) = - Sin (A)Cos ( - A) = cos (A)Sin (A) 2 + Cos A 2 = 1 Cos (A) 2 =(1 + Cos (2A) ) /2 Sin (A) 2 =(1 - Cos (2A) ) /2 Sin (A) Cos(A) = Sin (2A) /2 Sin (A + / - B) = Sin (A) Cos(B) +/ - Cos(A) Sin(B) Cos (A + / - B) = Cos (A) Cos(B) - /+ Sin(A) Sin(B) 1 + tan(A) 2 = sec(A) 21 + cot(A) 2 = cosec(A) 2
Hyperbolic Functions
sinh x = (e x - e - x) / 2cosh x = (e x + e - x) / 2tanh x = sinh x / cosh x = (e x - e - x) / (e x + e - x)sech x = 1 / sinh x = 2 / (e x - e - x) cosech x = 1 / cosh x = 2 / (e x + e - x)coth x = cosh x / sinh x = (e x + e - x) / (e x - e - x)
ejx = cos x + j sin xex = cosh x + j sinh xsin x = (e jx - e - jx) /2.jcos x = (e jx + e - ix) /2sin jx = j.sinh xcos jx = cohs x
Quadratic EquationA quadratic equation is generally of the form...
ax2 +b x + c = 0The general solution of this equation is
x = ( - b ± Ö (b2 - 4 a c ) /2a
Expansions
sin x = x / 1 - x3/3! + x 5/5! - x 7 / 7! +
cos x = 1 - x 2/2! + x 4/4! - x 6/6!...
ex = 1 + x / 1 + x2/2! + x3/3! +x 4/4!...
sinh x = x / 1! + x 3/3! + x 5/5! + x7 / 7! +
cosh x = 1 + x 2/2! + x 4/4! +x 6/6!...
log(1+ x ) = x - x 2/2! + x 3/3! - x 4/4! + ...
( x + 1)n = 1 + n . x + n .( n - 1 ) x 2 / 2! + n .( n - 1 ). ( n - 2 ) x3 / 3! + ...(n / r ) x r +... ................................( for x < 1 and all real n; all x, n a positive integer)
Derivatives..
f (x)
f '(x)= df(x) / dx
sin x
cos x
cos x
- sin x
tan x
sec2 x
cotan x
- cosec2 x
sec x
sec x. tan x
cosec x ...
- cosec x. cot x.
sinh x ...
cosh x.
cosh x ...
sinh x.
tanh x ...
sech2 x.
cosech x ...
- coth x cosech x
sech x ...
- tanh x sech x
coth x ...
- cosech 2 x
u .v
u . dv/dx + v . du/dx
u / v
(v . du / dx - u . dv / dx ) / v 2
a. x n
a. n . x n - 1
e a x
a . e a x
a x
a x. ln a
x x
x x /(1 + ln x)
ln x
1 / x
log a x
1 / x . log a e
sin - 1( x /a)
1 / Sqrt(a 2 - x 2 )
cos - 1( x /a)
- 1 / Sqrt(a 2 - x 2 )
tan - 1( x /a)
a / (a 2 + x 2 )
Indefinite Integrals..
f(x)
The constant of integration C is ommitted from the table of indefinite integrals below


xa
x a+1 / (a + 1)

1 / (x 2 + a2)
(1 / a) . tan - 1 (x / a)

1 / (x 2 - a2)
(1 /2 a) . ln ( ( x - a ) /(x + a))

( a + b x ) n (n not - 1)
(a + b x)n + 1 / b (n + 1)

( a + b x ) - 1
1 / b .ln ( a + b x )

x / (a x +b)
(a x + b - b ln(ax +b) ) / a2

1 / x
ln x

1 / Sqrt (x 2 - a 2)
cosh - 1 (x / a)

1 / Sqrt (x 2 + a 2)
sinh - 1 (x / a)

ex
ex

1 / ( a2 - x2)
(1 / a). tanh - 1 ( x / a ) = 1 /( 2. a) . log(a + x/a - x )

ax
ax / ln a

x ax
(a x / ln a ) - (a x /( ln a ) 2 )

x ea x
e a x (a x - 1) / a2

1 /(a + b e c x )
(x / a) - ln (a + b ec x ) / a c

ln x
x (ln x - 1 )

( ln x )2
x [ (ln x )2 - 2 ln x +2 ]

1 / x ln x
ln ( ln x )

sin x
- cos x

cos x
sin x

tan x
- ln cos x

cotan x
ln sin x

sec x
ln ( sec x + tan x ) = ln (tan (x/ 2 + π/ 4) )

cosec x
log tan x/ 2

1 / Sqrt( x 2 + a 2)
sinh - 1( x / a ) = log ( (x/a) +Sqrt(x2 /a2 +1))

1 / Sqrt( x2 - a 2)
cosh - 1( x / a )

1 / Sqrt( a2 - x 2)
sin - 1( x / a )

sinh x
cosh x

cosh x
sinh x

tanh x
ln cosh x

cosech x
ln tanh (x / 2 )

sech x
tan - 1 ( sinh x )

coth x
ln sinh x

sinh 2 x
( - x + ( sinh (2 x)) /2 ) / 2

cosh 2 x
( x + ( sinh (2 x) ) /2 ) / 2

sech 2 x
tanh x

cosech 2 x
- coth x

tanh 2 x
x - tanh x
Moments Of Inertia of Plane SectionsI = moment of Inertia about the identified axis.J = Polar moment of inertia about the centroid of section
For More detailed information refer..
Properties of Plane Areas Properties of solids
Parallel axis Theory.. If the second moment of an area (A) about an axis x - x = I xx. Then the second moment of Area about a parallel axis y - y which is distance x from x - x =
I yy = I xx + A . x 2
Complex Numbers..
In mathematics it is necessary to provide a method of identifying the root of a negative number i.e p = √ ( - 4). p is clearly not real number it is an imaginary number. Again an equation x2 - 2x +5 = 0 results in (x - 1)2 = - 4 so that (x - 1) = ± √ ( - 4). The roots are therefore x = 1 - √ ( - 4), and 1 + √ ( - 4), These roots which are a combination of a real number and an imaginary number are called complex numbers.The symbol i (j in electical work) is used to represent √ - 1. Therefore √ ( - 4) = 2i. The number i, or 1i , or xi are called purely imaginary numbers. The complex number solution of the above equation = and 1 + 2i and 1 - 2i,Powers of complex are identified below
i 1 =
+i

i - 1 =
- i
i 2 =
- 1

i - 2 =
- 1
i 3 =
- i

i - 3 =
+i
i 4 =
+1

i - 4 =
- 1
...
...
i...
...
i 5 =
+i

i - 5 =
- i
etc .
For two complex numbers (a 1 + ib 1) & (a 2 +ib 2)to be equal it can be easily proved that a 1 must equal a 2, and b 1 must equal b 2Complex numbers are conveniently represented using an argand diagrams as shown below.
Complex numbers can be manipulated using the Cartesian system as follows;
z = a + i bz 1 + z 2 = (a 1 + a 2) + i (b 1 + b 2)z 1 - z 2 = (a 1 + a 2) - i (b 1 + b 2)z 1 . z 2 = (a 1 . a 2 - b 1 .b 2) + i (a 1 . b 2 + a 2 .b a)a2 + a2 = r2 = (a +ib) (a - ib)
Complex numbers can be manipulated using the polar co - ordinate system as follows;
z = a + i b = r (cos φ + i। sin φ )r = √(a2 + b2 ),φ = arctan (b/a) = tan - 1 (b/a)sin φ = b/r, cos φ = a/r, tan φ = b/az 1. z 2 = r 1.r 2 [cos (φ 1 + φ 2 ) + i (sin (φ 1 + φ 2 ) ]z n = r n [cos (n φ) + i sin (n φ) ] z > 0 , Integere i φ = cos ( φ) + i sin ( φ ) Eulers formula i φ =ln [ (cos ( φ) + i sin ( φ ) ]e - i φ = cos ( φ) - i sin ( φ ) = 1/ [ cos ( φ) + i sin ( φ ) ].
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Albert Einstein
March 14, 1879 - April 18, 1955Physicist and MathematicianNobel Laureate for Physics 1921
"There are only two ways to live your life. One is as though nothing is a miracle. The other is as if everything is."- Albert Einstein -
Albert Einstein was a German-born theoretical physicist who is widely considered one of the greatest physicists of all time.
While best known for the theory of relativity (and specifically mass-energy equivalence, E=mc2), he was awarded the 1921 Nobel Prize in Physics for his 1905 (Annus Mirabilis) explanation of the photoelectric effect and "for his services to Theoretical Physics". In popular culture, the name "Einstein" has become synonymous with great intelligence and genius. Einstein was named Time magazine's "Man of the Century."
He was known for many scientific investigations, among which were: his special theory of relativity which stemmed from an attempt to reconcile the laws of mechanics with the laws of the electromagnetic field, his general theory of relativity which extended the principle of relativity to include gravitation, relativistic cosmology, capillary action, critical opalescence, classical problems of statistical mechanics and problems in which they were merged with quantum theory, leading to an explanation of the Brownian movement of molecules; atomic transition probabilities, the probabilistic interpretation of quantum theory, the quantum theory of a monatomic gas, the thermal properties of light with a low radiation density which laid the foundation of the photon theory of light, the theory of radiation, including stimulated emission; the construction of a unified field theory, and the geometrization of physics.
Einstein was born on March 14, 1879, to a Jewish family, in Ulm, Württemberg, Germany. His father was Hermann Einstein, a salesman who later ran an electrochemical works, and his mother was Pauline née Koch. They were married in Stuttgart-Bad Cannstatt.
At his birth, Albert's mother was reputedly frightened that her infant's head was so large and oddly shaped. Though the size of his head appeared to be less remarkable as he grew older, it's evident from photographs of Einstein that his head was disproportionately large for his body throughout his life, a trait regarded as "benign macrocephaly" in large-headed individuals with no related disease or cognitive deficits. His parents also worried about his intellectual development as a child due to his initial language delay and his lack of fluency until the age of nine, though he was one of the top students in his elementary school.
In 1880, shortly after Einstein's birth the family moved to Munich, where his father and his uncle founded a company manufacturing electrical equipment (Elektrotechnische Fabrik J. Einstein & Cie). This company provided the first lighting for the Oktoberfest as well as some cabling in the suburb of Schwabing.
Albert's family members were all non-observant Jews and he attended a Catholic elementary school. At the insistence of his mother, he was given violin lessons. Though he initially disliked the lessons, and eventually discontinued them, he would later take great solace in Mozart's violin sonatas.
When Einstein was five, his father showed him a small pocket compass, and Einstein realized that something in "empty" space acted upon the needle; he would later describe the experience as one of the most revelatory events of his life. He built models and mechanical devices for fun and showed great mathematical ability early on.
In 1889, a medical student named Max Talmud (later: Talmey), who regularly visited the Einsteins, introduced Einstein to key science and philosophy texts, including Kant's Critique of Pure Reason.
Einstein attended the Luitpold Gymnasium, where he received a relatively progressive education. In 1891, he taught himself Euclidean geometry from a school booklet and began to study calculus; Einstein realized the power of deductive reasoning from Euclid's Elements, which Einstein called the "holy little geometry book" (given by Max Talmud). At school, Einstein clashed with authority and resented the school regimen, believing that the spirit of learning and creative thought were lost in such endeavors as strict rote learning.
From 1894, following the failure of Hermann Einstein's electrochemical business, the Einsteins moved to Milan and proceeded to Pavia after a few months. Einstein's first scientific work, called "The Investigation of the State of Aether in Magnetic Fields", was written contemporaneously for one of his uncles. Albert remained in Munich to finish his schooling, but only completed one term before leaving the gymnasium in the spring of 1895 to join his family in Pavia. He quit a year and a half before the final examinations, convincing the school to let him go with a medical note from a friendly doctor, but this meant that he had no secondary-school certificate. That same year, at age 16, he performed a famous thought experiment by trying to visualize what it would be like to ride alongside a light beam. He realized that, according to Maxwell's equations, light waves would obey the principle of relativity: the speed of the light would always be constant, no matter what the velocity of the observer. This conclusion would later become one of the two postulates of special relativity.
Rather than pursuing electrical engineering as his father intended for him, he followed the advice of a family friend and applied at the Federal Polytechnic Institute in Zurich in 1895. Without a school certificate he had to take an admission exam, which he - at the age of 16 being the youngest participant ­ did not pass. He had preferred travelling in northern Italy over the required preparations for the exam. Still, he easily passed the science part, but failed in general knowledge.
After that he was sent to Aarau, Switzerland to finish secondary school. He lodged with Professor Jost Winteler's family and became enamoured with Sofia Marie-Jeanne Amanda Winteler, commonly referred to as Sofie or Marie, their daughter and his first sweetheart. Einstein's sister, Maja, who was perhaps his closest confidant, was to later marry their son, Paul. While there, he studied Maxwell's electromagnetic theory and received his diploma in September 1896. Einstein subsequently enrolled at the Federal Polytechnic Institute in October and moved to Zurich, while Marie moved to Olsberg, Switzerland for a teaching post. The same year, he renounced his Württemberg citizenship to avoid military service.
In the spring of 1896, Mileva Maric started as a medical student at the University of Zurich, but after a term switched to the Federal Polytechnic Institute. She was the only woman to study in that year for the same diploma as Einstein. Maric's relationship with Einstein developed into romance over the next few years, though his mother objected because she was too old, not Jewish, and physically defective.
In 1900, Einstein was granted a teaching diploma by the Federal Polytechnic Institute. Einstein then submitted his first paper to be published, on the capillary forces of a straw, titled "Consequences of the observations of capillarity phenomena". In this paper his quest for a unified physical law becomes apparent, which he followed throughout his life. Through his friend Michele Besso, Einstein was presented with the works of Ernst Mach, and would later consider him "the best sounding board in Europe" for physical ideas. Einstein and Maric had a daughter, Lieserl Einstein, born in January 1902. Her fate is unknown; some believe she died in infancy, while others believe she was given out for adoption.
Works and Doctorate
Einstein could not find a teaching post upon graduation, mostly because his brashness as a young man had apparently irritated most of his professors. The father of a classmate helped him obtain employment as a technical assistant examiner at the Swiss Patent Office[8] in 1902. His main responsibility was to evaluate patent applications relating to electromagnetic devices. He also learned how to discern the essence of applications despite sometimes poor descriptions, and was taught by the director how "to express [him]self correctly". He occasionally corrected their design errors while evaluating the practicality of their work.
His friend from Zurich, Michele Besso, also moved to Bern and took a job at the patent office, and he became an important sounding board. Einstein also joined with two friends he made in Bern, Maurice Solovine and Conrad Habicht, to create a weekly discussion club on science and philosophy, which they grandly and jokingly named "The Olympia Academy." Their readings included Poincare, Mach, Hume, and others who influenced the development of the special theory of relativity.
Einstein married Mileva Maric on January 6, 1903. Einstein's marriage to Maric who was a mathematician, was both a personal and intellectual partnership: Einstein referred to Mileva as "a creature who is my equal and who is as strong and independent as I am". Ronald W. Clark, a biographer of Einstein, claimed that Einstein depended on the distance that existed in his marriage to Mileva in order to have the solitude necessary to accomplish his work; he required intellectual isolation. In an obituary of Einstein Abram Joffe wrote: "The author of [the papers of 1905] wasŠ a bureaucrat at the Patent Office in Bern, Einstein-Maric which has been taken as evidence of a collaborative relationship. However, most probably Joffe referred to Einstein- Maric ecause he believed that it was a Swiss custom at the time to append the spouse's surname to the husband's name. The extent of her influence on Einstein's work is a controversial and debated question.
In 1903, Einstein's position at the Swiss Patent Office had been made permanent, though he was passed over for promotion until he had "fully mastered machine technology". He obtained his doctorate under Alfred Kleiner at the University of Zürich after submitting his thesis "A new determination of molecular dimensions" ("Eine neue Bestimmung der Moleküldimensionen") in 1905.
During 1905, in his spare time, he wrote four articles that participated in the foundation of modern physics, without much scientific literature he could refer to or many fellow scientists with whom he could discuss the theories. Most physicists agree that three of those papers (on Brownian motion, the photoelectric effect, and special relativity) deserved Nobel Prizes. Only the paper on the photoelectric effect would be mentioned by the Nobel committee in the award; at the time of the award, it had the most unchallenged experimental evidence behind it, although the Nobel committee expressed the opinion that Einstein's other work would be confirmed in due course.
Some might regard the award for the photoelectric effect ironic, not only because Einstein is far better-known for relativity, but also because the photoelectric effect is a quantum phenomenon, and Einstein became somewhat disenchanted with the path quantum theory would take.
Einstein submitted this series of papers to the "Annalen der Physik". They are commonly referred to as the "Annus Mirabilis Papers" (from Annus mirabilis, Latin for 'year of wonders')।
In the last years of Albert Einstein's life, he amused himself by telling jokes to his parrot, and avoided visitors by feigning illness, according to a newly discovered diary written by the woman known around Princeton as his last girlfriend. While Einstein also talked about the travails of his continuing work in physics, most of Johanna Fantova's diary recalls his views on world politics and his personal life. The writings are an unvarnished portrait of Einstein struggling bravely with the manifold inconveniences of sickness and old age, Freeman Dyson, a mathematician at the Institute for Advanced Study in Princeton, told The New York Times in Saturday¹s editions.
The 62-page diary, written in German, was discovered in February in Fantova¹s personnel files at Princeton University¹s Firestone Library, where she had worked as a curator. The manuscript is the subject of an article to be published next month in The Princeton University Library Journal. According to the article, the new manuscript is the only one kept by someone close to Einstein in the final years of his life.
"There is surprisingly little about physics in the diary," Donald Skemer, Firestone Library¹s curator of manuscripts, told The Times of Trenton. Fantova wrote that she recorded her time with the renowned physicist to "cast some additional light on our understanding of Einstein, not on the great man who became a legend in his lifetime, not on Einstein the renowned scientist, but on Einstein the humanitarian." Fantova was 22 years younger than Einstein. Although the two spent considerable time together starting in the 1940s, her journal only records their relationship from October 1953 until his death in April 1955 at age 76. She died in 1981 at age 80.
Princeton already had a collection of the poems, letters and photos Einstein sent to Fantova, who sold them after his death to Gillett G. Griffin, a retired curator at Princeton¹s Art Museum. He gave those documents to the library. Griffin, invited many times to Einstein¹s home for dinner, said Fantova was a fixture there. "Reading what she left gives me an immediate connection with my own experience and gives everyone the immediacy of knowing Einstein himself," Griffin said.
The diary recounts Einstein speaking about the politics of the day and portrays him as critical of speeches of Adlai Stevenson, the nuclear arms race and the anti-communist attack on the scientist J. Robert Oppenheimer by Sen. Joseph McCarthy. "This political persecution of his associate was a source of bitter disillusionment," Fantova wrote. Besides his politics, Fantova wrote of Einstein's popularity and how he tried to write back to strangers, some of whom tried to convert him to Christianity. He said, "All the maniacs in the world write to me," she wrote.
Lighter moments recounted The diary also recounts how, on his 75th birthday, Einstein received a parrot as gift. After deciding the bird was depressed, Einstein tried alter its mood by telling bad jokes. At times, Einstein would pretend to be sick in bed so he would not have to pose with visitors who wanted photographs. Einstein still enjoyed himself even when real illness did take hold. Einstein¹s health began to fail, but he continued to indulge in what remained his favorite of all pastimes, sailing. Seldom did I see him so gay and in so light a mood as in this strangely primitive little boat, Fantova wrote. Einstein also wrote Fantova poems, some of which are in the diary.
Einstein, with his second wife Elsa, had arrived in Princeton in 1933 at the newly formed Institute for Advanced Study. Elsa died three years later. Fantova first met Einstein in 1929 in Berlin. She arrived in the United States alone in 1939 and, at Einstein¹s urging, attended library school at the University of North Carolina.
ESP lab sees doors close Guardian - February 12, 2007 "A laboratory dedicated to extra-sensory perception and telekinesis at the prestigious Princeton University in New Jersey is to close after nearly 30 years of research."
Einstein the Greatest
November 29, 1999 - BBC
Albert Einstein has been voted the greatest physicist of all time in an end of the millennium poll, pushing Sir Isaac Newton into second place.
The survey was conducted among 100 of today's leading physicists.
All-time top ten:
1. Albert Einstein2. Isaac Newton3. James Clerk Maxwell4. Niels Bohr5. Werner Heisenberg6. Galileo Galilei7. Richard Feynman8. Paul Dirac9. Erwin Schrödinger10. Ernest Rutherford
"Einstein's special and general theories of relativity completely overturned previous conceptions of a universal, immutable space and time, and replaced them with a startling new framework in which space and time are fluid and malleable," said physicist Brian Greene from Columbia University, US, who participated in the poll for Physics World magazine.
Peter Rodgers, Editor of Physics World, said: "Einstein and Newton were always going to be one and two but what was surprising about the top 10 was that there were seven out and out theorists."
The top 10 includes three British scientists: Newton, James Clerk Maxwell and Paul Dirac. New Zealander Ernest Rutherford, who did much of his work in the UK, also makes the list, at 10.
Hawking and Archimedes
A parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton and also included Michael Faraday.
Neither list included any living scientist, but Stephen Hawking was rated at 16 by PhysicsWeb users, just behind Archimedes.
Paul Guinnessy, editor of PhysicsWeb, said: "My two biggest surprises were the inclusion of Stephen Hawking, as I think more time is needed to see whether his scientific contributions will last, and the low number of votes for Marie Curie and Ernest Rutherford.
"Both these physicists had a dramatic impact not only on scientific achievements but in the students they taught and drew into physics. Rutherford's lab in particular had a number of students who were awarded Nobel prizes at a later date."
Big science
The three most important discoveries in physics are quantum mechanics, Einstein's theory of general relativity and Newton's mechanics and gravitation.
Quantum computation pioneer David Deutsch of Oxford University said: "In each of these three cases, the discovery in question not only revolutionised the branch of physics that it nominally addressed, but also provided a framework so deep and universal that all subsequent theories in physics have been formulated within it."
Asked about their careers, the physicists said they were mostly happy. Over 70% of respondents said they would study physics if they were starting university this year. But 17% said they would not, with one Japanese researcher commenting: "I worked too hard. I want to enjoy life next time."
However, asked for the biggest problem in physics, one respondent joked "getting tenure or quantum gravity".
As is traditional, the physicists had a high opinion of their subject, calling it "the most grandiose science", "the most fascinating activity for our brain" and "still the most fundamental of all sciences".
But the biological sciences did appeal to some. Michael Green, a particle theorist at Cambridge University, said: "There is something attractive about a subject that is still in a relatively primitive state."
Einstein's brain found to be anatomically distinct


Einstein allowed his brain to be studied after his death
AP - June 17, 1999
We always thought something must have made Albert Einstein smarter than the rest of us. Now, scientists have found that one part of his brain was indeed physically extraordinary.
In the only study ever conducted of the overall anatomy of Einstein's brain, scientists at McMaster University in Ontario, Canada, discovered that the part of the brain thought to be related to mathematical reasoning - the inferior parietal region - was 15 percent wider on both sides than normal.
Furthermore, they found that the groove that normally runs from the front of the brain to the back did not extend all the way in Einstein's case. That finding could have applications even to those with more pedestrian levels of intelligence.

Einstein thought in images
"That kind of shape was not observed in any one of our brains and is not depicted in any atlas of the human brain," said Sandra Witelson, a neuroscientist who led the study, published in this week's issue of The Lancet, a British medical journal.
"But it shouldn't be seen as anatomy is destiny," she added. "We also know that environment has a very important role to play in learning and brain development. But what this is telling us is that environment isn't the only factor."
The findings may point to the importance of the inferior parietal region, Witelson said.
While the differences may be extraordinary between Einstein and everyone else, there may be more subtle, even microscopic, differences when the anatomies of the brains of people who don't fall into the genius category are compared with each other, she said.
The researchers compared the founder of the theory of relativity's brain with the preserved brains of 35 men and 56 women known to have normal intelligence when they died.
With the men's brains, they conducted two separate comparisons - first between Einstein's brain and all the men, and next between his brain and those of the eight men who were similar in age to Einstein when they died.
They found that, overall, Einstein's brain was the same weight and had the same measurements from front to back as all the other men, which Witelson said confirms the belief of many scientists that focusing on overall brain size as an indicator of intelligence is not the way to go.
Witelson theorized that the partial absence of the groove in Einstein's brain may be the key, because it might have allowed more neurons in this area to establish connections between each other and work together more easily.
She said it is likely that the groove, known as the sulcus, was always absent in that part of Einstein's brain, rather than shrinking away as a result of his intelligence, because, as one of the two or three landmarks in the human brain, it appears very early in life.
"We don't know if every brilliant physicist and mathematician will have this same anatomy," Witelson said. "It fits and it makes a compelling story, but it requires further proof."
John Gabrieli, an associate professor of psychology at Stanford University who was not connected with the study, said the finding relating to the groove and connections between the neurons in the brain may be the key.
"We don't have a clue, so anything that is suggested is interesting," he said. "There must have been something about his brain that made him so brilliant."
Brilliance of the kind Einstein possessed is so extreme, however, that although the findings may give a clue to the neurology of genius, whether they could apply to normal differences in intelligence is more doubtful, Gabrieli said.
Witelson said the next stage is to scan the brains of living mathematicians and look for minute differences.
Witelson and her team acquired Einstein's brain after they were contacted by its keeper, scientist John Harvey, who had read about the university's brain research.
Harvey was a pathologist working at a small hospital in Princeton, N.J., when Einstein died in 1955 at the age of 76. Harvey performed the autopsy, determined Einstein died of natural causes and took the brain home with him.
Some parts of the brain were given to scientists, but no major study was ever conducted, until now.
The Einstein Cross Gravitational Lens - NASA
Albert Einstein - Wikipedia
Another Biography
Einstein Photos
NOVA Online
Einstein Archives Online
Einstein speaking about the Holocaust: (MP3 File)
Einstein's Theories on Science and Religion - 1941

Quote about Crustal Displacement
"In a polar region there is a continual deposition of ice, which is not symmetrically distributed about the pole. The earth's rotation acts on these unsymmetrically deposited masses [of ice], and produces centrifugal momentum that is transmitted to the rigid crust of the earth. The constantly increasing centrifugal momentum produced in this way will, when it has reached a certain point, produce a movement of the earth's crust over the rest of the earth's body, and this will displace the polar regions toward the equator. " From The Path of the Pole by Charles Hapgood.
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थॉमस अल्वा edison

Thomas Alva Edison
Thomas Alva Edison - born February 11, 1847, Milan, Ohio, U.S. d. Oct. 18, 1931, West Orange, N.J. American inventor who, singly or jointly, held a world record 1,093 patents. In addition, he created the world's first industrial research laboratory.
Edison was the quintessential American inventor in the era of Yankee ingenuity.
He began his career in 1863, in the adolescence of the telegraph industry, when virtually the only source of electricity was primitive batteries putting out a low-voltage current.
Before he died, in 1931, he had played a critical role in introducing the modern age of electricity. From his laboratories and workshops emanated the phonograph, the carbon-button transmitter for the telephone speaker and microphone, the incandescent lamp, a revolutionary generator of unprecedented efficiency, the first commercial electric light and power system, an experimental electric railroad, and key elements of motion-picture apparatus, as well as a host of other inventions.
Edison was the seventh and last child--the fourth surviving--of Samuel Edison, Jr., and Nancy Elliot Edison. At an early age he developed hearing problems, which have been variously attributed but were most likely due to a familial tendency to mastoiditis. Whatever the cause, Edison's deafness strongly influenced his behaviour and career, providing the motivation for many of his inventions.
Early years
In 1854 Samuel Edison became the lighthouse keeper and carpenter on the Fort Gratiot military post near Port Huron, Mich., where the family lived in a substantial home. Alva, as the inventor was known until his second marriage, entered school there and attended sporadically for five years. He was imaginative and inquisitive, but because much instruction was by rote and he had difficulty hearing, he was bored and was labeled a misfit.
To compensate, he became an avid and omnivorous reader. Edison's lack of formal schooling was not unusual. At the time of the Civil War the average American had attended school a total of 434 days--little more than two years' schooling by today's standards.
In 1859 Edison quit school and began working as a trainboy on the railroad between Detroit and Port Huron. Four years earlier, the Michigan Central had initiated the commercial application of the telegraph by using it to control the movement of its trains, and the Civil War brought a vast expansion of transportation and communication. Edison took advantage of the opportunity to learn telegraphy and in 1863 became an apprentice telegrapher.
Messages received on the initial Morse telegraph were inscribed as a series of dots and dashes on a strip of paper that was decoded and read, so Edison's partial deafness was no handicap. Receivers were increasingly being equipped with a sounding key, however, enabling telegraphers to "read" messages by the clicks.
The transformation of telegraphy to an auditory art left Edison more and more disadvantaged during his six-year career as an itinerant telegrapher in the Midwest, the South, Canada, and New England. Amply supplied with ingenuity and insight, he devoted much of his energy toward improving the inchoate equipment and inventing devices to facilitate some of the tasks that his physical limitations made difficult.
By January 1869 he had made enough progress with a duplex telegraph (a device capable of transmitting two messages simultaneously on one wire) and a printer, which converted electrical signals to letters, that he abandoned telegraphy for full-time invention and entrepreneurship.
Edison moved to New York City, where he initially went into partnership with Frank L. Pope, a noted electrical expert, to produce the Edison Universal Stock Printer and other printing telegraphs. Between 1870 and 1875 he worked out of Newark, N.J., and was involved in a variety of partnerships and complex transactions in the fiercely competitive and convoluted telegraph industry, which was dominated by the Western Union Telegraph Company. As an independent entrepreneur he was available to the highest bidder and played both sides against the middle. During this period he worked on improving an automatic telegraph system for Western Union's rivals.
The automatic telegraph, which recorded messages by means of a chemical reaction engendered by the electrical transmissions, proved of limited commercial success, but the work advanced Edison's knowledge of chemistry and laid the basis for his development of the electric pen and mimeograph, both important devices in the early office machine industry, and indirectly led to the discovery of the phonograph.
Under the aegis of Western Union he devised the quadruplex, capable of transmitting four messages simultaneously over one wire, but railroad baron and Wall Street financier Jay Gould, Western Union's bitter rival, snatched the quadruplex from the telegraph company's grasp in December 1874 by paying Edison more than $100,000 in cash, bonds, and stock, one of the larger payments for any invention up to that time. Years of litigation followed.
Menlo Park
Although Edison was a sharp bargainer, he was a poor financial manager, often spending and giving away money more rapidly than he earned it.
In 1871 he married 16-year-old Mary Stilwell, who was as improvident in household matters as he was in business, and before the end of 1875 they were in financial difficulties.
To reduce his costs and the temptation to spend money, Edison brought his now-widowed father from Port Huron to build a 2 1/2-story laboratory and machine shop in the rural environs of Menlo Park, N.J.--12 miles south of Newark--where he moved in March 1876.
Accompanying him were two key associates, Charles Batchelor and John Kruesi.
Batchelor, born in Manchester in 1845, was a master mechanic and draftsman who complemented Edison perfectly and served as his "ears" on such projects as the phonograph and telephone. He was also responsible for fashioning the drawings that Kruesi, a Swiss-born machinist, translated into models.
Edison experienced his finest hours at Menlo Park. While experimenting on an underwater cable for the automatic telegraph, he found that the electrical resistance and conductivity of carbon (then called plumbago) varied according to the pressure it was under.
This was a major theoretical discovery, which enabled Edison to devise a "pressure relay" using carbon rather than the usual magnets to vary and balance electric currents.
In February 1877 Edison began experiments designed to produce a pressure relay that would amplify and improve the audibility of the telephone, a device that Edison and others had studied but which Alexander Graham Bell was the first to patent, in 1876.
By the end of 1877 Edison had developed the carbon-button transmitter that is still used in telephone speakers and microphones.
The phonograph
Edison invented many items, including the carbon transmitter, in response to specific demands for new products or improvements. But he also had the gift of serendipity: when some unexpected phenomenon was observed, he did not hesitate to halt work in progress and turn off course in a new direction.
This was how, in 1877, he achieved his most original discovery, the phonograph.
Because the telephone was considered a variation of acoustic telegraphy, Edison during the summer of 1877 was attempting to devise for it, as he had for the automatic telegraph, a machine that would transcribe signals as they were received, in this instance in the form of the human voice, so that they could then be delivered as telegraph messages. (The telephone was not yet conceived as a general, person-to-person means of communication.)
Some earlier researchers, notably the French inventor Léon Scott, had theorized that each sound, if it could be graphically recorded, would produce a distinct shape resembling shorthand, or phonography ("sound writing"), as it was then known. Edison hoped to reify this concept by employing a stylus-tipped carbon transmitter to make impressions on a strip of paraffined paper.
To his astonishment, the scarcely visible indentations generated a vague reproduction of sound when the paper was pulled back beneath the stylus.
Edison unveiled the tinfoil phonograph, which replaced the strip of paper with a cylinder wrapped in tinfoil, in December 1877. It was greeted with incredulity. Indeed, a leading French scientist declared it to be the trick device of a clever ventriloquist.
The public's amazement was quickly followed by universal acclaim. Edison was projected into worldwide prominence and was dubbed the Wizard of Menlo Park, although a decade passed before the phonograph was transformed from a laboratory curiosity into a commercial product.
The electric light
Another offshoot of the carbon experiments reached fruition sooner. Samuel Langley, Henry Draper, and other American scientists needed a highly sensitive instrument that could be used to measure minute temperature changes in heat emitted from the Sun's corona during a solar eclipse along the Rocky Mountains on July 29, 1878. To satisfy those needs Edison devised a "microtasimeter" employing a carbon button.
This was a time when great advances were being made in electric arc lighting, and during the expedition, which Edison accompanied, the men discussed the practicality of "subdividing" the intense arc lights so that electricity could be used for lighting in the same fashion as with small, individual gas "burners."
The basic problem seemed to be to keep the burner, or bulb, from being consumed by preventing it from overheating. Edison thought he would be able to solve this by fashioning a microtasimeter-like device to control the current.
He boldly announced that he would invent a safe, mild, and inexpensive electric light that would replace the gaslight.
The incandescent electric light had been the despair of inventors for 50 years, but Edison's past achievements commanded respect for his boastful prophecy. Thus, a syndicate of leading financiers, including J.P. Morgan and the Vanderbilts, established the Edison Electric Light Company and advanced him $30,000 for research and development.
Edison proposed to connect his lights in a parallel circuit by subdividing the current, so that, unlike arc lights, which were connected in a series circuit, the failure of one light bulb would not cause a whole circuit to fail. Some eminent scientists predicted that such a circuit could never be feasible, but their findings were based on systems of lamps with low resistance - the only successful type of electric light at the time.
Edison, however, determined that a bulb with high resistance would serve his purpose, and he began searching for a suitable one.
He had the assistance of 26-year-old Francis Upton, a graduate of Princeton University with an M.A. in science. Upton, who joined the laboratory force in December 1878, provided the mathematical and theoretical expertise that Edison himself lacked. (Edison later revealed, "At the time I experimented on the incandescent lamp I did not understand Ohm's law."
On another occasion he said, "I do not depend on figures at all. I try an experiment and reason out the result, somehow, by methods which I could not explain.")
By the summer of 1879 Edison and Upton had made enough progress on a generator--which, by reverse action, could be employed as a motor--that Edison, beset by failed incandescent lamp experiments, considered offering a system of electric distribution for power, not light.
By October Edison and his staff had achieved encouraging results with a complex, regulator-controlled vacuum bulb with a platinum filament, but the cost of the platinum would have made the incandescent light impractical.
While experimenting with an insulator for the platinum wire, they discovered that, in the greatly improved vacuum they were now obtaining through advances made in the vacuum pump, carbon could be maintained for some time without elaborate regulatory apparatus.
Advancing on the work of Joseph Wilson Swan, an English physicist, Edison found that a carbon filament provided a good light with the concomitant high resistance required for subdivision. Steady progress ensued from the first breakthrough in mid-October until the initial demonstration for the backers of the Edison Electric Light Company on December 3.
It was, nevertheless, not until the summer of 1880 that Edison determined that carbonized bamboo fibre made a satisfactory material for the filament, although the world's first operative lighting system had been installed on the steamship Columbia in April.
The first commercial land-based "isolated" (single-building) incandescent system was placed in the New York printing firm of Hinds and Ketcham in January 1881.
In the fall a temporary, demonstration central power system was installed at the Holborn Viaduct in London, in conjunction with an exhibition at the Crystal Palace. Edison himself supervised the laying of the mains and installation of the world's first permanent, commercial central power system in lower Manhattan, which became operative in September 1882.
Although the early systems were plagued by problems and many years passed before incandescent lighting powered by electricity from central stations made significant inroads into gas lighting, isolated lighting plants for such enterprises as hotels, theatres, and stores flourished--as did Edison's reputation as the world's greatest inventor.
One of the accidental discoveries made in the Menlo Park laboratory during the development of the incandescent light anticipated the British physicist J.J. Thomson's discovery of the electron 15 years later.
In 1881-82 William J. Hammer, a young engineer in charge of testing the light globes, noted a blue glow around the positive pole in a vacuum bulb and a blackening of the wire and the bulb at the negative pole. This phenomenon was first called "Hammer's phantom shadow," but when Edison patented the bulb in 1883 it became known as the "Edison effect."
Scientists later determined that this effect was explained by the thermionic emission of electrons from the hot to the cold electrode, and it became the basis of the electron tube and laid the foundation for the electronics industry.
Edison had moved his operations from Menlo Park to New York City when work commenced on the Manhattan power system. Increasingly, the Menlo Park property was used only as a summer home.
In August 1884 Edison's wife, Mary, suffering from deteriorating health and subject to periods of mental derangement, died there of "congestion of the brain," apparently a tumour or hemorrhage. Her death and the move from Menlo Park roughly mark the halfway point of Edison's life.
The Edison laboratory
A widower with three young children, Edison, on Feb. 24, 1886, married 20-year-old Mina Miller, the daughter of a prosperous Ohio manufacturer. He purchased a hilltop estate in West Orange, N.J., for his new bride and constructed nearby a grand, new laboratory, which he intended to be the world's first true research facility.
There, he produced the commercial phonograph, founded the motion-picture industry, and developed the alkaline storage battery. Nevertheless, Edison was past the peak of his productive period.
A poor manager and organizer, he worked best in intimate, relatively unstructured surroundings with a handful of close associates and assistants; the West Orange laboratory was too sprawling and diversified for his talents. Furthermore, as a significant portion of the inventor's time was taken up by his new role of industrialist, which came with the commercialization of incandescent lighting and the phonograph, electrical developments were passing into the domain of university-trained mathematicians and scientists.
Above all, for more than a decade Edison's energy was focused on a magnetic ore-mining venture that proved the unquestioned disaster of his career.
The first major endeavor at the new laboratory was the commercialization of the phonograph, a venture launched in 1887 after Alexander Graham Bell, his cousin Chichester, and Charles Tainter had developed the graphophone--an improved version of Edison's original device--which used waxed cardboard instead of tinfoil.
Two years later, Edison announced that he had "perfected" the phonograph, although this was far from true. In fact, it was not until the late 1890s, after Edison had established production and recording facilities adjacent to the laboratory, that all the mechanical problems were overcome and the phonograph became a profitable proposition.
In the meantime, Edison conceived the idea of popularizing the phonograph by linking to it in synchronization a zoetrope, a device that gave the illusion of motion to photographs shot in sequence. He assigned the project to William K.L. Dickson, an employee interested in photography, in 1888.
After studying the work of various European photographers who also were trying to record motion, Edison and Dickson succeeded in constructing a working camera and a viewing instrument, which were called, respectively, the Kinetograph and the Kinetoscope.
Synchronizing sound and motion proved of such insuperable difficulty, however, that the concept of linking the two was abandoned, and the silent movie was born. Edison constructed at the laboratory the world's first motion-picture stage, nicknamed the "Black Maria," in 1893, and the following year Kinetoscopes, which had peepholes that allowed one person at a time to view the moving pictures, were introduced with great success.
Rival inventors soon developed screen-projection systems that hurt the Kinetoscope's business, however, so Edison acquired a projector developed by Thomas Armat and introduced it as "Edison's latest marvel, the Vitascope."
Another derivative of the phonograph was the alkaline storage battery, which Edison began developing as a power source for the phonograph at a time when most homes still lacked electricity.
Although it was 20 years before all the difficulties with the battery were solved, by 1909 Edison was a principal supplier of batteries for submarines and electric vehicles and had even formed a company for the manufacture of electric automobiles.
In 1912 Henry Ford, one of Edison's greatest admirers, asked him to design a battery for the self-starter, to be introduced on the Model T. Ford's request led to a continuing relationship between these two Americans, and in October 1929 he staged a 50th-anniversary celebration of the incandescent light that turned into a universal apotheosis for Edison.
Most of Edison's successes involved electricity or communication, but throughout the late 1880s and early 1890s the Edison Laboratory's top priority was the magnetic ore-separator. Edison had first worked on the separator when he was searching for platinum for use in the experimental incandescent lamp.
The device was supposed to cull platinum from iron-bearing sand. During the 1880s iron ore prices rose to unprecedented heights, so that it appeared that, if the separator could extract the iron from unusable low-grade ores, then abandoned mines might profitably be placed back in production.
Edison purchased or acquired rights to 145 old mines in the east and established a large pilot plant at the Ogden mine, near Ogdensburg, N.J.
He was never able to surmount the engineering problems or work the bugs out of the system, however, and when ore prices plummeted in the mid-1890s he gave up on the idea. By then he had liquidated all but a small part of his holdings in the General Electric Company, sometimes at very low prices, and had become more and more separated from the electric lighting field.
Failure could not discourage Edison's passion for invention, however. Although none of his later projects were as successful as his earlier ones, he continued to work even in his 80s.
Assessment
The thrust of Edison's work may be seen in the clustering of his patents: 389 for electric light and power, 195 for the phonograph, 150 for the telegraph, 141 for storage batteries, and 34 for the telephone. His life and achievements epitomize the ideal of applied research.
He always invented for necessity, with the object of devising something new that he could manufacture. The basic principles he discovered were derived from practical experiments, invariably by chance, thus reversing the orthodox concept of pure research leading to applied research.
Edison's role as a machine shop operator and small manufacturer was crucial to his success as an inventor.
Unlike other scientists and inventors of the time, who had limited means and lacked a support organization, Edison ran an inventive establishment.
He was the antithesis of the lone inventive genius, although his deafness enforced on him an isolation conducive to conception. His lack of managerial ability was, in an odd way, also a stimulant.
As his own boss, he plunged ahead on projects more prudent men would have shunned, then tended to dissipate the fruits of his inventiveness, so that he was both free and forced to develop new ideas. Few men have matched him in the positiveness of his thinking.
Edison never questioned whether something might be done, only how.
Edison's career, the fulfillment of the American dream of rags-to-riches through hard work and intelligence, made him a folk hero to his countrymen.
In temperament he was an uninhibited egotist, at once a tyrant to his employees and their most entertaining companion, so that there was never a dull moment with him.
He was charismatic and courted publicity, but he had difficulty socializing and neglected his family. His shafts at the expense of the "long-haired" fraternity of theorists sometimes led formally trained scientists to deprecate him as anti-intellectual; yet he employed as his aides, at various times, a number of eminent mathematical physicists, such as Nikola Tesla and A.E. Kennelly.
The contradictory nature of his forceful personality, as well as such eccentricities as his ability to catnap anywhere, contributed to his legendary status.
By the time he was in his middle 30s Edison was said to be the best-known American in the world.
When he died he was venerated and mourned as the man who, more than any other, had laid the basis for the technological and social revolution of the modern electric world.
Patents
Edison executed the first of his 1,093 successful U.S. patent applications on 13 October 1868, at the age of 21. (For a graphic representation of his annual output.
He filed an estimated 500–600 unsuccessful or abandoned applications as well. Unfortunately, the names given Edison's patents are too irregular to make simple word searches an accurate means of finding patents for particular technologies. His issued patents are presented here in three listsóby execution date, patent date, and subject. The execution and patent date lists are each presented in six parts to make the files less cumbersome.
He execution date of a patent application is the date on which the inventor signs the application, and hence is the date closest to the actual inventive activity. However, in his early years Edison did not always rush to his patent lawyer with an invention, especially if there was little competition for the invention or he was feeling broke and unable to pay the various fees involved in an application. In a few cases Edison removed some of the claims from an original application and filed a new application to cover those claims. The execution date of such a patent can be considerably later than that of the original application even though the patent covers designs from the earlier date.
The subject lists are necessarily somewhat arbitrary. They are arranged by execution date. A few patents appear in two listsófor example, Patent 142,999 is for a battery Edison developed for telegraphy, and it is under "Batteries" and "Telegraphy and Telephony."