Pythagoras theorem , thePythagorean theorem, also known asPythagoras's theorem, is a relation inEuclidean geometryamong the three sides of aright triangle. It states that the square of thehypotenuse(the side opposite the right angle) is equal to the sum of the squares of the other two sides. Thetheoremcan be written as anequation relating the lengths of the sidesa,bandc, often called the "Pythagorean equation":[1] wherecrepresents the hypotenuse andaandbthe lengths of the triangle's other two sides.Although it is often argued that knowledge of the theorem predates him,[citation needed]the theorem is named after the ancient GreekmathematicianPythagoras(c.570 c.495 BC) as it is he who, by tradition, is credited with its first recordedproof .He was aIonianGreekphilosopher,mathematician, and founder of the religious movement calledPythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him.Pythagoras made influential contributions tophilosophyandreligionin the late 6th century BC. He is often revered as a greatmathematician,mystic, andscientistand is best known for thePythagorean theoremwhich bears his name. However, because legend and obfuscation cloud his work even more than that of the otherpre-Socratic philosophers, one can give only a tentative account of his teachings, and some have questioned whether he contributed much tomathematicsornatural philosophyFundamental thorem of calculusGottfried Leibnizwas a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.Leibniz is credited, along with SirIsaac Newton, with the discovery ofcalculus(that comprises differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on November 11, 1675, when he employed integral calculus for the first time to find the area under the graph of a functiony=(x).[56]He introduced several notations used to this day, for instance theintegral sign representing an elongated S, from the Latin wordsumma and thedused fordifferentials, from the Latin worddifferentia. This cleverly suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684.[57]Theproduct ruleofdifferential calculusis still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called theLeibniz integral ruleTheoremThefundamental theorem of calculusis a theorem that links the concept of thederivativeof afunctionwith the concept of theintegral.The first part of the theorem, sometimes called thefirst fundamental theorem of calculus, is that thedefinite integrationof a function[1]is related to itsantiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence ofantiderivativesforcontinuous functions.[2]The second part, sometimes called thesecond fundamental theorem of calculus, is that thedefinite integralof a function can be computed by using any one of its infinitely manyantiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation ofdefinite integrals.For a continuous functiony=f(x)whose graph is plotted as a curve, each value ofxhas a corresponding area functionA(x), representing the area beneath the curve between 0 andx. The functionA(x) may not be known, but it is given that it represents the area under the curve.The area under the curve betweenxandx+hcould be computed by finding the area between 0 andx+h,then subtracting the area between 0 andx. In other words, the area of this sliver would beA(x+h) A(x).

Thefundamental theoremof algebraThefundamental theoremof algebrastates that every non-constantsingle-variablepolynomialwithcomplexcoefficientshas at least one complexroot. This includes polynomials with real coefficients, since every real number is a complex number with zeroimaginary part.Equivalently (by definition), the theorem states that thefieldofcomplex numbersisalgebraically closed.The theorem is also stated as follows: every non-zero, single-variable,degreenpolynomial with complex coefficients has, counted withmultiplicity, exactlynroots. The equivalence of the two statements can be proven through the use of successivepolynomial division.Johann Carl Friedrich Gauss(/as/;German:Gau,pronounced[as](listen);Latin:Carolus Fridericus Gauss) (30 April 1777 23 February 1855) was aGermanmathematician, who contributed significantly to many fields, includingnumber theory,algebra,statistics,analysis,differential geometry,geodesy,geophysics,electrostatics,astronomy,Matrix theory, andoptics.Gauss pursued his studies at the University of Gottingen. While there he submitted a proof that every algebraic equation has at least one root or solution. This theorem had challengeFour squares theoremLagrange's four-square theorem, also known asBachet's conjecture, states that anynatural numbercan be represented as the sum of four integersquares.

where the four numbersare integers. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows:

This theorem was proven byJoseph Louis Lagrangein 1770.Joseph-Louis Lagrange(bornGiuseppe Lodovico Lagrangia[1][2][3](also reported asGiuseppe Luigi Lagrangia[4]), 25 January 1736 inTurin,Piedmont-Sardinia; died 10 April 1813 inParis) was anItalianEnlightenment Eramathematicianandastronomer. He made significant contributions to the fields ofanalysis,number theory, and bothclassicalandcelestial mechanics.In 1766, on the recommendation ofEulerandd'Alembert, Lagrange succeeded Euler as the director of mathematics at thePrussian Academy of SciencesinBerlin,Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of theFrench Academy of Sciences. Lagrange's treatise onanalytical mechanics(Mcanique Analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 188889), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics sinceNewtonand formed a basis for the development of mathematical physics in the nineteenth century.Infitude of primesEuclid's theoremis a fundamental statement innumber theorythat asserts that there areinfinitelymanyprimenumbers. There are several well-known proofs of thetheorem.Euclid offered the following proof published in his workElements(Book IX, Proposition 20),[1]which is paraphrased here.Consider any finite list of prime numbersp1,p2,...,pn. It will be shown that at least one additional prime number not in this list exists. LetPbe the product of all the prime numbers in the list:P=p1p2...pn. Letq=P+1. Thenqis either prime or not: Ifqis prime, then there is at least one more prime than is in the list. Ifqis not prime, then someprime factorpdividesq. If this factorpwere on our list, then it would divideP(sincePis the product of every number on the list); butpdividesP+1=q. IfpdividesPandq,thenpwould have to divide the difference[2]of the two numbers, which is (P+1)Por just1. Since no prime number divides 1, this would be a contradiction and sopcannot be on the list. This means that at least one more prime number exists beyond those in the list.This proves that for every finite list of prime numbers there is a prime number not on the list, and therefore there must be infinitely many prime numbers.EuclidEuclid /;Greek:Eukleids;fl.300 BC), sometimes calledEuclid of Alexandriato distinguish him fromEuclid of Megara, was aGreek mathematician, often referred to as the "Father of Geometry". He was active inAlexandriaduring the reign ofPtolemyI(323283 BC). HisElementsis one of the most influential works in thehistory of mathematics, serving as the main textbook for teachingmathematics(especiallygeometry) from the time of its publication until the late 19th or early 20th century.[1][2][3]In theElements, Euclid deduced the principles of what is now calledEuclidean geometryfrom a small set ofaxioms. Euclid also wrote works onperspective,conic sections,spherical geometry,number theoryandrigor.

Pythagoras

Pythagoras theoram

.

Gottfreid lebinz

Fundamental theorem of algebra Lagrange

Infinitude of primes