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Natural numbers : the numbers which starts from 1 these numbersare callled natural numbers.Whole Numbers : The numbers which starts from 0 are calledwhole numbers.Rational Numbers : A numbers r is called a rational number, if itcan be written in the form of p/q , where p & r integers & q is notequal to 0.Irrational numbers : A number s is called a irrational number, if itcan not be written in the form p/q, where p and q are integersand q is not equal to 0.The decimal expansion of a rational number is either terminatingor non terminating recurring moreover a number whose decimalexpansion is terminating or non-terminating recurring is rational .
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PYTHAGORASPythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras hoSámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about 570 – d. about 495 BC[1][2]) was an Ionian Greekphilosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the informationabout Pythagoras was written down centuries after he lived, so very little reliable information is known about him. Hewas born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seekingknowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect.His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophicaltheories. The society took an active role in the politics of Croton, but this eventually led to their downfall. ThePythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended hisdays in Metapontum.Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is oftenrevered as a great mathematician, mystic and scientist, but he is best known for the Pythagorean theorem whichbears his name. However, because legend and obfuscation cloud his work even more than with the other pre-Socraticphilosophers, one can give account of his teachings to a little extent, and some have questioned whether hecontributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras mayactually have been accomplishments of his colleagues and successors. Whether or not his disciples believed thateverything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that hewas the first man to call himself a philosopher, or lover of wisdom,[3] and Pythagorean ideas exercised a markedinfluence on Plato, and through him, all of Western philosophy.
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Chapter 2 Polinomial A polynomial have one term is called monomial. A polynomial have 2 terms is called binomial. A polynomial have 3 terms is called a trinomial. A polynomial of degree one is called linear polynomial. A polynomial of degree 2 is called a quadratic polynomial. A polynomial of degree 3 is called a cubic polynomial. A real number ‘a’ is a zero of polynomial p(x) if p(a) = 0. In this case , a isalso called a root of the equation p(x) = 0. Remainder theorem : If p(x) is any polynomial of degree greater than orequal to 1 and p(x) is divided by the linear polynomial x – a, than theremainder is p(a). Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, ifx –a is a factorof p(x), then p(a) = 0.
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R.DedekindWhile teaching calculus for the first time at the Polytechnic, Dedekind came up withthe notion now called a Dedekind cut (German: Schnitt), now a standard definition ofthe real numbers. The idea behind a cut is that an irrational number divides therational numbers into two classes (sets), with all the members of one class (upper)being strictly greater than all the members of the other (lower) class. For example, thesquare root of 2 puts all the negative numbers and the numbers whose squares areless than 2 into the lower class, and the positive numbers whose squares are greaterthan 2 into the upper class. Every location on the number line continuum containseither a rational or an irrational number. Thus there are no empty locations, gaps, ordiscontinuities. Dedekind published his thoughts on irrational numbers and Dedekindcuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrationalnumbers");[1] in modern terminology, Vollständigkeit, completeness.In 1874, while on holiday in Interlaken, Dedekind met Cantor. Thus began an enduringrelationship of mutual respect, and Dedekind became one of the very firstmathematicians to admire Cantors work on infinite sets, proving a valued ally inCantors battles with Kronecker, who was philosophically opposed to Cantorstransfinite numbers.If there existed a one-to-one correspondence between two sets, Dedekind said thatthe two sets were "similar." He invoked similarity to give the first precise definition ofan infinite set: a set is infinite when it is "similar to a proper part of itself," in modernterminology, is equinumerous to one of its proper subsets. (This[clarification needed] isknown as Dedekinds theorem.[citation needed]) Thus the set N of natural numbers can beshown to be similar to the subset of N whose members are the squares of everymember of N, (N → N2):
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Georg Ferdinand Ludwig Philipp Cantor ( /ˈ kæntɔr/ KAN-tor; German: [ɡeˈ ɔʁk ˈfɛʁdinantˈ luˈtv ˈ ɪp ˈ ɪç fiˈl kantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a Germanmathematician, best known as the inventor of set theory, which has become a fundamental theoryin mathematics. Cantor established the importance of one-to-one correspondence between themembers of two sets, defined infinite and well-ordered sets, and proved that the real numbers are"more numerous" than the natural numbers. In fact, Cantors method of proof of this theoremimplies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers andtheir arithmetic. Cantors work is of great philosophical interest, a fact of which he was well aware. [2]Cantors theory of transfinite numbers was originally regarded as so counter-intuitive—evenshocking—that it encountered resistance from mathematical contemporaries such as LeopoldKronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while LudwigWittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantors work as a challenge to the uniqueness of the absolute infinity in thenature of God - [4] on one occasion equating the theory of transfinite numbers with pantheism[5] - aproposition which Cantor vigorously refuted. The objections to his work were occasionally fierce:Poincaré referred to Cantors ideas as a "grave disease" infecting the discipline of mathematics,[6]and Kroneckers public opposition and personal attacks included describing Cantor as a "scientificcharlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantors proofsthat the algebraic numbers are countable, and that the transcendental numbers are uncountable,results now included in a standard mathematics curriculum. Writing decades after Cantors death,Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idiomsof set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong". [8] Cantorsrecurring bouts of depression from 1884 to the end of his life have been blamed on the hostileattitude of many of his contemporaries,[9] though some have explained these episodes as probablemanifestations of a bipolar disorder.[10]The harsh criticism has been matched by later accolades. In 1904, the Royal Society awardedCantor its Sylvester Medal, the highest honor it can confer for work in mathematics.[11] It has beensuggested that Cantor believed his theory of transfinite numbers had been communicated to him byGod.[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us fromthe Paradise that Cantor has created."[13]
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Made by --- Blossom ShrivastavaClass --- 9 ‘ A ’Roll no : 16
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Definition of a Linear Equation A linear equation in two variable x isan equation that can be written in theform ax + by + c = 0, where a ,b and c arereal numbers and a and b is not equal to0. An example of a linear equation in xis .
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Equations of the form ax + by = c arecalled linear equations in two variables.Equations of the form ax + by = c are (0,4)called linear equations in two variables. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept. -2 2
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Solution of an Equation in Two Variables Example: Given the equation 2x + 3y = 18, determine if the ordered pair (3, 4) is a solution to the equation. We substitute 3 in for x and 4 in for y. 2(3) + 3 (4) ? 18 6 + 12 ? 18 18 = 18 True. Therefore, the ordered pair (3, 4) is a solution to the equation 2x + 3y = 18.
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CARTESIAN PLANEQuadrant II Quadrant I ( - ,+) (+,+) Quadrant III Quadrant IV (-,-) (+, - )
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Finding Solutions of an EquationFind five solutions to the equation y = 3x + 1.Start by choosing some x values and then computing thecorresponding y values. If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5) If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2) If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1) If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4) If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7)