who discovered Pascal's triangle and binomial expansion before Pascal?
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The first reference to this triangle occurs in Indian mathematician Pingala's book on Sanskrit poetics that may be as early as 450 BC as Meru-prastaara, the "staircase of Mount Meru". The commentators of this book were also aware that the shallow diagonals of the triangle sum to the Fibonacci numbers. It was also known to Chinese mathematicians. It is said that the triangle was called "Yang Hui's triangle" by the Chinese. Later, the Persian mathematician Karaji and the Persian astronomer-poet Omar Khayyám; thus the triangle is referred to as the "Khayyam triangle" in Iran. Several theorems related to the triangle were known, including the binomial theorem. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. Finally, in Italy, it is referred to as "Tartaglia's triangle", named for the Italian algebraist Niccolo Fontana Tartaglia who lived a century before Pascal; Tartaglia is credited with the general formula for solving cubic polynomials.\n\nIn modern times, Pascal's triangle takes its name from the Traité du triangle arithmétique (1655) by Blaise Pascal. In that work, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The "arithmetical" triangle was later called after Pascal by Pierre Raymond de Montmort (1708) and Abraham de Moivre (1730).
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In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after Blaise Pascal, even though others studied it centuries before him.\nhttp://en.wikipedia.org/wiki/Pascal_triangle
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Bionomial expansion\n\nthis has got n+1 terms\n\nthe terms are (nCk) a^kb^n-k where k varies from 0 to n\n\nfor example (a+b)^3\none term (3C0) a^0b^3 = b^3\nanother term (3C1)a^1b^2 = 3ab^2\nanother (3C2)a^2b^1 = 3a^2b\nanother (3c3)a^3b^0 = a^3\n\nadd them and get a^3+3a^2b+3ab^2+ b^3\n\n(nCk) = n!/(k!(n-k)!)\n\nThe n'th line of the pascal triangle gives you the coeffiecients . Starting from 0.
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this may be somewhat subjective, eh?\n\n1) Euclid's work\n2) Isaac Newton's calculus\n3) ___________________ (open for discussion) The Pythagoreans? Archimedes? The development of Algebra from al-jah-bra...dang - i'm anxious to see the other answers
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Jacob (Jacques) Bernoulli\n\n1654 - 1705\n\nJacob Bernoulli was a Swiss mathematician who was the first to use the term integral. He studied the catenary, the curve of a suspended string. He was an early user of polar coordinates and discovered the isochrone.
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The formula is credited to Heron of Alexandria in the 1st century, and a proof can be found in his book Metrica. It is now believed that Archimedes already knew the formula, and it is of course possible that it has been known long before.\n\nLet the sides of a triangle have lengths a,b and c. Introduce the semiperimeter p = (a + b + c)/2 and the area S. Then Heron's formula asserts that\n S2 = p(p - a)(p - b)(p - c)\n\nW.Dunham analyzes the original Heron's proof in his Journey through Genius.\n\nFor the right triangle with hypotenuse c, we have S = ab/2. We'll modify the right hand side of the formula by noting that\n p - a = (- a + b + c)/2, p - b = (a - b + c)/2, p - c = (a + b - c)/2\n\nIt takes a little algebra to show that\n \n(a + b + c)(- a + b + c)(a - b + c)(a + b - c)\n= 2a2b2 + 2a2c2 + 2b2c2 - (a4 + b4 + c4)\n\nFor the right triangle, this expression is equal to 16S2 = 4a2b2. So we have\n 4a2b2= 2a2b2 + 2a2c2 + 2b2c2 - (a4 + b4 + c4)\n\nTaking all terms to the left side and grouping them yields\n (a4 + 2a2b2 + b4) - 2a2c2 - 2b2c2 + c4 = 0\n\nWith a little more effort\n (a2 + b2)2 - 2c2(a2 + b2) + c4 = 0\n\nAnd finally\n [(a2 + b2) - c2]2 = 0\nRemark\n\nFor a quadrilateral with sides a, b, c and d inscribed in a circle there exists a generalization of Heron's formula discovered by Brahmagupta. In this case, the semiperimeter is defined as p = (a + b + c + d)/2. Then the following formula holds\n S2 = (p - a)(p - b)(p - c)(p - d)\n\nSince any triangle is inscribable in a circle, we may let one side, say d, shrink to 0. This leads to Heron's formula.\n\nFor more information you can go to www.wikipedia.com  \nhttp://mathworld.wolfram.com/HeronsFormula.html\nhttp://www.cut-the-knot.org/pythagoras/herons.shtml\n\nI hope this helps! =-)
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Johann Castillon used the name cardioid for the first time, in a paper in the Philosophical Transactions of the Royal Society (1741).\n\nHowever, he was not the first person who used it.  For example, it was studied by Roemer (1674) in an investigation for the best form of gear teeth. \n\nCardioid is a special case of limacon of Pascal: a family of curves studied and named after Etienne Pascal (1588-1640), father of Blaise Pascal (1623-1662). \n\nI don't know whether they originally used polar co-ordinates which followed after the Cartesian system developed by Descartes in 1637. But given that the general formula for the limacon is a ± bcos(theta) or a ± bsin(theta), it's certainly possible.
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Of course. The pattern is well known, and is manifested as the binomial distribution and Pascal's triangle.\n\nIt is well formulated mathematically. There is a formula that can tell you, for example, what the 18th term is in the expansion of (a+b)^30.
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p1v1t1=p2v2t2\n\nSee Bernoulli's theorem and Pascals Law
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Pierre de fermat because of his intelligent works including his last theorem.
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Descartes made one of the greatest advances in geometry by connecting algebra and geometry. A myth is that he was watching a fly on the ceiling when he conceived of locating points on a plane with a pair of numbers. Maybe this has something to do with the fact that he stayed in bed everyday until 11:00 A.M. Fermat also discovered coordinate geometry, but it's Descartes' version that we use today.\n\nCo-ordinate Geometry\nCo-ordinate geometry was introduced by Fermat and Descartes, ignoring rather than solving the foundational problems which had prevented the Greeks from taking this step (viz: the lack of any well understood number system which could account for incommensurable ratios). \nThis development is important to science because it makes geometry quantitative and permits the use of algebraic methods. Geometry must be quantitative for it to be useful in science and engineering, and algebraic methods permit more rapid development of mathematics than the less systematic (if more rigourous) methods required by the Greek axiomatic approach to geometry. \n \nIn co-ordinate geometry the points of a plane are labelled by 2 co-ordinates, its distances from the 2 axes, and familiar figures become the loci of points satisfying an algebraic equation. Thus, geometry is incorporated into algebra and the way is opened for the introduction of spaces of arbitrarily large dimension, important in mechanics. Curvature, in its simplest form, is a number attached to a surface at a point. The flatter the surface, the smaller the number (positive for a sphere, negative for a saddle-shaped surface). \nFrench mathematician who is often called the founder of the modern theory of numbers. Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Independently of Descartes, Fermat discovered the fundamental principle of analytic geometry. His methods for finding tangents to curves and their maximum and minimum points led him to be regarded as the inventor of the differential calculus. Through his correspondence with Blaise Pascal he was a co-founder of the theory of probability.\n\n\nhttp://people.bath.ac.uk/jm255/Geometry.htm this website will help u aswell!\n\nAnalytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.\n\nRené Descartes is popularly regarded as having introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native language (French), and its philosophical principles, provided the foundation for calculus in Europe.\n\nThe relationship between algebra and geometry is nontrivial. There exist proofs of this relationship based on the theory of geometry and notions of geometric length.\n\n\n\nwrite this down and im finding more info ......
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Blaise Pascal was a French mathematician, scientist, and theologian who lived between 1623-1662. In 1645, Pascal invented the first digital calculating machine, making him one of the founding fathers of microcomputer technology. In 1646, he invented the syringe and hydraulic press. \n\nIn 1648, Pascal proved that the level of mercury in a barometer is determined by a change in atmospheric pressure. This led to new insights in the studies of meteorology and fluid dynamics. It also led to Pascal's law, which states that fluids transmit pressure equally in all directions. \n\nThe International Standard (SI) unit for pressure was named after Pascal. Pascal is also honored in having one of the first computer programming languages named after him. \n\nPascal died at 39 years of age from stomach and brain cancer. One of his theological writings, Pensees (Thoughts) is still famous throughout the world.
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Aryabhatta, the Indian mathematician who invented 0.
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http://gcc.gnu.org/    C/C++\nhttp://www.freepascal.org/      Pascal
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check this site\n\nhttp://scienceworld.wolfram.com/biography/Fermat.html
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Blaise Pascal? He invented the calculator. Maybe Isaac Newton, the inventor of calculus, or Charles Babbage, the inventor of the computer. Perhaps John Napier, who discovered how to add, subtract, multiply, divide, and even extract square roots simply by moving pieces on a chessboard? Tell you what, pick René Descartes, he's the inventor of coordinate geometry. These are all excellent choices!
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the first can be shown by induction.\n\nthe 2nd is.. pretty obvious\n\nI couldn't do the 2nd one with Binomial Theorem... btw, what is Binomial theorem? but it is very easy to see 99^50+100^50<101^50\n\nu can write 99^50+100^50<101^50 as 0.99^50+1<1.01^50 and use a calculator, or\n\nTo prove this is true, show 101^50-(99^50)>100^50\n\nWith Binomial expansion\n101^50=(100+1)^50=100^50+50(100^49) +a +b +c +d +...\n99^50=(100-1)^50=100^50-50(100^49) +a -b +c -d +....\n\nwith a,b,c,d,e,f... being some positive real numbers.\n\nu can see that the difference is (50+50)100^49 +2b +2d +2f +... =(100)100^49 +2b +2d +2f +... =100^50 +2b +2d +2f +...> 100^50\n\nI'm not sure if this has anything to do with the Binomial theorem u mentioned.. but.. it has Binomial expansion