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He was first to prove Heron's formula for the area of a triangle.
For example, the area of any right triangle is equal to thesum of the areas of the two lunes formed when semi-circlesare drawn on each of the three edges of the triangle.
However, Duff 1986 and Hájek 2003 argue that the argument isin fact invalid. Their point is that there are strategies besideswagering for God that also have infinite expectation—namely,mixed strategies, whereby you do not wager for or against Godoutright, but rather choose which of these actions to perform on thebasis of the outcome of some chance device. Consider the mixedstrategy: “Toss a fair coin: heads, you wager for God; tails, you wageragainst God”. By Pascal’s lights, with probability 1/2 your expectationwill be infinite, and with probability 1/2 it will be finite. Theexpectation of the entire strategy is:
Pascal's Triangle(short vers.) - Mega Essays
Pascal discovered new properties of the triangle and solved problems using it, therefore the triangle became known as "Pascal's Triangle." Pascal became interested in probability when given a gambling question.
He also solved the problems of the volume and surface area of the solid of revolution formed by rotating the cycloid about the x-axis. Pascal died at the age of 39 from a malignant stomach ulcer.
Pascal's Triangle(short vers.) ..
Pascal employed his arithmetical triangle in 1653, butno account of his method was printed till 1665. The triangleis constructed as in the figure below, each horizontal linebeing formed form the one above it by making every number in itequal to the sum of those above and to the left of it in therow immediately above it; ex. gr. the fourth number in thefourth line, namely, 20, is equal to 1 + 3 + 6 + 10.The numbers in each line are what are now called figuratenumbers. Those in the first line are called numbers of the firstorder; those in the second line, natural numbers or numbers of thesecond order; those in the third line, numbers of the third order,and so on. It is easily shewn that the th number in the throw is (m+n-2)! / (m-1)!(n-1)!
This relationship exists between every number and its corresponding level of Pascal's Triangle.
Note that the total number of permutations at each level is always a power of 2 ie 1, 2, 4, 8, 16, 32, 64 etc.
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Math exploration of Pascals Triangle - Essay Example
Take a look at Pascal's triangle. From the fourth row, we know our coefficients will be 1, 4, 6, 4, and 1. That negative sign means that the first term of our expansion will be positive, and the following terms will alternate signs. The exponents will start at x4y0 and move to x3y1, etc.
This triangle is similar to the Pascal’s triangle that was ..
With Pascal on our side, we feel invincible. However, that isn't quite the case. We're still weak to bullets, being dipped in lava while a villain monologues at us, or a well-timed chocolate. We also have trouble dealing with very large exponents; for every row we add to Pascal's triangle, it takes longer and longer to find the next. We also can't afford to skip any rows, or the usher will escort us out.
MATH IB SL INT ASS1 - Pascal's Triangle
Pascal's Triangle gives us the coefficients for an expanded binomial of the form (a + b)n, where n is the row of the triangle. The Binomial Theorem tells us we can use these coefficients to find the entire expanded binomial, with a couple extra tricks thrown in.
Math Forum: Ask Dr. Math FAQ: Pascal's Triangle
Blaise Pascal was born at Clermont on June 19, 1623, and diedat Paris on Aug. 19, 1662. His father, a local judge at Clermont, andhimself of some scientific reputation, moved to Paris in 1631, partlyto prosecute his own scientific studies, partly to carry on theeducation of his only son, who had already displayed exceptionalability. Pascal was kept at home in order to ensure his not beingoverworked, and with the same object it was directed that his educationshould be at first confined to the study of languages, and should notinclude any mathematics. This naturally excited the boy's curiosity,and one day, being then twelve years old, he asked in what geometryconsisted. His tutor replied that it was the science of constructingexact figures and of determining the proportions between their differentparts. Pascal, stimulated no doubt by the injunction against reading it,gave up his play-time to this new study, and in a few weeks had discoveredfor himself many properties of figures, and in particular the propositionthat the sum of the angles of a triangle is equal to two right angles.I have read somewhere, but I cannot lay my hand on the authority, thathis proof merely consisted in turning the angular points of a triangularpiece of paper over so as to meet in the centre of the inscribed circle:a similar demonstration can be got by turning the angular points overso as to meet at the foot of the perpendicular drawn from the biggestangle to the opposite side. His father, struck by this display ofability, gave him a copy of Euclid's , a book whichPascal read with avidity and soon mastered.
Pascal's triangle | Free Homework Help
From Pascal's Triangle, we can see that our coefficients will be 1, 3, 3, and 1. The positive sign between the terms means that everything our expansion is positive.
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