Download PDF by Philip J. Davis, William G. Chinn (auth.): 3.1416 And All That

By Philip J. Davis, William G. Chinn (auth.)

ISBN-10: 0817633049

ISBN-13: 9780817633042

ISBN-10: 1461585198

ISBN-13: 9781461585190

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It is. Also, if it is not raining, but the ground is wet (someone tossed a bucket of water), the compound statement remains true. Finally, if it is both raining and the ground is wet, the truth value is obviously true. Using 1 to stand for a true statement and 0 for a false statement, the result of the compound sentence involving the linking word "or" is shown in Table la, where the left-hand column shows the truth value of the first phrase x, the top row the truth value of the second phrase y, and the entry shows the result of the compound statement.

Hamilton called such a number a triplet; just as Wessel represented complex numbers by a point on a two-dimensional plane, the triplets were to be represented by a point in three-dimensional space. The problem was a hard nut to crack. It was continually on Hamilton's mind, and his family worried over it with him. " One day in 1843, while he was walking with his wife along a canal in Dublin, Hamilton suddenly conceived a way to multiply triplets. He was so elated that he took out a penknife then and there and carved on Brougham Bridge the key to the problem, which certainly must have mystified passersby who read it: i2 = j2 = k 2 = ijk = -1 The letters i, j and k represent hypercomplex numbers Hamilton called quaternions (the general form of a quaternion being a bi cj dk, with a, b, c and d denoting real numbers).

As we have just shown that this club must have exactly 7 members and as we were told that it was necessary to expel 2 in order to fulfill the requirements, there were originally 9 ants at the meeting. The example just given illustrates a topic in mathematics known as finite geometry ( also called miniature geometry). Here, the mathematical system consists of a finite number of points, lines, and so on. It may apply to the above examples, or to telephone trunk lines, utility lines, or to electrical circuits.

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3.1416 And All That by Philip J. Davis, William G. Chinn (auth.)

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