Download e-book for iPad: An Imaginary Tale: The Story of ?-1 by Paul J. Nahin

By Paul J. Nahin

ISBN-10: 0691127980

ISBN-13: 9780691127989

Today advanced numbers have such frequent useful use--from electric engineering to aeronautics--that few humans might count on the tale at the back of their derivation to be jam-packed with event and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old historical past of 1 of mathematics' such a lot elusive numbers, the sq. root of minus one, often referred to as i. He recreates the baffling mathematical difficulties that conjured it up, and the colourful characters who attempted to resolve them.

In 1878, while brothers stole a mathematical papyrus from the traditional Egyptian burial web site within the Valley of Kings, they led students to the earliest identified prevalence of the sq. root of a damaging quantity. The papyrus provided a selected numerical instance of the way to calculate the quantity of a truncated sq. pyramid, which implied the necessity for i. within the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate undertaking, yet fudged the mathematics; medieval mathematicians stumbled upon the idea that whereas grappling with the that means of unfavorable numbers, yet disregarded their sq. roots as nonsense. by the point of Descartes, a theoretical use for those elusive sq. roots--now referred to as "imaginary numbers"--was suspected, yet efforts to unravel them resulted in extreme, sour debates. The infamous i eventually received reputation and was once positioned to take advantage of in advanced research and theoretical physics in Napoleonic times.

Addressing readers with either a basic and scholarly curiosity in arithmetic, Nahin weaves into this narrative interesting ancient evidence and mathematical discussions, together with the applying of complicated numbers and capabilities to special difficulties, resembling Kepler's legislation of planetary movement and ac electric circuits. This ebook could be learn as an interesting heritage, virtually a biography, of 1 of the main evasive and pervasive "numbers" in all of mathematics.

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Additional info for An Imaginary Tale: The Story of ?-1

Example text

Next, he used the well-known method for bisecting a line segment to locate K, the midpoint of FH. Then, using K as the center, he constructed the semicircle FIH with radius KH ϭ FK. Finally, at G he erected a perpendicular that intersects the semicircle at I (and so IK is a radius, too). From all this we can write that FG + GH = 2 IK , 1 + GH = 2 IK , 1 (1 + GH ) = IK . 1. Constructing the square root of a line (IG ϭ ͙GH). Also, FG + GK = IK , 1 GK = IK − FG = IK − 1 = (1 + GH ) − 1, 2 GK = 1 (GH − 1).

5. The two imaginary roots are p ϩ iq and p Ϫ iq. 1 Ren´e Descartes Despite the success of Bombelli in giving formal meaning to ͙Ϫ1 when it appeared in the answers given by Cardan’s formula, there still lacked a physical interpretation. Mathematicians of the sixteenth century were very much tied to the Greek tradition of geometry, and they felt uncomfortable with concepts to which they could not give a geometric meaning. , are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.

After having given a correct formula for the determination of the volume of a frustum of a pyramid with square base and applied it successfully to the case where the side of the lower base is 10, of the upper 2, and the edge 9, the author endeavors to solve the problem where the side of the lower base is 28, of the upper 4, and the edge 15. Instead of the square root of 81 Ϫ 144 required by the formula, he takes the square root of 144 Ϫ 81 . . , he replaces ͙Ϫ1 by 1, and fails to observe that the problem as stated is impossible.

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An Imaginary Tale: The Story of ?-1 by Paul J. Nahin


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