By Marshall Clagett

This quantity keeps Marshall Clagett's stories of a number of the points of the technological know-how of historical Egypt. the amount offers a discourse at the nature and accomplishments of Egyptian arithmetic and in addition informs the reader as to how our wisdom of Egyptian arithmetic has grown because the booklet of the Rhind Mathematical Papyrus towards the tip of the nineteenth century. the writer prices and discusses interpretations of such authors as Eisenlohr, Griffith, Hultsch, Peet, Struce, Neugebauer, Chace, Glanville, van der Waerden, Bruins, Gillings, and others. He additionally additionally considers stories of newer authors resembling Couchoud, Caveing, and Guillemot.

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It begins to look as though finding complex numbers through the square root of −1 is a lucky accident. And yet . . 2) is kind of an accident too. Who would have guessed in advance that the product of sums of two squares is a sum of two squares? Is this true for three squares? Four? Five? See Chapter 6 for more on these questions. 6 Geometry of Multiplication Complex numbers appeared sporadically in mathematics for more than 200 years after Bombelli’s Algebra, but without being fully accepted or recognized.

When p(x) has degree 5 the equation p(x) = 0 cannot in general be reduced to equations of lower degree. This delayed the proof of the fundamental theorem of algebra for a long time, while futile attempts were made to solve the equation of degree 5 via equations of lower degree. Finally, in 1799, Gauss tried a new approach: he set out to prove only the existence of solutions, rather than trying to construct them by square roots, cube roots, and so on. Gauss’s first attempt had a serious gap, but the general idea was sound, and he later presented satisfactory proofs.

The most important feature of the number plane is that length or distance represents absolute value. The Pythagorean theorem says that the distance from O to a + bi is a 2 + b 2 = |a + bi |, and it follows easily that the distance between any two complex numbers is the absolute value of their difference. Now suppose that u is some complex number and that we multiply all the numbers in the plane by u. This sends numbers v and w to the numbers uv and uw. The distance between uv and uw is |uw − uv|, and |uw − uv| = |u(w − v))| by the distributive law = |u||w − v| by the multiplicative property of absolute value = |u| × distance between v and w.