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Vector

One of the perplexing problems confronting early mathematicians involved the imaginary number i, which represents the square root of -1. Mathematicians had been able to conceive of real numbers as points on a line. Addition, subtraction, multiplication, and division could then be interpreted as movements along that line. But complex numbers of the form a + bi could not be located along the line and thus had no geometric interpretation. A possible solution to this problem had been suggested in a remarkable paper by the Danish surveyor and mathematician Caspar Wessel (1745-1818) in 1798. Wessel proposed that a complex number be represented by a pair of lines at an angle to each other, one representing the real part of the number (a) and the other representing the imaginary part (bi). A third line joining these two could then represent the complex number. Unfortunately, Wessel's paper was published in an obscure mathematical journal and did not receive its proper recognition for nearly a century. Instead, credit for this method for representing complex numbers is often given to Carl Friedrich Gauss, who apparently developed the same concept at about the same time, but published nothing on the subject until 1831. Some credit is also due to Robert Argand for his paper on the topic, published in 1806.

The geometrical representation of complex numbers is an unusually significant development because of its many important practical applications. Any number of physical phenomena involve the operation of two or more forces at once. For example, the flight of an airplane is determined both by the thrust of its own engines and by wind forces.

The resultant of those two forces can be calculated by vector analysis, the technique introduced by Wessel, Argand, and Gauss. The term vector itself was first suggested by the brilliant Irish mathematician William Rowan Hamilton. It refers to any number that has both magnitude and direction. For example, wind velocity is a vector because it blows with a particular speed (its magnitude) and a particular direction. Hamilton's work on vector analysis resulted from his interest in extending Gauss' two-dimensional analysis to the three dimensions of the real world. He was stymied in these efforts, however, until he made one of the great intuitive discoveries in mathematical history, namely that the commutative law of mathematics may not apply in all cases. That is, it may not always be true that (a) x (b) = (b) x (a). Starting from this revolutionary assumption, Hamilton developed a way of dealing with complex numbers in three dimensions using a technique known as an algebra of quaternions. Hamilton's analysis eventually became enormously useful in the analysis of many physical phenomena as, for example, in understanding atomic structure. The final stage in the development of basic vector theory evolved from the work of Hermann Günther Grassman (1809-1877) in the 1840s. Grassman found a method for dealing with complex numbers in more than three dimensions, numbers that became known as hypercomplex numbers. His work became the basis of geometrical studies in any number of dimensions, or n-dimensional geometry. Unfortunately, his writing was so obscure that its initial impact was less widespread than that of Hamilton's, and it continues to be a difficult challenge even for modern mathematicians.