It was in the nineteenth century that algebra acquired the very abstract nature it has today. The key event in its development was the overthrow of the law of commutativity for multiplication (that is, for all x and y, xy = yx). This was accomplished by William Rowan Hamilton (1805–1865) who, incidentally, was the first person to conceive of complex numbers as ordered pairs. Just as many people before Lobachevsky thought that Euclid’s parallel postulate was a kind of sacred truth, so many people before Hamilton thought that the law of commutativity for multiplication was ineluctable. For us it is a commonplace that this law need not hold, since we have a ready example of noncommutativity in matrix multiplication. Hamilton, however, made his discovery about fifteen years before matrix algebra had been discovered.
Unable to display preview. Download preview PDF.