text: Hypercomplex numbers, better known as Quaternions, were discovered by Hamilton in the nineteenth century. It had long been appreciated that the real numbers have a geometrical interpretation as points lying on a line, and that the complex numbers have a similar geometrical interpretation as points lying in a plane.\nAs Hamilton pointed out in a paper in 1833, the addition symbol used in the Cartesian representation of a complex number \n\na+ib,\n\nis somewhat misleading, since a real and purely imaginary number cannot be directly added together arithmetically. \nA more suitable representation might be as an ordered pair of real numbers \n\n(a,b),\n\ntogether with a set of manipulation rules that define how to perform operations like addition and multiplication of these pairs.