Remaining doubts about the legitimacy of complex numbers were finally dispelled when their geometric interpretation became widespread among mathematicians. This interpretation, initially and independently conceived by the Norwegian surveyor Caspar Wessel and the French bookkeeper Jean-Robert Argand (see Argand diagram), was made known to a larger audience of mathematicians mainly through its explicit use by Gauss in his 1848 proof of the fundamental theorem of algebra. Under this interpretation, every complex number appeared as a directed segment on the plane, characterized by its length and its angle of inclination with respect to the x-axis. The number i thus corresponded to the segment of length 1 that was perpendicular to the x-axis. Once a proper arithmetic was defined on these numbers, it turned out that i2 = −1, as expected.
An alternative interpretation, very much within the spirit of the British school of symbolic algebra, was published in 1837 by Hamilton. Hamilton defined a complex number a + bi as a pair (a, b) of real numbers and gave the laws of arithmetic for such pairs. For example, he defined multiplication as:
(a, b)(c, d) = (ac − bd, bc + ad).
In Hamilton’s notation i = (0, 1) and by the above definition of complex multiplication (0, 1)(0, 1) = (−1, 0)—that is, i2 = −1 as desired. This formal interpretation obviated the need to give any essentialist definition of complex numbers.
Starting in 1830, Hamilton pursued intensely, and unsuccessfully, a scheme to extend his idea to triplets (a, b, c), which he expected to be of great utility in mathematical physics. His difficulty lay in defining a consistent multiplication for such a system, which in hindsight is known to be impossible. In 1843 Hamilton finally realized that the generalization he was looking for had to be found in the system of quadruplets (a, b, c, d), which he named quaternions. He wrote them, in analogy with the complex numbers, as a + bi + cj + dk, and his new arithmetic was based on the rules: i2 = j2 = k2 = ijk = −1 and ij = k, ji = −k, jk = i, kj = −i, ki = j, and ik = −j. This was the first example of a coherent, significant mathematical system that preserved all of the laws of ordinary arithmetic, with the exception of commutativity.
In spite of Hamilton’s initial hopes, quaternions never really caught on among physicists, who generally preferred vector notation when it was introduced later. Nevertheless, his ideas had an enormous influence on the gradual introduction and use of vectors in physics. Hamilton used the name scalar for the real part a of the quaternion, and the term vector for the imaginary part bi + cj + dk, and defined what are now known as the scalar (or dot) and vector (or cross) products. It was through successive work in the 19th century of the Britons Peter Guthrie Tait, James Clerk Maxwell, and Oliver Heaviside and the American Josiah Willard Gibbs that an autonomous theory of vectors was first established while developing on Hamilton’s initial ideas. In spite of physicists’ general lack of interest in quaternions, they remained important inside mathematics, although mainly as an example of an alternate algebraic system.
An alternative interpretation, very much within the spirit of the British school of symbolic algebra, was published in 1837 by Hamilton. Hamilton defined a complex number a + bi as a pair (a, b) of real numbers and gave the laws of arithmetic for such pairs. For example, he defined multiplication as:
(a, b)(c, d) = (ac − bd, bc + ad).
In Hamilton’s notation i = (0, 1) and by the above definition of complex multiplication (0, 1)(0, 1) = (−1, 0)—that is, i2 = −1 as desired. This formal interpretation obviated the need to give any essentialist definition of complex numbers.
Starting in 1830, Hamilton pursued intensely, and unsuccessfully, a scheme to extend his idea to triplets (a, b, c), which he expected to be of great utility in mathematical physics. His difficulty lay in defining a consistent multiplication for such a system, which in hindsight is known to be impossible. In 1843 Hamilton finally realized that the generalization he was looking for had to be found in the system of quadruplets (a, b, c, d), which he named quaternions. He wrote them, in analogy with the complex numbers, as a + bi + cj + dk, and his new arithmetic was based on the rules: i2 = j2 = k2 = ijk = −1 and ij = k, ji = −k, jk = i, kj = −i, ki = j, and ik = −j. This was the first example of a coherent, significant mathematical system that preserved all of the laws of ordinary arithmetic, with the exception of commutativity.
In spite of Hamilton’s initial hopes, quaternions never really caught on among physicists, who generally preferred vector notation when it was introduced later. Nevertheless, his ideas had an enormous influence on the gradual introduction and use of vectors in physics. Hamilton used the name scalar for the real part a of the quaternion, and the term vector for the imaginary part bi + cj + dk, and defined what are now known as the scalar (or dot) and vector (or cross) products. It was through successive work in the 19th century of the Britons Peter Guthrie Tait, James Clerk Maxwell, and Oliver Heaviside and the American Josiah Willard Gibbs that an autonomous theory of vectors was first established while developing on Hamilton’s initial ideas. In spite of physicists’ general lack of interest in quaternions, they remained important inside mathematics, although mainly as an example of an alternate algebraic system.
