While Euler advanced the theory of complex numbers by incorporating them into analysis, and
simplifying calculations by writing i instead of root
‐
1, there were many who would not be satisfied with
them until they could be explained geometrically. This was looked at by Caspar Wessel (1745
‐
1818) and
Jean
‐
Robert Argand (1768
‐
1822), who represented imaginary numbers as vectors in the x
‐
y plane.
However Wessel and Argand were not regarded as great mathematicians, and so it was Gauss and his
great reputation that finally allowed imaginary numbers (which Gauss referred to as “complex numbers”
to remove some of the mystique) to become accepted more broadly by mathematicians. Instead of
using vectors to denote the number a + bi, he used a point (a,b), and commented in an 1831 paper that
“[Imaginary numbers] have hitherto been considered from the wrong point of view and surrounded by a
mysterious obscurity. If for instance +1,
‐
1, root
‐
1 had been called direct, inverse, and lateral units,
instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out
of the question.”
Both Cauchy and Riemann made further advances, deriving the main results that are seen in a complex
variables class today, so that by the end of the 19th century the existence and theory of complex
numbers was generally beyond question. However resistance did still persist, especially in England.
Recalling his time as a student at Cambridge in the 1880‘s, A.R. Forsyth (1858
‐
1942) wrote that “It was
an age when the use of root
‐
1 was suspect, even in trigonometric formulae. The imaginary i was
suspiciously regarded as an untrustworthy intruder.”
Given how Gauss represented the complex number a + bi by the point (a,b) in the plane, we can write
their properties in purely algebraic terms, analogous to the way that was done for real numbers. For
example:
Addition: (a,b) + (c,d) = (a+c, b+d)
Multiplication: (a,b) x (c,d) = (ac
‐
bd, ad +bc)
Additive identity: (a,b) + (0,0) = (a,b)
Multiplicative identity: (a,b) x (1,0) = (a,b)
Commutative property for addition: (a,b) + (c,d) = (c,d) + (a,b)
Commutative property for multiplication: (a,b) x (c,d) = (c,d) x (a,b)