has no roots, and such an equation as
has only one. Every generalisation of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise
would be meaningless if
were less than
; fractions were needed in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible. But extensions of number are not created by the mere need for them: they are created by the definition, and it is to the definition of complex numbers that we must now turn our attention.
A complex number may be regarded and defined as simply an ordered couple of real numbers. Here, as elsewhere, many definitions are possible. All that is necessary is that the definitions adopted shall lead to certain properties. In the case of complex numbers, if they are defined as ordered couples of real numbers, we secure at once some of the properties required, namely, that two real numbers are required to determine a complex number, and that among these we can distinguish a first and a second, and that two complex numbers are only identical when the first real number involved in the one is equal to the first involved in the other, and the second to the second. What is needed further can be secured by defining the rules of addition and multiplication. We are to have