Complex numbers, though capable of a geometrical interpretation, are not demanded by geometry in the same imperative way in which irrationals are demanded. A "complex" number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number. Using the letter for the square root of , any number involving the square root of a negative number can be expressed in the form , where and are real. The part is called the "imaginary" part of this number, being the "real" part. (The reason for the phrase "real numbers" is that they are contrasted with such as are "imaginary.") Complex numbers have been for a long time habitually used by mathematicians, in spite of the absence of any precise definition. It has been simply assumed that they would obey the usual arithmetical rules, and on this assumption their employment has been found profitable. They are required less for geometry than for algebra and analysis. We desire, for example, to be able to say that every quadratic equation has two roots, and every cubic equation has three, and so on. But if we are confined to real numbers, such an equation as 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 [Pg 74] 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