THE COMPLEX NUMBER SYSTEM As soon as equations of the second and higher degree are con sidered, a further and (in some ways) final extension of the number system is suggested. In fact the general formula for the solution of an equation of the second degree involves the extrac tion of the square root of a quantity which may be negative. But no real quantity has a negative square. Hence we are again led to meaningless symbols. However the symbol for the solution can be reduced formally to a+bi where i stands for -\,/ —1 so that — 1. Here bi is an abbreviation for b Xi.
The question then arises as to whether these symbols can be regarded as an appropriate generalization of the real numbers. If we write (a+bi)+(c+di)=(a+b)+(c+d)i, (a+bi) X (c+di) = ac + (ad+ bc)i+ = (ac— bd) + (ad+ bc)i, we define thereby the operations of addition and multiplication of these complex or imaginary numbers. It may then be verified that the usual formal laws will be satisfied in all cases. Conse quently the further extension of the number system must be regarded as legitimate.
The above treatment may be clarified as follows. Let us begin
by defining a complex number as a pair of associated real num bers (a, b). The definitions of addition e and multiplication 0 may be written Furthermore as a means of abbreviation we may write (a, o) = a and call these pairs a the real numbers of the extended system; in particular we have (o, o) = o, (I, o) =1. Likewise we write i= (o, and find from the definition of multiplication We are now in a position to deduce (a, b) = (a, o) (o, b) = (a, 0)e (b, ei= a (b 0i), which is essentially the form of expression a+bi with which we started.
When this complex number system is adopted the solution of all algebraic equations of any degree may be accomplished, even when the coefficients are themselves complex numbers. Thus from an algebraic standpoint no further generalization is called for. The so-called Argand diagram represents complex numbers by means of the points of a plane, and makes it evident graphically that these numbers satisfy the general and special laws. (See COMPLEX NUMBERS.)