Real The system just developed is very rich in numbers. In fact, it can be proved that no matter how near any two frac tions are to one another, there is a fraction lying between them in magnitude. Neverthe less, the series of fractions is everywhere full of gaps which prevent some of the very simplest equations from having solutions within it. For example, it can be shown that xl= 2 has no fractional or rational solution. If it had such a solution, it would necessarily be of the form nt n m and n can be so chosen as to have no factor other than ± 1 in common. ' Therefore, we can so choose m and n that at least one of them will be odd, if any values whatever exist m _ for m and n. Now, by hypothesis, 2, or m'=2 n', and m is consequently even. If m is even, it is of the form 2k. We thus get 4k'=2n', or n'=2k', whereby n is shown to be even, so that our previous hypothesis is con tradicted.
Though 2 has no rational square root, there are numbers whose squares exceed 2 by as little as we please, and numbers whose squares fall short of 2 by as little as we please. The ap proximate values we obtain in extracting the square root of 2 constitute a family of num bers which exemplifies this property : 2 l'=1, 2l.4=.04, 2 1.41'=.0119, 2l.414'= .009,064 and so on indefinitely, while 2' 2= 2, 1.5' 2=25, 2= .0168, 1.415' 2 .002,625, and so ad infinitum. That is, the proc ess which we call extracting the square root of 2 consists in finding sequences of pairs of num bers whose squares bracket 2 by intervals which decrease beyond any assignable degree of minuteness. To put it crudely, we mark the place where the square root of 2 ought to be by a definite dividing-line in the scale of rationale between those whose squares fall short of 2 and those whose squares exceed 2.
These dividing-lines between the numbers that are larger than a given standard and those that are not have been termed by Dedekind cuts. Some of these cuts are immediately bounded above or below by a rational number, as for instance the cut which separates all numbers less than 3 from 3 and greater num bers. Others, such as the cut discussed in the last paragraph, have not this property.
By considering only those cuts which do not immediately follow a rational number, we obtain one cut and one only corresponding to each rational number, and a cut, to put it crudely, where a number ought to be, but isn't. For instance, if we take our cuts as a larger system of numbers, we shall find that there will be a cut which we might call the square root of 2. If we define the sum of the two cuts C. and C, as the cut which divides all the sums of fractions on the lower side of C, with frac tions on the lower side of C, from all other fractions, and if we define C,C, as the cut dividing all products of fractions on the side of C, more remote from 0 with fractions on the side of C, more remote from 0 from all other fractions, we shall be able to prove both that all the convenient laws of addition and multi plication are satisfied and also that there will be no lacuna: in our system such as that which prevented .e.= 2 from being soluble in the
universe of rational numbers. If we define the order of magnitude of our cuts in the natural manner, making one cut precede another if the second cut contains on its lower side fractions larger than any on the lower side of the first, we shall not only find that this order is a series, and a series between any two of whose members there lies a third, but also that every cut in this series is bounded on one side or the other by one of the terms of the series. These properties go to make up what is known to mathematicians as continuity,* and the ordinary properties which we have learned to attribute to the number system of real algebra are those that belong to continuous series. Therefore, taking all these considerations to gether, we can define the real numbers of ordinary algebra as the cuts which we have just discussed.
Complex Even in the system of real numbers we meet with equations that have no solution. While x' 2 =0 has real roots, s' + 2 = 0 is satisfied by no real value of x. However, there is a system easily obtained from that of the real numbers in which this equation and many like it are soluble. Let us consider a system made up of ordered pairs of real numbers, such as (a,b). Let us define (a,b)+(c,d) as (a+c, b+d) and (a,b).(c,d) as (ac-bd, ad+bc). It will be found that the usual laws of commutativity, associativity and distributivity for addition and multiplication will be fulfilled in this system, and that the ordeded pairs of the form fa, 0) will have the same formal properties among themselves as the real numbers a. Further more the equation (0,1/2).(0,VT)=-(-2,0) is valid, so that the analogue of -2 will have a square root in this system, and x'+2=0 will be soluble, but it is still further demonstrable that if we define as x.x.x...x, every equation of n. times the form (hrn alxn-1 an-ix + an will have solution in the algebra of number-couples. The couple (a,b) is usually written a + ib, and the theorem just discussed, to the effect that every equation of the form given (called an algebraic equa tion) has a solution in the algebra of complex numbers is known as the Fundamental Theo rem of Algebra. It is due to Gauss.