In ordinary algebra we find that there are ways of forming classes by picking out one term from each of v mutually exclusive assemblages of P terms each. As this process may be applied to all assemblages whether finite or infinite, we make it the basis for our definition of involution. The ordinary laws of involution — that µ v +.7r x that /iv X = (g" )", and that Cu x lor X v — follow readily for all the num bers in our algebra.
Extension of the If we limit our discussion to what we have called the natural numbers, the system which we have obtained has identically the same algorithm as the algebra of our high-school days, in so far as we limit its application to the positive in tegers and 0. This system is called Number Theory (see NUMBERS, THEORY OF). Now, number-theory is famous alike for the dif ficulty and the lack of generality of its methods and for its very slight practical value, so that a more regular and a more useful algebra becomes a desideratum. We want to be able to solve the equations x+5=..-3, 2x=7, x' - 2=0, and s' + 2=0, all of which lack a solution of the algebra so far at our disposal. Be it noticed, the mere fact that these equations are insoluble within the present system does not permit us to assign to them arbitrary solu tions; there is no more justification for saying that x + 5 = 3 °must have' a solution than there would be for saying that there °must be' an even prime greater than two. The absence of a solution to a problem is never a sufficient ex cuse for building that solution out of whole cloth. Our search for a more complete algebra must consequently take the form of a hunt for systems that actually do obey more universal laws than those of the signless integers, rather than that of filling out the system of the sign less integers by the haphazard introduction of heterogeneous material.
Integers with Here the problem is that of finding a system in which such equa tions as .r + 3 = 2 are soluble. When we have once defined addition, the definition of subtrac tion follows without any further ado: P - v is defined as that number which when added to v gives L.• p v and p - v may be regarded as the results of performing on p the opera tions of adding or subtracting v. Let us call these operations + v and — v, respec tively. We shall define the sum of two operations of this sort as the operation consisting in their consecutive application. The word °sum° will here have a significance different from that which it possesses with reference to p and v. Its use, however, is natural, for the adding of numerical °steps* is closely analogous to the process whereby we add, say, two linear steps or lengths by the re peated application of a yardstick. Furthermore,
when only entities of the form +11 are con sidered, the formal properties of the new sum mation will be the same as those of the old, When operations of the form - p are admitted, our new additions will have all the properties we are accustomed to associate with addition among integers with sign, among which are the properties of associativity and commutativity. Similarly, if we define (+ it) (+ v) and (- p) (-v) as + (# X v), and (-I- #) ( ( -p) ( +v) as - ( p X v ), we shall find that the multiplication so defined will have the same properties intrinsically and with re spect to addition that we should naturally expect multiplication to possess in the uni verse of integers with sign. Among these are the claws' of associativity, com mutativity and distributivity. Consequently, putting all things together, there is no rea son why we should not call our + p's. and —p's positive and negative integers, respec tively, provided v t 0. In this system, we shall find that every equation of the form x + a= b will be satisfied by some value of x. It is note worthy that the system of integers for which this is true does not contain any of the integers of previous paragraphs, so that a positive in teger turns out to differ from the corresponding signless or absolute integer.' Fractions or Rational Equa tions such as 3 x =2 are rendered soluble in a way quite analogous to the procedure of the last paragraph. Let m and n be any two posi tive or negative integers. We shall define as the relation which subsists between any two integers p and q if mq=np. We shall define II .2 as mq+ np and m. p as m. p We , — — .13 - .
n q nq n q n. q shall say that > q if ti, q, and mq—np are positive. When this has been done, it is demonstrable that the formal laws of tivity, commutativity and distributivity hold in the proper manner for addition and tion, that every equation of the form ax=b has a solution when the entities concerned are expressions such as , or as they are called, fractions or rational numbers, and that the relation > determines an order among the fractions such that (a) we never have.x > x; (b) if x > y and > z, then x > z; (c) if x and y are two distinct fractions, either x > y or y > x. This latter set of facts is expressed by the statement that the fractions arranged in order of magnitude form a series. Be it noticed that though the fraction x may be taken to represent the integer x and is usually expressed by the symbol x, it is altogether dis tinct from that integer.