The Real Number System

THE REAL NUMBER SYSTEM Positive or Ordinary Numbers.—The concept of quantity expressed in terms of a suitable unit of measure is of almost the same intuitive nature as the concept of the integer. This is particularly true of geometric quantities such as lengths, areas and volumes. In the case in which the given quantity contains the unit an exact number of times, it is clear that the quantity may be designated by means of the corresponding integer.

For similar reasons a fraction fl (m and n being integers) may be taken to represent the nth part of m units.

Fractions were employed as early as 1700 B.C. by the ancient Egyptians, but the concept of ordinary number was first ade quately presented by Euclid. In the tenth book of his Elements he considered geometric magnitudes, and distinguished between commensurable and incommensurable quantities. For instance he established that the hypotenuse of an isosceles right triangle is incommensurable with the other two equal sides. In other words the number V 2, representing the hypotenuse in terms of one of the equal sides taken as unit, is an irrational number, not expressible as a fraction -"I. How then shall such irrational num n bers be represented by means of marks? This must be accom plished with the aid of a sequence of fractions which approach the irrational number as a limit. For instance when V 2 is represented in the form of an infinite decimal : V 2 = 1.4142 . . . , this amounts to a specification of V2 by means of the sequence of fractions The fundamental operation of addition of two such numbers may be interpreted geometrically, and in this way it is intuitively evident that the first two general laws must continue to hold.

Zero and Negative Numbers.

The real number system includes not only the positive or ordinary numbers, but zero and the so-called negative numbers, and is the system of most theoretical importance in the actual application of number to scientific questions. It might be shown how in various fields such a further extension is very natural ; for instance in dealing with temperature we must select an initial or zero point of the scale and must differentiate between temperatures above and below this point. Instead of doing so we propose to indicate how more purely mathematical considerations suggest this further extension. Similar considerations also suggest the extension already made from the integers to the fractions, but this extension can be based so directly upon experience as to make the statement of the purely mathematical considerations seem superfluous. The mathematical considerations entering are the following : For the free manipulation of numbers the use of the inverse operations is an indispensable adjunct. We may deduce the gen eral laws for such operations from the five fundamental laws. One illustration of this must suffice. The usual algebraic rule for the addition of two fractions with a common denominator is embodied in the formula To establish this law we may write a - = b - = y, whence we find d d xd= a, yd=b, in virtue of the definition of division. By addition we infer that whence follows which it was desired to prove. d In this free manipulation of quantities with the aid of these inverse operations, such symbols arise as i 1— 2, which do not represent ordinary numbers. Yet it is soon found that if

formally correct manipulations are made regardless of the mean ing of the symbols, the correct result is always obtained. This situation suggests inevitably that such symbols in reality repre sent a valid kind of number.

Let us assume that this is the case and consider the conclusions which follow. For brevity let us write a—a=o. This definition will be legitimate since we can establish formally that the equality a—a=b—b holds whatever a and b may be. The two further special laws a x o= o, a-Fo=a are obeyed by this symbol o, and are of the same nature as the special law a X already mentioned. Next write for the sake of brevity o —a= —a in case a is a positive number. Such a quantity may be called negative. With this extension of the number system to include o and the negative quantities, all symbols are found to be formally reducible to the same type, provided that division by o is excluded. Consequently the extension of the ordinary numbers by the intro duction of zero and the negative numbers follows from the free formal use of the laws of operation. Furthermore the rules for dealing with such numbers are obtained in the same way; for example the conclusion that the product of —a and —b is ab follows in this way.

Up to this point the method of approach has been heuristic rather than logical. It is now a simple matter to explain the method by which the real numbers can be satisfactorily defined in terms of ordinary numbers. We propose to consider ordinary numbers, the number o, and ordinary numbers with a — symbol prefixed. The first type of mark will be referred to as a positive number and the last type as a negative number. The number o will be regarded as a member of either type; i.e., —o=o by con vention. The rules of combination of these marks under the operations e and 0 (which are not to be confused with the operations + and X) will be defined in the following manner: If a and b are positive a e b is the positively taken sum; if either a or b is negative a eb is their difference considered as positive if the positive term is numerically greater, and negative in the contrary case; if both a and b are negative, aeb is their ordinary sum taken as negative. Likewise if a and b are both positive or both negative a eb is the positively taken product; and if one is positive and the other negative, a eb is the negatively taken product. Finally we define aeo and °Oa to be a in all cases, and a0o and o®a to be o in all cases.

It may then be verified, upon the basis of the properties of ordinary numbers, that the extended real number system satisfies the general and special laws of operation in all cases whatsoever; for example it may be verified that the commutative law of addi tion holds always. Furthermore it is readily verified that the inverse operations (except division by o) can be performed. These inverse operations will of course obey the usual formal laws which can be deduced from the give general laws for the direct operations. Hence we are fully justified in regarding the real number system so obtained as a complete generalization of the ordinary number system.