Page:EB1911 - Volume 19.djvu/881

this by ${\displaystyle \upsilon }$ we have ${\displaystyle [a,b]+\upsilon =[a+b,b]>[a,b],[a,b]\times \upsilon =[a,b]}$, so that ${\displaystyle \upsilon }$ is the ground element of the new aggregate.

Again ${\displaystyle 2\upsilon =\upsilon +\upsilon =(2,1)}$, and by induction ${\displaystyle m\upsilon =[m,1]}$. Moreover, if ${\displaystyle a}$ is a multiple of ${\displaystyle b}$, say ${\displaystyle mb}$, we may denote ${\displaystyle [a,b]}$ by ${\displaystyle m\upsilon }$.

11. The new aggregate of couples will be denoted by ${\displaystyle {\mbox{R}}}$. It differs from ${\displaystyle {\mbox{N}}}$ and ${\displaystyle {\bar {\mbox{N}}}}$ in one very important respect, namely, that when its elements are arranged in order of magnitude (that is to say, by the rule above given) they are not isolated from each other. In fact if ${\displaystyle [a,b]=\alpha }$, and ${\displaystyle [a',b']=\alpha '}$, the element ${\displaystyle [a+a',b+b']}$ lies between ${\displaystyle \alpha }$ and ${\displaystyle \alpha '}$; hence it follows that between any two different elements of ${\displaystyle {\mbox{R}}}$ we can find as many other elements as we please. This property is expressed by saying that ${\displaystyle {\mbox{R}}}$ is in close order when its elements are arranged in order of magnitude. Strange as it appears at first sight, ${\displaystyle {\mbox{R}}}$ is a countable aggregate; a theorem first proved by G. Cantor. To see this, observe that every element of R may be represented by a “reduced” couple ${\displaystyle [a,b]}$, in which ${\displaystyle a,b}$ are prime to each other. If ${\displaystyle [a,b],[c,d]}$ are any two reduced couples, we will agree that ${\displaystyle [a,b]}$ is anterior to ${\displaystyle [c,d]}$ if either (1) ${\displaystyle a+b<c+d}$, or (2) ${\displaystyle a+b<c+d}$ but ${\displaystyle a<c}$. This gives a new criterion by which all the elements of R can be arranged in the succession

which is similar to the natural scale.

The aggregate ${\displaystyle {\mbox{R}}}$, arranged in order of magnitude, agrees with ${\displaystyle {\bar {\mbox{N}}}}$ in having no least and no greatest element; for if ${\displaystyle \alpha }$ denotes any element ${\displaystyle [a,b]}$, then ${\displaystyle [2a-1,2b]<\alpha }$.

12. The division of one element of ${\displaystyle {\mbox{R}}}$ by another is always possible; for by definition

and consequently ${\displaystyle [a,b]\div [c,d]}$ is always interpretable as ${\displaystyle [ad,bc]}$. As a particular case ${\displaystyle [m,1]\div [n,1]=[m,n]}$, so that every element of ${\displaystyle {\mbox{R}}}$ is expressible in one of the forms ${\displaystyle m\upsilon ,m\upsilon /n\upsilon }$. It is usual to omit the symbol ${\displaystyle \upsilon }$ altogether, and to represent the element ${\displaystyle [m,n]}$ by ${\displaystyle m/n}$, whether ${\displaystyle m}$ is a multiple of ${\displaystyle n}$ or not. Moreover, ${\displaystyle m/1}$ is written ${\displaystyle m}$, which may be done without confusion, because ${\displaystyle m/1+n/1=(m+n)/1}$, and ${\displaystyle m/1\times n/1=mn/1}$, by the rules given above.

13. Within the aggregate ${\displaystyle \mathrm {R} }$ subtraction is not always practicable; but this limitation may be removed by constructing an aggregate ${\displaystyle {\bar {\mathrm {R} }}}$ related to ${\displaystyle \mathrm {R} }$ in the same way as ${\displaystyle {\bar {\mathrm {N} }}}$ to ${\displaystyle \mathrm {N} }$. This may be done in two ways which lead to equivalent results. We may either form symbols of the type ${\displaystyle (\alpha ,\beta )}$ , where ${\displaystyle \alpha ,\beta }$ denote elements of ${\displaystyle \mathrm {R} }$, and apply the rules of § 7 ; or else form symbols of the type ${\displaystyle [\alpha ,\beta ]}$, where ${\displaystyle \alpha ,\beta }$ denote elements of ${\displaystyle {\bar {\mathrm {N} }}}$, and apply the rules of § 10. The final result is that ${\displaystyle {\bar {\mathrm {R} }}}$ contains a zero element, ${\displaystyle 0}$, а ground element ${\displaystyle \upsilon }$, an element ${\displaystyle \upsilon '}$ such that ${\displaystyle \upsilon +\upsilon '=0}$, and a set of elements representable by the symbols (${\displaystyle m/n)\upsilon ,(m/n)\upsilon '}$. In this notation the rules of operation are

Here ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ denote any two elements of ${\displaystyle {\bar {\mbox{R}}}}$. If ${\displaystyle \beta =(m/n)\upsilon }$, then ${\displaystyle \beta '=(m/n)\upsilon '}$, and if ${\displaystyle \beta =(m/n)\upsilon '}$, then ${\displaystyle \beta '=(m/n)\upsilon }$. If ${\displaystyle \beta =0}$, then ${\displaystyle \beta '=0}$

14. When ${\displaystyle {\bar {\mbox{R}}}}$ is constructed by means of couples taken from ${\displaystyle {\bar {\mbox{N}}}}$, we must put ${\displaystyle [m\iota ,n\iota ]=[m\iota ',n\iota ']=(m/n)\upsilon }$, ${\displaystyle [m\iota ,n\iota ']=[m\iota ',n\iota ]=(m/n)\upsilon '}$, and ${\displaystyle [0,\alpha ]=0}$, if ${\displaystyle \alpha }$ is any element of ${\displaystyle {\bar {\mbox{N}}}}$ except ${\displaystyle 0}$. The symbols ${\displaystyle [0,0]}$ and ${\displaystyle [\alpha ,0]}$ are inadmissible; the first because it satisfies the definition of equality (§ 10) with every symbol ${\displaystyle [\alpha ,\beta ]}$, and is therefore indeterminate; the second because, according to the rule of addition,

which is inconsistent with ${\displaystyle \xi +\upsilon >\xi }$

In the same way, if ${\displaystyle 0}$ denotes the zero element of ${\displaystyle {\bar {\mbox{R}}}}$, and ${\displaystyle \xi }$ any other element, the symbol ${\displaystyle 0/0}$ is indeterminate, and ${\displaystyle \xi /0}$ inadmissible, because, by the formal rules of operation, ${\displaystyle \xi /0+\upsilon =\xi /0}$, which conflicts with the definition of the ground element ${\displaystyle \upsilon }$. It is usual to write ${\displaystyle +{\frac {m}{n}}}$ (or simply ${\displaystyle {\frac {m}{n}}}$) for ${\displaystyle {\frac {m}{n}}\upsilon }$, and ${\displaystyle -{\frac {m}{n}}}$ for ${\displaystyle {\frac {m}{n}}\upsilon '}$. Each of these elements is said to have the absolute value ${\displaystyle m/n}$. The criterion for arranging the elements of ${\displaystyle {\bar {\mbox{R}}}}$ in order of magnitude is that, if ${\displaystyle \xi ,\eta }$ are any two elements of it, ${\displaystyle \xi >\eta }$ when ${\displaystyle \xi -\eta }$ is positive; that is to say, when it can be expressed in the form ${\displaystyle (m/n)\upsilon }$.

15. The aggregate ${\displaystyle {\bar {\mbox{R}}}}$ is very important, because it is the simplest type of a field of rationality, or corpus. An algebraic corpus is an aggregate, such that its elements are representable by symbols ${\displaystyle \alpha ,\beta }$, &c., which can be combined according to the laws of ordinary algebra; every algebraic expression obtained by combining a finite number of symbols, by means of a finite chain of rational operations, being capable of interpretation as representing a definite element of the aggregate, with the single exception that division by zero is inadmissible. Since, by the laws of algebra, ${\displaystyle \alpha -\alpha =0}$, and ${\displaystyle \alpha /\alpha =1}$, every algebraic field contains ${\displaystyle {\bar {\mbox{R}}}}$, or, more properly, an aggregate which is an image of ${\displaystyle {\bar {\mbox{R}}}}$.

16. Irrational Numbers.—Let ${\displaystyle \alpha }$ denote any element of ${\displaystyle {\bar {\mbox{R}}}}$; then ${\displaystyle \alpha }$ and all lesser elements form an aggregate, ${\displaystyle {\mbox{A}}}$ say; the remaining elements form another aggregate ${\displaystyle {\mbox{A}}'}$, which we shall call complementary to ${\displaystyle {\mbox{A}}}$, and we may write ${\displaystyle {\bar {\mbox{R}}}={\mbox{A}}+{\mbox{A}}'}$. Now the essence of this separation of ${\displaystyle {\bar {\mbox{R}}}}$ into the parts ${\displaystyle {\mbox{A}}}$ and ${\displaystyle {\mbox{A}}'}$ may be expressed without any reference to ${\displaystyle \alpha }$ as follows:—

I. The aggregates ${\displaystyle {\mbox{A}},{\mbox{A}}'}$ are complementary; that is, their elements, taken together, make up the whole of ${\displaystyle {\bar {\mbox{R}}}}$.

II. Every element of ${\displaystyle {\mbox{A}}}$ is less than every element of ${\displaystyle {\mbox{A}}'}$.

III. The aggregate ${\displaystyle {\mbox{A}}'}$ has no least element. (This condition is artificial, but saves a distinction of cases in what follows.)

Every separation ${\displaystyle {\bar {\mbox{R}}}={\mbox{A}}+{\mbox{A}}'}$ which satisfies these conditions is called a cut (or section), and will be denoted by ${\displaystyle ({\mbox{A}},{\mbox{A}}')}$. We have seen that every rational number ${\displaystyle \alpha }$ can be associated with a cut. Conversely, every cut ${\displaystyle ({\mbox{A}},{\mbox{A}})}$ in which ${\displaystyle {\mbox{A}}}$ has a last element ${\displaystyle \alpha }$ is perfectly definite, and specifies ${\displaystyle \alpha }$ without ambiguity. But there are other cuts in which ${\displaystyle {\mbox{A}}}$ has no last element. For instance, all the elements (${\displaystyle \alpha }$) of ${\displaystyle {\bar {\mbox{R}}}}$ such that either ${\displaystyle \alpha \leq 0}$, or ${\displaystyle \alpha ^{2}<2}$, form an aggregate ${\displaystyle {\mbox{A}}}$, while those for which ${\displaystyle \alpha >0}$ and ${\displaystyle \alpha ^{2}>2}$, form the complementary aggregate ${\displaystyle {\mbox{A}}'}$. This separation is a cut in which ${\displaystyle {\mbox{A}}}$ has no last element; because if ${\displaystyle p/q}$ is any positive element of ${\displaystyle {\mbox{A}}}$, the element ${\displaystyle (3p+4q)/(2p+3q)}$ exceeds ${\displaystyle p/q}$, and also belongs to ${\displaystyle {\mbox{A}}}$. Every cut of this kind is said to define an irrational number. The justification of this is contained in the following propositions:—

(1) A cut is a definite concept, and the assemblage of cuts is an aggregate according to definition; the generic quality of the aggregate being the separation of ${\displaystyle {\bar {\mbox{R}}}}$ into two complementary parts, without altering the order of its elements.

(2) The aggregate of cuts may be arranged in order by the rule that ${\displaystyle ({\mbox{A}},{\mbox{A}}')<({\mbox{B}},{\mbox{B}}')}$ if ${\displaystyle {\mbox{A}}}$ is a part of ${\displaystyle {\mbox{B}}}$.

(3) This criterion of arrangement preserves the order of magnitude of all rational numbers.

(4) Cuts may be combined according to the laws of algebra, and, when the cuts so combined are all rational, the results are in agreement with those derived from the rational theory.

As a partial illustration of proposition (4) let ${\displaystyle ({\mbox{A}},{\mbox{A}}'),({\mbox{B}},{\mbox{B}}')}$ be any two cuts ; and let ${\displaystyle {\mbox{C}}'}$ be the aggregate whose elements are obtained by forming all the values of ${\displaystyle \alpha '+\beta '}$, where ${\displaystyle \alpha '}$ is any element of ${\displaystyle {\mbox{A}}'}$ and ${\displaystyle \beta '}$ is any element of ${\displaystyle {\mbox{B}}'}$. Then if ${\displaystyle {\mbox{C}}}$ is the complement of ${\displaystyle {\mbox{C}}'}$, it can be proved that ${\displaystyle ({\mbox{C}},{\mbox{C}}')}$ is a cut; this is said to be the sum of ${\displaystyle ({\mbox{A}},{\mbox{A}}')}$ and ${\displaystyle ({\mbox{B}},{\mbox{B}}')}$. The difference, product and quotient of two cuts may be defined in a similar way. If ${\displaystyle {\mathfrak {n}}}$ denotes the irrational cut chosen above for purposes of illustration, we shall have ${\displaystyle {\mathfrak {n}}^{2}=({\mbox{C}},{\mbox{C}}')}$ where ${\displaystyle {\mbox{C}}'}$ comprises all the numbers ${\displaystyle \alpha '\beta '}$ obtained by multiplying any two elements, ${\displaystyle \alpha ',\beta '}$ which are rational and positive, and such that ${\displaystyle \alpha '^{2}>2,\beta '^{2}>2}$. Since ${\displaystyle \alpha '^{2}>2\beta '^{2}>2>4}$ it follows that ${\displaystyle \alpha '\beta '}$ is positive and greater than ${\displaystyle 2}$; it can be proved conversely that every rational number which is greater than ${\displaystyle 2}$ can be expressed in the form ${\displaystyle \alpha '\beta '}$. Hence ${\displaystyle {\mathfrak {n}}^{2}=2,}$ so that the cut ${\displaystyle {\mathfrak {n}}}$ actually gives a real arithmetical meaning to the positive root of the equation ${\displaystyle x^{2}=2}$; in other words we