Chapter 2
Subtraction and the Negative Integer

8. Numerical Subtraction. Corresponding to every mathematical operation there is another, commonly called its inverse, which exactly undoes what the operation itself does. Subtraction stands in this relation to addition, and division to multiplication.

To subtract b from $a$ is to find a number to which if $b$ be added, the sum will be $a$. The result is written $a-b$; by definition, it identically satisfies the equation

VI. ;

that is to say, $a-b$ is the number belonging to the group which with the $b$-group makes up the $a$-group.

Obviously subtraction is always possible when $b$ is less than $a$, but then only. Unlike addition, in each application of this operation regard must be had to the relative size of the two numbers concerned.

9. Determinateness of Numerical Subtraction. Subtraction, when possible, is a determinate operation. There is but one number which will satisfy the equation $x+b=a$, but one number the sum of which and $b$ is $a$. In other words, $a-b$ is one-valued.

For if $c$ and $d$ both satisfy the equation $x+b=a$, since then $c+b=a$ and $d+b=a$, $c+b=d+b$; that is, a one-to-one correspondence may be set up between the individuals of the and groups (§4). The same sort of correspondence, however, exists between any $b$ individuals of the first group and any $b$ individuals of the second; it must, therefore, exist between the remaining $c$ of the first and the remaining $d$ of the second, or $c=d$.

This characteristic of subtraction is of the same order of importance as the commutative and associative laws, and we shall add to the group of laws I–V and definition VI—as being, like them, a fundamental principle in the following discussion—the theorem

VII. $\left\{\begin{array}{ccc}\hfill \mathrm{\text{ If}}a+c& \hfill =\hfill & b+c\hfill \\ \hfill a& \hfill =\hfill & b,\hfill \end{array}\right.$

which may also be stated in the form: If one term of a sum changes while the other remains constant, the sum changes. The same reasoning proves, also, that

VIII. $\left\{\begin{array}{ccc}\hfill \mathrm{\text{ As}}a+c>& \hfill \mathrm{\text{ or}}\hfill & <b+c\hfill \\ \hfill a& \hfill \mathrm{\text{ or}}\hfill & b,\hfill \end{array}\right.$

10. Formal Rules of Subtraction. All the rules of subtraction are purely formal consequences of the fundamental laws I–V, VII, and definition VI. They must follow, whatever the meaning of the symbols $a$, $b$, $c$, $+$, $-$, $=$; a fact which has an important bearing on the following discussion.

It will be sufficient to consider the equations which follow. For, properly combined, they determine the result of any series of subtractions or of any complex operation made up of additions, subtractions, and multiplications.

1. .
2. .
3. $a+b-b=a$.
4. .
5. .

For

1. $a-b-c$ is the form to which if first $c$ and then $b$ be added; or, what is the same thing (by I), first $b$ and then $c$; or, what is again the same thing (by II), $b+c$ at once,—the sum produced is $a$ (by VI). $a-b-c$ is therefore the same as $a-c-b$, which is as it stands the form to which if $b$, then $c$, be added the sum is $a$; also the same as , which is the form to which if $b+c$ be added the sum is $a$.
2. 
   
3. 
   
   $$\begin{array}{ccc}\hfill a+b-b+b& =a+b.\hfill & \hfill \mathrm{\text{ Def. VI.}}\\ \hfill \mathrm{\text{ But }}a+b& =a+b.\hfill \\ \hfill \therefore a+b-b& =a.\hfill & \hfill \mathrm{\text{ Law VII.}}\end{array}$$
4. 
   
5. 
   

Equation 3 is particularly interesting in that it defines addition as the inverse of subtraction. Equation 1 declares that two consecutive subtractions may change places, are commutative. Equations 1, 2, 4 together supplement law II, constituting with it a complete associative law of addition and subtraction; and equation 5 in like manner supplements law V.

11. Limitations of Numerical Subtraction. Judged by the equations 1–5, subtraction is the exact counterpart of addition. It conforms to the same general laws as that operation, and the two could with fairness be made to interchange their rôles of direct and inverse operation.

But this equality proves to be only apparent when we attempt to interpret these equations. The requirement that subtrahend be less than minuend then becomes a serious restriction. It makes the range of subtraction much narrower than that of addition. It renders the equations 1–5 available for special classes of values of $a$, $b$, $c$ only. If it must be insisted on, even so simple an inference as that is equal to $b$ cannot be drawn, and the use of subtraction in any reckoning with symbols whose relative values are not at all times known must be pronounced unwarranted.

One is thus naturally led to ask whether to be valid an algebraic reckoning must be interpretable numerically and, if not, to seek to free subtraction and the rules of reckoning with the results of subtraction from a restriction which we have found to be so serious.

12. Symbolic Equations. Principle of Permanence. Symbolic Subtraction. In pursuance of this inquiry one turns first to the equation , which serves as a definition of subtraction when $b$ is less than $a$.

This is an equation in the primary sense (§ 4) only when $a-b$ is a number. But in the broader sense, that

An equation is any declaration of the equivalence of definite combinations of symbols—equivalence in the sense that one may be substituted for the other,— may be an equation, whatever the values of $a$ and $b$.

And if no different meaning has been attached to $a-b$, and it is declared that $a-b$ is the symbol which associated with $b$ in the combination is equivalent to $a$, this declaration, or the equation

is a definition² of this symbol.

By the assumption of the permanence of form of the numerical equation in which the definition of subtraction resulted, one is thus put immediately in possession of a symbolic definition of subtraction which is general.

The numerical definition is subordinate to the symbolic definition, being the interpretation of which it admits when $b$ is less than $a$.

But from the standpoint of the symbolic definition, interpretability—the question whether $a-b$ is a number or not—is irrelevant; only such properties may be attached to $a-b$, by itself considered, as flow immediately from the generalized equation

In like manner each of the fundamental laws I–V, VII, on the assumption of the permanence of its form after it has ceased to be interpretable numerically, becomes a declaration of the equivalence of certain definite combinations of symbols, and the formal consequences of these laws—the equations 1–5 of § 10—become definitions of addition, subtraction, multiplication, and their mutual relations—definitions which are purely symbolic, it may be, but which are unrestricted in their application.

These definitions are legitimate from a logical point of view. For they are merely the laws I–VII, and we may assume that these laws are mutually consistent since we have proved that they hold good for positive integers. Hence, if used correctly, there is no more possibility of their leading to false results than there is of the more tangible numerical definitions leading to false results. The laws of correct thinking are as applicable to mere symbols as to numbers.

What the value of these symbolic definitions is, to what extent they add to the power to draw inferences concerning numbers, the elementary algebra abundantly illustrates.

One of their immediate consequences is the introduction into algebra of two new symbols, zero and the negative, which contribute greatly to increase the simplicity, comprehensiveness, and power of its operations.

13. Zero. When $b$ is set equal to $a$ in the general equation

it takes one of the forms

It may be proved that

$a-a$ is therefore altogether independent of $a$ and may properly be represented by a symbol unrelated to a. The symbol which has been chosen for it is 0, called zero.

Addition is defined for this symbol by the equations

1. 
   
   $$\begin{array}{ccc}\hfill 0+a& =a,\hfill & \hfill \mathrm{\text{ definition of 0.}}\\ \hfill a+0& =a.\hfill & \hfill \mathrm{\text{ Law I.}}\end{array}$$

   Subtraction (partially), by the equation

2. 
   

   Multiplication (partially), by the equations

3. 
   

14. The Negative. When $b$ is greater than $a$, equal say to $a+d$, so that $b-a=d$, then

For $0-d$ the briefer symbol $-d$ has been substituted; with propriety, certainly, in view of the lack of significance of 0 in relation to addition and subtraction. The equation $0-d=-d$, moreover, supplies the missing rule of subtraction for 0. (Compare § 13, 2.)

The symbol $-d$ is called the negative, and in opposition to it, the number $d$ is called positive.

Though in its origin a sign of operation (subtraction from 0), the sign $-$ is here to be regarded merely as part of the symbol $-d$.

$-d$ is as serviceable a substitute for $a-b$ when $a<b$, as is a single numeral symbol when $a>b$.

The rules for reckoning with the new symbol—definitions of its addition, subtraction, multiplication—are readily deduced from the laws I–V, VII, definition VI, and the equations 1–5 of § 10, as follows:

1. 
   

   $-b$ may therefore be defined as the symbol the sum of which and $b$ is 0.

2. 
   
3. 
   
4. 
   
5. 
   
6. 
   
7. 
   
8. 
   
   By this method one is led, also, to definitions of equality and greater or lesser inequality of negatives. Thus
9. 
   

15. Recapitulation. The nature of the argument which has been developed in the present chapter should be carefully observed.

From the definitions of the positive integer, addition, and subtraction, the associative and commutative laws and the determinateness of subtraction followed. The assumption of the permanence of the result $a-b$, as defined by , for all values of $a$ and $b$, led to definitions of the two symbols $0$, $-d$, zero and the negative; and from the assumption of the permanence of the laws I–V, VII were derived definitions of the addition, subtraction, and multiplication of these symbols,—the assumptions being just sufficient to determine the meanings of these operations unambiguously.

In the case of numbers, the laws I–V, VII, and definition VI were deduced from the characteristics of numbers and the definitions of their operations; in the case of the symbols $0$, $-d$, on the other hand, the characteristics of these symbols and the definitions of their operations were deduced from the laws.

With the acceptance of the negative the character of arithmetic undergoes a radical change.⁴ It was already in a sense symbolic, expressed itself in equations and inequalities, and investigated the results of certain operations. But its symbols, equations, and operations were all interpretable in terms of the reality which gave rise to it, the number of things in actually existing groups of things. Its connection with this reality was as immediate as that of the elementary geometry with actually existing space relations.

But the negative severs this connection. The negative is a symbol for the result of an operation which cannot be effected with actually existing groups of things, which is, therefore, purely symbolic. And not only do the fundamental operations and the symbols on which they are performed lose reality; the equation, the fundamental judgment in all mathematical reasoning, suffers the same loss. From being a declaration that two groups of things are in one-to-one correspondence, it becomes a mere declaration regarding two combinations of symbols, that in any reckoning one may be substituted for the other.