Chapter 1
The Positive Integer, and the Laws which Regulate the Addition and Multiplication of Positive Integers

1. Number. We say of certain distinct things that they form a group¹ when we make them collectively a single object of our attention.

The number of things in a group is that property of the group which remains unchanged during every change in the group which does not destroy the separateness of the things from one another or their common separateness from all other things.

Such changes may be changes in the characteristics of the things or in their arrangement within the group. Again, changes of arrangement may be changes either in the order of the things or in the manner in which they are associated with one another in smaller groups.

We may therefore say:

The number of things in any group of distinct things is independent of the characters of these things, of the order in which they may be arranged in the group, and of the manner in which they may be associated with one another in smaller groups.

2. Numerical Equality. The number of things in any two groups of distinct things is the same, when for each thing in the first group there is one in the second, and reciprocally, for each thing in the second group, one in the first.

Thus, the number of letters in the two groups, $A$, $B$, $C$; $D$, $E$, $F$, is the same. In the second group there is a letter which may be assigned to each of the letters in the first: as $D$ to $A$, $E$ to $B$, $F$ to $C$; and reciprocally, a letter in the first which may be assigned to each in the second: as $A$ to $D$, $B$ to $E$, $C$ to $F$.

Two groups thus related are said to be in one-to-one (1–1) correspondence.

Underlying the statement just made is the assumption that if the two groups correspond in the manner described for one order of the things in each, they will correspond if the things be taken in any other order also; thus, in the example given, that if $E$ instead of $D$ be assigned to $A$, there will again be a letter in the group $D$, $E$, $F$, viz. $D$ or $F$, for each of the remaining letters $B$ and $C$, and reciprocally. This is an immediate consequence of § 1, foot-note.

The number of things in the first group is greater than that in the second, or the number of things in the second less than that in the first, when there is one thing in the first group for each thing in the second, but not reciprocally one in the second for each in the first.

3. Numeral Symbols. As regards the number of things which it contains, therefore, a group may be represented by any other group, e. g. of the fingers or of simple marks, $|$’s, which stands to it in the relation of correspondence described in § 2. This is the primitive method of representing the number of things in a group and, like the modern method, makes it possible to compare numerically groups which are separated in time or space.

The modern method of representing the number of things in a group differs from the primitive only in the substitution of symbols, as 1, 2, 3, etc., or numeral words, as one, two, three, etc., for the various groups of marks $|$, $\left|\right|$, $\left|\right||$, etc. These symbols are the positive integers of arithmetic.

A positive integer is a symbol for the number of things in a group of distinct things.

For convenience we shall call the positive integer which represents the number of things in any group its numeral symbol, or when not likely to cause confusion, its number simply,—this being, in fact, the primary use of the word “number” in arithmetic.

In the following discussion, for the sake of giving our statements a general form, we shall represent these numeral symbols by letters, $a$, $b$, $c$, etc.

4. The Equation. The numeral symbols of two groups being $a$ and $b$; when the number of things in the groups is the same, this relation is expressed by the equation

5. Counting. The fundamental operation of arithmetic is counting.

To count a group is to set up a one-to-one correspondence between the individuals of this group and the individuals of some representative group.

Counting leads to an expression for the number of things in any group in terms of the representative group: if the representative group be the fingers, to a group of fingers; if marks, to a group of marks; if the numeral words or symbols in common use, to one of these words or symbols.

There is a difference between counting with numeral words and the earlier methods of counting, due to the fact that the numeral words have a certain recognized order. As in finger-counting one finger is attached to each thing counted, so here one word; but that word represents numerically not the thing to which it is attached, but the entire group of which this is the last. The same sort of counting may be done on the fingers when there is an agreement as to the order in which the fingers are to be used; thus if it were understood that the fingers were always to be taken in normal order from thumb to little finger, the little finger would be as good a symbol for 5 as the entire hand.

6. Addition. If two or more groups of things be brought together so as to form a single group, the numeral symbol of this group is called the sum of the numbers of the separate groups.

If the sum be $s$, and the numbers of the separate groups $a$, $b$, $c$, etc., respectively, the relation between them is symbolically expressed by the equation

The operation of finding $s$ when $a$, $b$, $c$, etc., are known, is addition.

Addition is abbreviated counting.

Addition is subject to the two following laws, called the commutative and associative laws respectively, viz.:

Or,

Both these laws are immediate consequences of the fact that the sum-group will consist of the same individual things, and the number of things in it therefore be the same, whatever the order or the combinations in which the separate groups are brought together (§1).

7. Multiplication. The sum of $b$ numbers each of which is $a$ is called the product of $a$ by $b$, and is written $a\times b$, or $a\cdot b$, or simply $ab$.

The operation by which the product of $a$ by $b$ is found, when $a$ and $b$ are known, is called multiplication.

Multiplication is an abbreviated addition.

Multiplication is subject to the three following laws, called respectively the commutative, associative, and distributive laws for multiplication, viz.:

Or,

These laws are consequences of the commutative and associative laws for addition. Thus,

III. The Commutative Law. The units of the group which corresponds to the sum of $b$ numbers each equal to $a$ may be arranged in $b$ rows containing $a$ units each. But in such an arrangement there are $a$ columns containing $b$ units each; so that if this same set of units be grouped by columns instead of rows, the sum becomes that of $a$ numbers each equal to $b$, or $ba$. Therefore $ab=ba$, by the commutative and associative laws for addition.

IV. The Associative Law.

V. The Distributive Law.

The commutative, associative, and distributive laws for sums of any number of terms and products of any number of factors follow immediately from I–V. Thus the product of the factors $a$, $b$, $c$, $d$, taken in any two orders, is the same, since the one order can be transformed into the other by successive interchanges of consecutive letters.