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INTRODUCTION

What

1. In the study of Common Algebra the notion number is fundamental, and it is therefore necessary first of all to define it. are the properties or characteristics of number?

Given a group of objects, as marbles, a party of boys, a herd of horses, a village, or the like, distinctness or separateness of the things in any one of these groups is an intuitive property of these objects which enables one to realize that there is a marble, boy, horse, or house which is different and distinct from another marble, boy, horse, or house of the same group of marbles, boys, horses, or houses. If each marble of the group of marbles were replaced by an apple, then each apple by a nail, and so on; or if the marbles were painted different colors, arranged differently; or finally if any change were made in the things of the group which would not destroy their distinctness, the group of objects would contain as many individuals after any such change as it did before the change was made.

The notion of number is based upon this property of the separateness of the things in a group, and is defined as that property of a group of different things which is unchanged no matter what change is made in the things of the group without destroying the distinctness of the individual things.

Such changes affect only the character or arrangement of the things and do not cause any individual thing to be divided into two or more, or two or more to be merged into one. These characteristics of number expressed in the form of a theorem constitute the fundamental postulate of Arithmetic:

The number of individual things in a group of things does not depend upon the order of their arrangement in the group, their characteristics, or the way they may be related to one another in smaller groups.

2. The Equality of Two Groups. Consider two groups of letters,

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On comparing the individual objects of the groups 1 and 2, we may assign A to a, B to b, C to c, D to d, and E to e, and reciprocally, a to A, b to B, e to C, d to D, and e to E, i. e., there are just as many things in group 1 as in group 2.

The number of things in two groups of things is the same if to every thing in the first there may be assigned one in the second, and, reciprocally, to every thing in the second there can be assigned one in the first. Such a relation is called a one-to-one correspondence. In case of the groups of letters 3 and 4:

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to every letter in 4 may be assigned a letter in 3, thus, a to A, b to B, c to C, but, reciprocally, a letter of 4 can not be assigned to each letter of 3, thus: A to a, B to b, C to c, since the letters of group 4 are exhausted and there remain no letters of this group which can be assigned to the remaining letters d and e of group 3.

The number of things in group 3 is greater than the number of things in group 4, or the number of things in group 4 is less than that in group 3, when there is one thing in the first group for each thing in the second, but not reciprocally one thing in the second for each thing in the first.

Reciprocally, the number of things in group 5 is less than the number of things in group 6:

a

b

с

5

A B C D etc.

for, while it is possible to assign a letter of group 6 to each letter of group 5, thus, A to a, B to b, C to c, it is not possible, reciprocally, to assign to each letter in group 6 a letter of group 5, because after a has been assigned to A, b to B, c to C, there remain no letters in group 5 to assign to the remaining letters, D, etc., of group 6.

3. Representation of Numbers by Symbols.-When it is desired to compare the number of things in several groups of objects (sheep, cattle, horses, potatoes, bricks, etc.), the convenience of practical affairs demands that symbols be used to represent numbers-the totalities of things in groups of objects.

The number of things in a group can be represented by another group, e. g., by the fingers or any set of simple marks, thus:

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The number of things in group I can be represented by any one of the groups 1, 2, 3, since there is a one-to-one correspondence between the objects of groups 1, 2, 3 and group I.

The difference between the primitive and modern methods of representing groups of things is this: that the symbols in the middle row below or the numeral words of the third row are respectively used,

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