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General groups

A

Groups of marks

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Modern symbols
1 2 3 4 5
- Numeral words
One two three four five

instead of the groups of marks in the first row, for the groups of things in the circles to which the groups of marks respectively correspond. The positive integer is the symbol used to represent the number of things in a group. For example, the symbols 1, 2, 3, 4, 5, in the illustration given above are positive integers. The primary use of the word number is that of the positive integer, a numerical symbol denoting the totality of the things in a group. In the paragraphs which immediately follow, 1, 2, 3, 4, etc., in particular discussions, and the letters a, b, c, etc., in general discussions, are used as positive integers.

4. The Equation.—If a and b are the numerical symbols which represent the number of things in two"groups, and there is a one-toOne correspondence between the objects of the groups, this relation of

a group' b group the two groups to each other is indicated by the symbolical relation, or - b; for example, 5 = 5,

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which is called an equation or equality. The symbol = is read equals, and the equation a = b is read a is equal to b. If the first group is greater than the second, the relation of the groups is

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If the first group is less than the second, the relation of the

A F B G II E. C D a group b group groups is represented by the inequality, a 3 b; for example 5 × 8.

A numerical equation simply declares, in terms of the symbols which represent the groups, the numerical relation which exists between these groups; and the symbol =, that these groups are in a one-to-one correspondence.

5. Counting.—To count the things in a given group of things is to seek a one-to-one correspondence between the things of this group and the individual things of some group (or groups) which is known.

The fundamental operation in Arithmetic is counting.

Counting the things in a group leads to a numerical expression in terms of the representative groups; if the representative group is a group of marks, in terms of this group of marks; if it is fingers, in terms of the group of fingers; if it is one of the numeral words or symbols in common use, to one of these words or symbols.

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For example, since there is a one-to-one correspondence between the letters of the group to the left and the marks of the group to the right, counting the letters in the first group leads to the

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group of marks (or 9) which may be taken to represent the number of objects in the group of letters.

BOOK I

CHA PTER I

ADDITION AND MULTIPLICATION

ADDITION 6. Two groups a and b A B C. D E F. G. H. (t. b

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C

by removing the vertical bar which separates them.

The numerical symbol 8 of group c is the number of letters in the groups a and b, or the sum of their numerical symbols, 3 and 5. This result written in the form of an equation is and is read eight is equal to the sum of three and five, or is equal to three plus five.

The sum-group 8 is formed by uniting group b, to which 5 belongs, to group a, to which 3 belongs.

In general, if a, b, c, etc., are the numerical symbols for the number of things in the groups 1, 2, 3, etc., then the number corresponding to the sum-group including the a's, b’s, c's, etc., is s = (a+b)+ c + etc. The sign of continuation, . . . . . . , is read and so on, thus, 1, 2, 3, 4, 5, 6, . . . . . . is read one, two, three, four, five, sir, and so on. The symbol ( ) is read parentheses, and when it encloses the sum or sums of two or more numbers it indicates that all within it is to be treated as a single number. s will be the numerical sum of the groups 1, 2, 3, etc., that is, the number corresponding to the group d.

al b, Cl as b, Ca a's b, 6s

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The sum-group is found by joining the group, 2, of b"s to the group, 1, of a's, giving a group with the numerical symbol (a + b); then the group, 3, of c's to the resulting group of the a's and bos, giving the numerical symbol (a + b) + c, and so on

(figure d).

Addition is the operation of finding 8 when 3 and 5 are given, or of finding s when a, b, c, etc., are given. That is, a group 8 has been found as the result of bringing together the groups 3 and 5, and a group s as the result of uniting the groups a, b, c, etc.

Thus it follows that addition is abbreviated counting.

Addition, in consequence of its definition (addition of groups a and b and of groups 1, 2, 3, etc.), is subject to the following laws, called the Commutative and Associative Laws respectively, viz.:

In particular In general I. Commutative 3 + 5 = 5 + 3 a + b = b + a To add 5 to 3 is the same as To add b to a is the same as to add 3 to 5. to add a to b.

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