CHAPTER I.

SYMBOLS, DEFINITIONS, AND FORMAL LAWS.

1. Arithmetic is pure or concrete. Pure arithmetic deals with abstract number or numerical quantity. Concrete arithmetic has relation to numbers of concrete objects or things.

Thus 3 is an abstract number, but 3 days is concrete. Algebra is primarily related to pure arithmetic, but its extension to concrete arithmetic is an easy matter.

The quantities which are the subject of arithmetic are of three kinds :

(1) Whole numbers or integers;

(2) Symbolized operations called fractions;

(3) Numerical quantities which cannot be exactly expressed as integers or fractions, but whose values may be expressed to any required degree of approximation. Such are the square roots of the non-square numbers, the cube roots of the non-cube numbers, etc. This third class goes under the general name of incommensurables. The expression numerical quantity, and frequently the word number, will be taken to denote any of the three classes.

2. Numbers are fundamentally subject to two operations increase and diminution; but convenience, drawn from experience, has led us to enumerate four elementary

operations, viz.: Addition, Subtraction, Multiplication, and Division.

All higher operations on numbers are but combinations of the four elementary ones.

3. Algebra originated in arithmetic, and elementary algebra is arithmetic generalized, the generalization being effected by employing symbols, usually non-numerical, to stand for and represent not only numbers or numerical quantities, but also the operations usually performed upon numbers.

Thus algebra becomes a symbolic language in which numbers and the operations upon them are written.

The symbols of algebra are thus primarily of two kinds :

(1) Quantitative symbols, which represent numerical quantities, and

(2) Operative symbols, which indicate operations to be performed upon the quantity denoted by the quantitative symbol.

A third class, called verbal symbols, may be enumerated, in which the symbol is a convenient contraction for a word or phrase.

4. The quantitative symbols are usually letters. The operative symbols, especially in elementary algebra and in arithmetic, are mostly marks or signs which are not letters. Relative position is employed to denote some operations, and in higher algebra very complex operations are often denoted by letters.

The verbal symbols do not denote quantity, and they cannot be said, in general, to denote operations.

The principal verbal symbols are:

(1) and, either of which denotes that all that precedes the symbol, taken in its totality, is equal to or is the same as all that follows the symbol, taken also in its totality.

(2) > and <. The first denotes that all that precedes the symbol, taken in its totality, is greater than all that follows the symbol, taken in its totality; and the second is like the first with less put for greater.

Other verbal symbols will be introduced as required.

5. From Art. 3 it is seen that operations in arithmetic must be special cases of more general operations in algebra. And hence it follows that arithmetic and algebra must proceed on similar principles, and must be subject to the same formal operative laws.

That the generalizing process of algebra should introduce new ideas into arithmetic is to be expected; and that this generalization should carry us beyond the necessarily limited field of arithmetic is also to be expected. Illustrations will occur hereafter.

6. The operative symbol + (plus) denotes addition, and tells us that the quantity before which it stands, and to which it belongs, is to be added to whatever precedes. Thus, 5+3 tells us that 3 is to be added to 5, and 0+3 is the same as the arithmetical number 3.

Similarly, a is the same as a, whatever a stands for; and for this reason the sign + is seldom written whenever it can be dispensed with without producing ambiguity.

a+b is the same as +a+b, and indicates that the

number denoted by b is to be added to the number denoted by a.

7. Any interpretable combination of quantitative and operative symbols is an algebraic expression. We shall, in the meantime, confine ourselves to expressions written in a single line, as

3 ab+2c+d2, etc.

In arithmetic we know that 3+5 is in its sum the same as 5+3, and 3+5+8 is the same as 3+8+5, the same as 8+5+3, etc. And as this must be a particular case of algebra (Art. 5), we must have a+b= b+a, a+b+c=a+c+b=b+c+a:

= etc.

This is the Commutative Law for Addition, and is expressed by saying that the order of adding quantities is arbitrary, or the sum is independent of the order of the addends.

8. The symbol

(minus) placed before a quantity indicates that the quantity is to be subtracted from whatever precedes the symbol.

Thus, 5-3 tells us that 3 is to be subtracted from 5; and ab tells us that the quantity denoted by b is to be subtracted from that denoted by a. Now, a and b denoting any numerical quantities, as long as a is greater than b the subtraction is arithmetically possible, and the result is an arithmetical quantity. But if a is less than b, the operation symbolized is not arithmetically possible. The expression ab is then a symbolic representation of an operation that cannot be arithmetically performed, and the result of the operation, whatever it may be, is not arithmetical.

Thus, 03, which is simply written 3, and which is called a negative number, does not belong to pure arithmetic, but is an idea introduced into algebraic arithmetic by generalizing the operation of subtraction.

And thus to every pure number, called now a positive number, corresponds an algebraical negative number, the relation between corresponding numbers being that their algebraic sum is zero or nothing.

Negative numbers are important in their relations to concrete arithmetic, and especially where geometric ideas are concerned. This matter will be dealt with in Chap

ter VIII.

9. It is said in Art. 2 that numbers are fundamentally capable of only increase or diminution. Hence + and - symbolize the two great operations in arithmetic and algebra. These are distinctively the signs of algebra.

By the sign of a quantity is meant that one of these two signs which precedes the quantity; and to change signs is to change to and to throughout.

Also, two quantities have like signs when both are preceded by + or both by ; otherwise they have unlike signs.

When no sign is written, + is understood.

10. That part of an expression included between two consecutive signs is called a term.

To indicate that any portion of an expression lying between two non-consecutive signs is to be taken in its totality as a single term, we enclose the portion within brackets.

Thus, in the expression a + 2 bc − 4 (3 c + 2 ab), a, 2bc