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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;
(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.