ALGEBRA.

I. DEFINITIONS AND SIGNS.

Algebra is the science which treats of numbers by means of letters of the alphabet and certain signs or symbols.

The letters are employed to denote numbers.

A letter may be employed to denote either some particular number, or any number whatever.

Thus a may denote a particular number, as 5, or it may denote any number whatever.

In Arithmetic the correctness of a certain operation can only be proved by taking a particular example; but in Algebra the general correctness of an operation can be proved by the employment of letters to denote any numbers whatever.

The numbers denoted by letters may be either known numbers, or unknown numbers, which have to be found. The former are generally represented by the first letters of the alphabet, a, b, c, etc.; and the latter by the last letters, x, y, z, etc. This distinction is not, however, always observed.

Signs or symbols are used to indicate either the mode in which the numbers denoted by the letters are to be employed, or the relation which subsists between them.

Some of these signs are common to Arithmetic and Algebra. The sign + (plus) denotes Addition, or that the number before which it stands is to be added.

Thus ab means that b is to be added to a. The sign — (minus) denotes Subtraction, or that the number before which it stands is to be subtracted.

III.

Thus a

- b means that b is to be subtracted from a.

A A

The sign x denotes Multiplication, or that the two numbers between which it stands are to be multiplied together.

Thus ab means that a and b are to be multiplied together. The sign denotes Division, or that the number which stands before it is to be divided by that which follows it.

Thus ab means that a is to be divided by b.

Thus is the sign of Addition.

The sign X is frequently omitted, and the process of multiplication is indicated by simply placing the numbers to be multiplied side by side.

Thus ab axb

7 ab=7× a x b

This mode of indicating multiplication must not, of course, be employed between ordinary arithmetical digits. Thus 75 cannot mean 7 X 5.

Multiplication is sometimes denoted by placing a dot (1) between the numbers to be multiplied together.

Thus ab axb.

Care should be taken never to employ this symbol where it can possibly be mistaken for a decimal point.

A fraction in Algebra is denoted in the same manner as a Vulgar Fraction in Arithmetic, by means of a numerator and