TREATISE ON ALGEBRA.

CHAPTER I.

DEFINITIONS AND PRELIMINARY RULES.

DEFINITIONS.

(Article 1.) (ALGEBRA is that branch of Mathematics) in which the calculations are performed by means of letters and signs or symbols.

(2.) In Algebra, quantities, whether given or required, are usually represented by letters. The first letters of the alphabet are, for the most part, used to represent known quantities; and the final letters are used for the unknown quantities.

(3.) (The symbol =, is called the sign of Equality and denotes that the quantities between which it is placed, are equal or equivalent to each other. Thus, $1 = 100 cents, which is read, one dollar equals one hundred cents. Again, ab, which is read, a equals b.

(4.) The symbol +, is called plus; and denotes that the quantities between which it is 'placed, are to be added together. Thus, a+b=c, which is read, a and b added, equals c. Again, which is read, a, b and

=

(5.) (The symbol —, is called minus and denotes that the

quantity which is placed at the right of it is to be subtracted from the quantity on the left. Thus, abc, which is read, a diminished by b equals c.

(6.) The symbol ×, is called the sign of multiplication and denotes that the quantities between which it is placed are to be multiplied together. Thus, a × b=c, which is read, a multiplied by b, equals c. Multiplication is also represented by placing a dot between the factors, or terms to be multiplied. Thus, a. b is the same as a x b. Another method, which is used as frequently as either of the above, is to unite the quantities in the form of a word. Thus, abc is the same

a xbx c, or a.b.c.

(7.) The symbol ÷, is called the sign of division); and denotes that the quantity on the left of it is to be divided by the quantity on the right. a divided by b equals c. cing the divisor under the

Thus, a÷bc, which is read, Division is also indicated by pladividend, with a horizontal line be

Ꮖ

-

tween them like a vulgar fraction. Thus, is the same as

a

(10.) When a quantity is added to itself several times, as c+c+o+c, we can write it but once, by placing before it a number to show how many times it has been taken. Thus, c+c+c+c= 4c. The number which is thus placed before the quantity is called the coefficient of the quantity. In the above example, 4 is the coefficient of c. A coefficient may consist, itself, of a letter. Thus, n is the coefficient of x in the expression nx; so also may x be regarded as the coefficient of n in the same expression.

(11.) The continued product of a quantity into itself is, usually, denoted by writing the quantity once and placing a number over the quantity, a little to the right. Thus, axaxa is the same as a'. The number thus placed over the quantity is called the exponent of the quantity. Thus, 5 is the exponent of a in the expression a', and denotes that a is to be multiplied into itself, as a factor, five times.

(12.) When a quantity is multiplied continually into itself, the result is called a power of the quantity. Thus, a® is the sixth power of a, and a' is the third power of a, the exponent always indicating the degree of the power.

When a quantity is written without any exponent, it is understood that its exponent is a unit.

Thus, a is the same as a?, and (x + y) x m is the same as (x + y)? x m2.

1. X
(13.) The symbol V, is called the radical sign; and de-
notes that a root of the quantity, over which it is placed, is to
be extracted. Thus,

ịu x or simply vx denotes the square root of x.
3x denotes the cube root of x.
vx denotes the fourth root of x.

The number placed over the radical is called the index of the root. Thus, 2, 3, and 4 are, respectively, the indices of the square root, cube root, and fourth root.

U

a

(14.) A root of a quantity may also be represented by means of a fractional exponent. Thus, the square root of a is až ; the cube root of a is ad; the fourth root of a is ał, and so on for other roots.

을 By the same notation, a is the cube root of the square of a, or the square of the cube root of a.

For the same reason a is the fifth root of the third

power

of Q, or the third power of the fifth root of a. (15.) The reciprocal of a quantity is a unit divided by 1

1 that quantity. Thus Thus - is the reciprocal of a, also

is the

3 reciprocal of 3.

(16.) The symbol ..., is equivalent to the phrase, therefore, or consequently.

(17.) When algebraic quantities are written without any sign prefixed, the sign plus is understood, and the quantities are said to be positive or affirmative ; and those having the sign minus prefixed are called negative quantities. Thus, a=ta, b=+b, are each positive quantities ; whilst -b are negative quantities.

(18.) An algebraic expression composed of two or more terms connected by + or -, is called a polynomial. A polynomial composed of but two terms, is called a binomial; one composed of three terms, is called a trinomial. Thus,

3a +46
7.x2 – 3y are binomials.

-
За2 + 46
4m

are trinomials. 5g

x + y (19.) Each of the literal factors which compose any term,

3a3

х

y ta

is called a dimension of this term: the degree of a term is the number of the dimensions or factors. Thus,

7aare terms of one dimension, or of
5b the first degree.

5axare terms of two dimensions, or
5xy of the second degree.

7a2b3=7aabbb are terms of five dimensions, or
3x5 =3xxxxx of the fifth degree.

(20.) A polynomial is said to be homogeneous, when all its terms are of the same degree.

Thus,

3a-5x+2y are homogeneous polynomials of
by+m the first degree.
b-

4a2 + 2x2-xy are homogeneous polynomials of
7am-c2 +a2 the second degree.

5a2b3 — 6a5 — 4x1y are homogeneous polynomials 3ab-b+4a362 of the fifth degree.

(21.) Any combination of letters, by the aid of algebraic signs, is called an algebraic expression. Thus,

is an algebraic expression, denoting seven times the quantity x.

a2 + 3√b

}

(a+b)2 }

is an algebraic expression, denoting that the quantity a is to be squared, and then added to three times the square root of b.

is an algebraic expression, denoting that the sum of a and b is to be squared.

The algebraic expression (a + b) (a —b) = a2 — b2 is read, "the sum of a and b, multiplied by the difference of a and b, equals the difference of the squares of a and b.”

(22.) We will give some identical algebraic expressions, which may serve to exercise the student in reading algebraic formulas.