ELEMENTS OF ALGEBRA.

CHAPTER I.

DEFINITIONS AND PRELIMINARY RULES.

DEFINITIONS AND SYMBOLS.

(Article 1.) ALGEBRA is that branch of Mathematics, in which the operations are performed by means of figures, 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. In the same way a=b, is read, a equals b. And the same for other similar expressions.

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

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(5.) The symbol, is called the sign of subtraction, or 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, a-b-c, is read, a diminished by b equals c.

(6.) When algebraic quantities are written without any sign prefixed, they are understood to be affected by the sign plus, and the quantities are said to be positive or affirmative; but quantities having the sign minus prefixed, are called negative quantities. Thus, a is the same as a, b is the same as +b, and each is called a positive quantity; while -a, -b, are called negative quantities.

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

(8.) 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. Thus, ab=c, is read, a divided by b equals c. Division is also indica

ted by placing the divisor under the dividend, with a horizontal line between them, like a vulgar fraction.

Thus, is the same as x÷y.

x y

((9.) When quantities are enclosed in a parenthesis, brace, or bracket, they are to be treated as a simple quantity. Thus, (a+b)÷÷c, indicates that the sum of a and b is to be divided by c. Again, each of the expressions, (x—y)÷z; {x—y}÷z; [x—y]÷z; is read, y subtracted from a, and the remainder divided by z. The same thing may be expressed by drawing a horizontal line, or bar, over the compound quantity, which line or bar is called a vinculum. Thus, x+yxz, denotes that the sum of x and y is to be multiplied by z.

(10.) When a quantity is added to itself several times, as c+c+c+c, we may write it but once, and prefix a number to show how many times its value has beer repeated by this addition. Thus, c+c+c is the same as 3c; d+d+d+d is the same as 4d. The numbers thus prefixed to the quantities are called coefficients. Thus, in the expressions 3c, 4d, the coefficients of c and d are 3 and 4 respectively. A coefficient may consist, itself, of a letter. Thus, in the expression nx, n may be regarded as the coefficient of x; so also may x be considered as the coefficient of n.

When no coefficient is written, the quantity is regarded as having 1 for its coefficient. Thus, a is the same as la, and xy is the same as lxy.

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(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 a3. The number thus placed over the quantity, is called the exponent of the quantity, and denotes the number of equal factors which

are to be multiplied together. Also, axaxaxa is the same as a1.

When a quantity is written without any exponent, it is understood that its exponent is 1. Thus, a is the same as a1, and (x+y)×m is the same as (x+y)1×m1.

(12.) The reciprocal of a quantity is the value of a

unit when divided by that quantity. Thus,

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reciprocal of a; also is the reciprocal of b. b

1

α

is the

(13.) An Algebraic expression is any combination of letters and numbers, formed by the aid of algebraic signs, in conformity with the foregoing definitions.

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.

ADDITION.

(14.) ADDITION, in Algebra, is finding the simplest expression for several algebraic quantities, connected by + or

Suppose we wish to find the sum of 3 apples, 7 apples, 4 apples, and 10 apples. From what we know of Arithmetic, we should proceed as in the following

OPERATION.

3 apples,

7 apples,

4 apples,

10 apples,

24 apples sum of all.

If, instead of writing the word apples, we write only a, its initial letter, we shall have this second

OPERATION.

3a

7a

4a

10a

24a=sum of all the a's.

If a, instead of representing an apple, stands for any other thing, then would the sum of 3a, 7a, 4a, and 10a, be accurately represented by 24a.

Frequently, in algebra, the quantities to be united or added, are all placed in the same horizontal line. The above quantities, when placed after this method, become: 3a+7a+4a+10a-24a.

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