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becomes acquainted with the ordinary algebraical processes without encountering too many of their difficulties; and he is learning at the same time something of the more attractive parts of the subject.

"As regards the early introduction of simple equations and problems, the experience of teachers favors the opinion that it is not wise to take a young learner through all the somewhat mechanical rules of Factors, Highest Common Factor, Lowest Common Multiple, Involution, Evolution, and the various types of Fractions, before making some effort to arouse his interest and intelligence through the medium of easy equations and problems. Moreover, this view has been amply sup ported by all the best text-books on Elementary Algebra which have been recently published."

The work will be found to meet the wants of all who do not require a knowledge of Algebra beyond Quadratic Equations — that portion of the subject usually covered in the examination for admission to the classical course of American Colleges.

JUNE, 1895.

FRANK L. SEVENOAK.

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XII. HIGHEST COMMON FACTOR, LOWEST COMMON MULTIPLE

OF SIMPLE EXPRESSIONS. FRACTIONS INVOLVING

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XVIII. HIGHEST COMMON FACTOR OF COMPOUND EXPRESSIONS
XIX. MULTIPLICATION AND DIVISION OF FRACTIONS

XX. LOWEST COMMON MULTIPLE OF COMPOUND EXPRESSIONS

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ALGEBRA.

CHAPTER I.

DEFINITIONS. SUBSTITUTIONS.

1. ALGEBRA treats of quantities as in Arithmetic, but with greater generality; for while the quantities used in arithmetical processes are denoted by figures which have one single definite value, algebraical quantities are denoted by symbols which may have any value we choose to assign to them.

The symbols employed are letters, usually those of our own alphabet; and, though there is no restriction as to the numerical values a symbol may represent, it is understood that in the same piece of work it keeps the same value throughout. Thus, when we say "let a=1," we do not mean that a must have the value 1 always, but only in the particular example we are considering. Moreover, we may operate with symbols without assigning to them any particular numerical value at all; indeed it is with such operations that Algebra is chiefly concerned.

We begin with the definitions of Algebra, premising that the symbols +, -, ×,÷, will have the same meanings as in Arithmetic.

2. An algebraical expression is a collection of symbols; it may consist of one or more terms, which are separated from each other by the signs + and Thus 7a+5b-3c-x+2y is

an expression consisting of five terms.

Note. When no sign precedes a term the sign + is understood.

3. Expressions are either simple or compound. A simple expression consists of one term, as 5a. A compound expression consists of two or more terms. Compound expressions may be

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further distinguished. Thus an expression of two terms, as 3a-2b, is called a binomial expression; one of three terms, as 2a-3b+c, a trinomial; one of more than three terms a multinomial.

4. When two or more quantities are multiplied together the result is called the product. One important difference between the notation of Arithmetic and Algebra should be here remarked. In Arithmetic the product of 2 and 3 is written 2×3, whereas in Algebra the product of a and b may be written in any of the forms axb, a. b, or ab. The form ab is the most usual. Thus, if a=2, b=3, the product ab=axb=2×3=6; but in Arithmetic 23 means "twenty-three," or 2 × 10+3.

5. Each of the quantities multiplied together to form a product is called a factor of the product. Thus 5, a, b are the factors of the product 5ab.

6. When one of the factors of an expression is a numerical quantity, it is called the coefficient of the remaining factors. Thus in the expression 5ab, 5 is the coefficient. But the word coefficient is also used in a wider sense, and it is sometimes convenient to consider any factor, or factors, of a product as the coefficient of the remaining factors. Thus in the product 6abc, 6a may be appropriately called the coefficient of bc. A coefficient which is not merely numerical is sometimes called a literal coefficient.

Note. When the coefficient is unity it is usually omitted. Thus we do not write la, but simply a.

7. If a quantity be multiplied by itself any number of times, the product is called a power of that quantity, and is expressed by writing the number of factors to the right of the quantity and above it. Thus

axa is called the second power of a, and is written a2;
axa xa...............third power

and so on.

of a,

a3;

The number which expresses the power of any quantity is called its index or exponent. Thus 2, 5, 7 are respectively the indices of a2, a5, a1.

Note. a2 is usually read "

a squared"; a3 is read " a cubed"; at is read "a to the fourth"; and so on.

When the index is unity it is omitted, and we do not write a1, but simply a. Thus a, la, a1, la1 all have the same meaning.

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