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SECTION 1. Reduction of Fractions..

II. Addition and Subtraction of Fractions.
III. Multiplication and Division of Fractions.
IV. Proportions.

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CHAPTER III.

EQUATIONS OF THE FIRST DEGREE.

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SECTION 1. Putting Problems into Equations.

II. Reduction and Classification of Equations.
III. Solution of Equations of the First Degree, with one

unknown quantity. ·
IV. Equations of the First Degree containing two or

more unknown quantities.

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CHAPTER IV.

NUMERICAL EQUATIONS.

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SECTION 1. Indeterminate Coefficients.

II. - Derivation. .
III. Numerical Equations.

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ALGEBRA.

CHAPTER I.

FUNDAMENTAL PROCESSES OF ALGEBRA.

SECTION I.

Definitions and Notation.

1. Algebra, according to the usual definition, is that branch of mathematics in which the quantities considered are represented by the letters of the alphabet, and the operations to be performed upon them are indicated by signs. In this sense it would embrace almost the whole science of mathematics, elementary geometry alone being excepted. It is, consequently, subject in common use to some limitations, which will be more easily understood, when we are advanced in the science.

2. The sign + is called plus or more, or the positive sign, and placed between two quantities denotes that they are to be added together.

Thus 3+5 is 3 plus or more 5, and denotes the sum of 3 and 5. Likewise a + b is the sum of a and b, or of the quantities which a and b represent.

Signs of Addition, Subtraction, Multiplication, and Division.

3. The sign is called minus or less, or the negative sign, and placed between two quantities denotes that the quantity which follows it is to be subtracted from the one which precedes it.

Thus 7-2 is 7 minus or less 2 and denotes the remainder after subtracting 2 from 7. Likewise ab is the remainder after subtracting b from a.

4. The sign X is called the sign of multiplication, and placed between two quantities denotes that they are to be multiplied together. A point is often used instead of this sign, or, when the quantities to be multiplied together are represented by letters, the sign may be altogether omitted.

Thus 3 x 5 x 7, or 3.5.7 is the continued product of 3, 5, and 7. Likewise 12 x a xb, or 12.a.b, or 12 ab, is the continued product of 12, a, and b.

5. The factor of a product is sometimes called its coefficient, and the numerical factor is called the numerical coefficient. When no coefficient is written, the coefficient may be considered to be unity.

Thus, in the expression 15 a b, 15 is the numerical coefficient of a b; and, in the expression ry, I may be regarded as the coefficient of x y.

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6. The continued product of a quantity multiplied repeatedly by itself, is called the power of the tity; and the number of times, which the quantity is taken as a factor, is called the exponent of the power.

The power is expressed by writing the quantity

Coefficient. Power. Root.

once with the exponent to the right of the quantity, and a little above it. When no exponent is written, the exponent may be considered to be unity.

Thus the fifth power of a is written as; but when a stands by itself, it may be regarded as a'.

7. The root of a quantity is the quantity which, multiplied a certain number of times by itself, produces the given quantity; and the inder of the root is the number of times which the root is contained as a factor in the given quantity.

The sign vis called the radical sign, and when prefixed to a quantity indicates that its root is to be extracted, the index of the root being placed to the left of the sign and a little above it. The index 2 is generally omitted, and the radical sign, without any index, is regarded as indicating the second or square root. Thus, a or va is the square root of a,

Va is the third or cube root of a,
va is the fourth root of a,

Pa is the nth root of a. 8. The signs • and : are called the signs of division, and either of them placed between two quantities denotes that the quantity which precedes it is to be divided by the one which follows it. The process of division is also indicated by placing the dividend over the divisor with a line between them.

Thus, a +b, or a :b, or a denotes the quotient of a divided by b.

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