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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 35 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 remain. der after subtracting 2 from 7. Likewise a· b 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 × 5 × 7, or 3.5.7 is the continued product of 3, 5, and 7. Likewise 12 x a × b, 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 ab; and, in the expression xy, 1 may be regarded as the coefficient of x y.

6. The continued product of a quantity multiplied repeatedly by itself, is called the power of the quantity; 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 a5; but when a stands by itself, it may be regarded as a1.

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

The sign is 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

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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 vided by b.

b

denotes the quotient of a di

Signs of Equality and Inequality. Algebraic Quantity.

9. The sign = is called equal to, and placed between two quantities denotes that they are equal to each other, and the expression in which this sign occurs is called an equation.

Thus, the equation ab denotes that a is equal to b.

10. The sign is called greater than, and the sign <is called less than; and the expression in which either of these signs occurs is called an inequality.

Thus, the inequality a> b denotes that a is greater than b; and the inequality a<b denotes that a is less than b; the greater quantity being always placed at the opening of the sign.

11. An algebraic quantity is any quantity written in algebraic language.

12. An algebraic quantity, in which the letters are not separated by the signs + and -, is called a monomial, or a quantity composed of a single term, or simply a term.

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13. An algebraic expression composed of several terms, connected together by the signs and is called a polynomial, one of two terms is called a binomial, one of three a trinomial, &c.

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14. The value of a polynomial is evidently not affected by changing the order of its terms.

Thus, a — · b · -cd is the same as a— c — b+d, or ad-b-c, or b+d+a-c, &c.

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