MANUAL OF ALGEBRA. . CHAPTER I. DEFINITIONS AND EXPLANATION OF SIGNS. Definitions, 1. Quantity is anything that can be measured, as number, time, or distance. A thing can be measured when it can be expressed in terms of some other thing of the same kind taken as a unit. The value of a quantity is an expression for that quantity in terms of some assumed unit; as 7 feet, 3 years, 4 pounds. 2. Mathematics is the science that treats of the relations of quantities, and of the operations that may be performed on them. 3. Algebra is a branch of Mathematics in which quantities to be considered are represented by letters, and operations to be performed on them are indicated by signs. The letters and signs are called symbols. In what follows, the expressions 1°, 2°, 3°, &c., stand for first, second, third, &c. Explanation of Symbols. 4. The quantities treated of in Algebra are of two kinds: 1o. Known quantities, those whose values are given; and, 2o. Unknown quantities, those whose values are required. Known quantities are generally represented by leading letters of the alphabet; as, a, b, c, &c. Unknown quantities are generally represented by final letters of the alphabet; as, x, y, z, w, &c. When, in the course of an operation, an unknown quantity becomes known, it is often convenient to represent it by one of the final letters, with one or more accents, as, X', y', Z", &c. These symbols are read, x prime, y second, z third, &c. 5. The signs employed in Algebra are of three kinds: signs of operation; signs of relation; and signs of abbreviation. The signs of operation are the following: 1°. The sign of addition, +, called plus. When placed between two quantities, it indicates that the second is to be added to the first. Thus, the expression, a + b, read, a plus b, indicates that b is to be added to a. 2°. Sign of subtraction, -, called minus. When placed between two quantities, it indicates that the second is to be subtracted from the first. Thus, the expression, c-d, read c minus d, indicates that d is to be subtracted from c. The double sign, #, read plus and minus, is used to indicate that the quantity before which it is placed, is first to be added to, and then to be subtracted from, the preceding quantity. Thus, the expression a b is equivalent to the two expressions, a + b, and a - b. If no sign is written before a quantity, the sign + is understood. 3°. The sign of multiplication, X. When placed between two quantities it indicates that one of them is to be multiplied by the other. Thus, the expression x x y indicates that ~ is to be multiplied by y, or that y is to be multiplied by a. The quantities x and y are called factors, and the result of the multiplication is called a product. If more than two factors are multiplied together, the result is called a continued product. Factors represented by letters are called literal factors; in this case the sign of multiplication may be replaced by a simple dot, or it may be omitted altogether. Thus, the continued product of x, y, and %, may be represented by any one of the expressions 49. The sign of division, -. When written between two quantities, it indicates that the first is to be divided by the second. Thus, the expression, p =9, indicates that p is to be divided by q. The operation may also be expressed by writing one quantity over the other, in the form of a fraction; or the sign of division may be replaced either by a straight, or by a curved line. Thus, the quo Jsju o U h. Even tient of p by q may be represented by any one of the P q? 5°. The exponential sign. The exponential sign is a number written on the right, and above a quantity, to show how many times that quantity is to be taken as a factor. Thus, in the expressions x2, 24, 2:”, the numbers 2, 4, and m, are exponents, indicating respectively that x is to be taken 2, 4, and m times, as a factor. The resulting products are called powers. Thus, 24 is called the fourth power of x. If no exponent is written, the exponent 1 is always understood. 6o. The radical sign, V. When placed over a quantity, it indicates that a root of that quantity is to be extracted. The nature of the root is indicated by a number placed over the radical sign, called an index. Thus, , the expressions, Va, Va, and "Vā, indicate that the square, cube, and nth roots of a, are to be extracted. If no index is written, the index 2 is always understood. The signs of relation are the following : 1°. The sign of equality, When written between two quantities, it indicates that they are equal to each other. Thus, the expression, a = nb, indicates that a is equal to the product of n and b. 2°. The sign of inequality, <, >. When written between two quantities, it indicates that they are unequal, the greater one being at the opening of the sign. Thus, the expressions, a > b, and c <d, indicate that a is greater than b, and that c is less than d. 3°. The signs of proportion, : :: :. The single colon stands for, is to; the double colon for, as. Thus, the expression, : 7 :: : d, is read, a is to b, as c is to d. a с The double colon is equivalent to the sign of equality and is often replaced by that sign. Thus, the preceding proportion may be written, a : b = c : d. The signs of abbreviation are the following: the bar, 1, and the parenthesis, or brackets, (), [], { }, are used to con nect several quantities, which are to be operated on as a single quantity. Thus, each of the expressions, indicates that the sum of a and b is to be multiplied by x. Other signs will be explained in their proper places. Additional Meaning of the Signs + and “. 6. The signs + and -, besides indicating addition and subtraction, are also used to show the sense in which a quantity is taken: |