ELEMENTS OF ALGEBRA. CHAPTER I. PRELIMINARY DEFINITIONS AND REMARKS. 1. QUANTITY is a general term applied to everything which can be estimated or measured. 2. MATHEMATICS is the science which treats of the properties and relations of quantities. 3. ALGEBRA is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. The letters and signs are called symbols. 4. The sign, is called plus, and when placed between two quantities indicates that they are to be added together. Thus, 95 is read, 9 plus 5, and indicates that the quantity represented by 5 is to be added to the quantity represented by 9. In like manner, a + b is read, a plus b, and denotes that the quantity represented by b is to be added to the quantity represented by a. 5. The sign - is called minus, and indicates that one quantity is to be subtracted from another. Thus, 95 is read, 9 minus 5 or 9 diminished by 5. -- b is read, a minus b, or a diminished by When In like manner, a 6. The sign X, is called the sign of multiplication. placed between two quantities, it denotes that they are to be multiplied together. Thus, 36 x 25, denotes that 36 is to be multiplied by 25. The multiplication of two quantities may also be indicated by placing a point between them. Thus, 36.25 is the same as 36 × 25, and is read, 36 multiplied by 25, or the product of 36 by 25. 7. The multiplication of quantities, which are represented by letters, is generally indicated by simply writing the letters one after the other, without interposing any sign. Thus, ab is the same as and abc, the same as ax b, or as a.b; a xbx c, or as a.b.c. It is plain that the notation ab, or abc, cannot be employed when the quantities are represented by figures. For, if it were required to express the product of 5 by 6, we could not write 5 6, without confounding the product with the number 56. 8. In the product of several letters, as abc, the single letters, a, b, and c, are called factors. Thus, in the product ab, there are two factors, a and b; in the a, c, and d. product acd, there are three, 9. There are three signs used to denote division. Thus, ab denotes that a is to be divided by b. alb denotes that a is to be divided by b. 10. The sign, is called the sign of equality, and is read, is equal to. When placed between two quantities, it denotes that they are equal to each other. Thus, 954: that is, 9 minus 5 is equal to 4: Also, a + b = c, indicates that the sum of the quantities represented by a and b, is equal to the quantity denoted by c. 11. The sign >, is called the sign of inequality, and is used to express that one quantity is greater or less than another. Thus, a > b is read, a greater than b; and a <b is read, a less than b; that is, the opening of the sign is turned toward the greater quantity. 12. If a quantity is added to itself several times, as a+a+a+a+a, it is generally written but once, and a figure is then placed before it, to show how many times it is taken. Thus, a + a + a + a + a = 5a. The number 5 is called the co-efficient of a, and denotes that a is taken 5 times. Hence, a co-efficient is a number prefixed to a quantity, denoting the number of times which the quantity is taken. The co-efficient also indicates the number of times plus one, that the quantity is added to itself. When no co-efficient is written, the co-efficient 1 is always understood. Thus, a = la. 13. If a quantity be multiplied continually by itself, as αχαχαχαχα, the product is generally expressed by writing the letter once, and placing a number to the right of, and a little above it: thus, a xaxax a × a = a5. The number 5 is called the exponent of a, and denotes the number of times which a enters into the product as a factor. Hence, the exponent of a quantity shows how many times the quantity is a factor. It also indicates the number of times, plus one, that the quantity is to be multiplied by itself. When no exponent is written, the exponent 1 is always understood. 14. The product resulting from the multiplication of a quantity by itself any number of times, is called the power of that quantity: and the exponent denotes the degree of the power. For example, ala is the first power of a, a2 = axa is the second power, or square of a, a3 = a × a × a is the third power, or cube of a, a4 = axa xa xa is the fourth power of a, a5 = a × a ×a×a×a is the fifth power of a, in which the exponents of the powers are, 1, 2, 3, 4, and 5; and the powers themselves, are the results of the multiplications. It should be observed that the exponent of a power is always greater by unity than the number of multiplications. 15. As an example of the use of the exponent in algebra, let it be required to express that a number a is to be multiplied three times by itself, that this product is then to be multiplied three times by b, and this new product twice by c; we should write axaxax a x b x b x bx c x c = a4b3c2. If it were further required to repeat this result a certain num ber of times, say seven times, that is, to add it to itself six times, we should simply write 7a4b3c2. This example shows the brevity of the algebraic language. 16. The root of a quantity, is a quantity which being multiplied by itself a certain number of times, will produce the given quantity. The sign, is called the radical sign, and when prefixed to a quantity, indicates that its root is to be extracted. Thus, 2 Va 3 α or simply √ denotes the square root of a. Va denotes the cube root of a. Va denotes the fourth root of a. The number placed over the radical sign is called the index of the root. Thus, 2 is the index of the square root, 3 of the cube root, 4 of the fourth root, &c. 17. The reciprocal of a quantity, is unity divided by that quantity. Thus, 18. Every quantity written in algebraic language, that is, with the aid of letters and signs, is called an algebraic quantity, or the algebraic expression of a quantity. Thus, is the algebraic expression of three times the quantity denoted by a; is the algebraic expression of five times the is the algebraic expression of seven times the |