8. The difference of two numbers is sometimes denoted by the sign; thus ab denotes the difference of the numbers denoted by a and b, and is equal to a -b or to b-a according as a is greater than b or less than b. 9. The sign > denotes greater than, and the sign <denotes less than; thus a>b denotes that the number represented by a is greater than the number represented by b, and b<a denotes that the number represented by b is less than the number represented by a. Thus in both signs the opening of the angle is turned towards the greater number. 10. The sign.. denotes then or therefore; the sign. denotes since or because. 11. When several numbers are to be taken collectively they are enclosed by brackets. Thus (a−b+c) x (d+e) signifies that the number represented by a-b+c is to be multiplied by the number represented by d + e. This may also be written thus (a−b+c) (d+e). The use of the brackets will be seen by comparing the above expressions with (a-b+c)d+e; the latter denotes that the number represented by a b+c is to be multiplied by d, and then e is to be added to the product. Sometimes instead of using brackets a line called a vinculum is drawn over quantities which are to be taken collectively. Thus a-b+cxd+e is used with the same meaning as (a − b + c) × (d+e). 12. The letters of the alphabet, and the signs or marks which we have already introduced and explained, together with those which may occur hereafter, are called Algebraical symbols, since they are used to represent the things about which we may be reasoning. Any collection of Algebraical symbols is called an Algebraical expression or a formula. 13. Those parts of an expression which are connected by the signs + or are called its terms. When an expression consists of two terms it is called a binomial expression; when it consists of three terms it is called a trinomial expression; any expression con sisting of several terms may be called a multinomial expression or a polynomial expression. When an expression does not contain parts connected by the sign + or the sign - it may be called a simple expression, or it may be said to contain only one term. Thus abc is a simple expression; abc + x is a binomial expression, of which abc is one term, and x is the other; ab + ac - bc is a trinomial expression, of which ab, ac, and bc are the terms. 14. When one number consists of the product of two or more numbers, each of the latter is called a factor of the product. Thus α, b and c are factors of the product abc. 15. A product may consist of one factor which is a number represented arithmetically, and of another factor which is a number represented algebraically, that is, by a letter or letters; in this case the former factor is said to be the coefficient of the latter. Thus in the product 7abc the factor 7 is called the coefficient of the factor abc. Where there is no arithmetical factor, we may supply unity; thus we may say that, in the product abc, the coefficient is unity. And when a product is represented entirely algebraically, any one factor may be called the coefficient of the product of the remaining factors. Thus, in the product abc, we may call a the coefficient of bc, or b the coefficient of ac, or c the coefficient of ab. If it be necessary to distinguish this use of the word coefficient from the former, we may call the latter coefficients literal coefficients, and the former numerical coefficients. 16. If a number be multiplied by itself any number of times, the product is called a power of that number. Thus a × a is called the second power of a; also a ×a×a is called the third power of a; and a×a×a×a is called the fourth power of a; and so on. The number a itself is often called the first power of a. 17. Any power of a quantity is usually expressed by placing above the quantity the number which represents how often it is repeated in the product. Thus a2 is used to express a × a; also of a as is used to express axaxa; and at is used to express axaxaxa; and so on. And a' may be used to denote the first power or a itself; that is, a' has the same meaning as a. Numbers placed above a quantity to express the powers of that quantity are called indices of the powers, or exponents of the powers; or more briefly indices or exponents. 18. Thus we may sum up the two preceding articles as follows "a×a×a× &c. to n factors is expressed by a”, and n is called the index or exponent of a", where n may denote any whole number." 19. The second power of a or a3 is often called the square of a, and the third power of a or a3 is often called the cube of a. The symbol a is read thus "a to the fourth power" or briefly "a to the fourth,;" and a" is read thus "a to the nth " 20. The square root of any proposed number is that number which has the proposed number for its square or second power. The cube root of any proposed number is that number which has the proposed number for its cube or third power. The fourth root of any proposed number is that number which has the proposed number for its fourth power. And so on. 21. The square root of a number a is denoted thus a, or simply thus Ja. The cube root of a is denoted thus /a. The fourth root of a is denoted thus a. And so on. The sign is said to be a corruption of the initial letter of the word radix. 22. Terms are said to be like or similar when they do not differ at all or differ only in their numerical coefficients; otherwise they are said to be unlike. Thus 4a, 6ab, 9a and 3a2bc are respectively similar to 15a, 3ab, 12a and 15abc. And ab, ab, ab2 and abc are all unlike. 23. Each of the letters which occurs in an algebraical product is called a dimension of the product, and the number of the letters is the degree of the product. Thus ab3c or a×a×b×b×bx c is said to be of six dimensions or of the sixth degree. A numerical coefficient is not counted; thus 9a3b and a3b are of the same dimensions, namely of seven dimensions. Thus the degree of a term or the number of dimensions of a term is the sum of the exponents provided we remember that if no exponent is expressed the exponent 1 must be understood as indicated in Art. 17. 24. An algebraical expression is said to be homogeneous when all its terms are of the same dimensions. Thus 7a3+ 3a2b+ 4abc is homogeneous, for each term is of three dimensions. The following examples will serve for an exercise in the preceding definitions. 13. Find the value of (9 − y) (x + 1) + (x + 5) (y + 7) – 112, when x=3 and y = 5. 14. Find the value of x √(x2 – 8y) + y √(x2 + 8y), when x=5 and y = 3. 15. Find the value of a √(x2 – 3a) + x √(x2 + 3a), when x and a = = 8. = 5 16. Find the value of a + b √(x + y) − (a − b) 3/(x − y), when a = 10, b=8, x = = 12 and y = 4. 17. If a = 16, b = 10, x=5 and y = 1, find the value of and of (b − x) (√a + b) + √{(a − b) (x + y)} . (a − y) {√(2bx) + x2 } + √{(a − x) (b +y)}. 18. If a = 2, b = 3, x= 6 and y=5, find the value of II. CHANGE OF THE ORDER OF TERMS. REDUCTION OF LIKE TERMS. ADDITION, SUBTRACTION, USE OF BRACKETS. 25. When the terms of an expression are connected by the sign + it is indifferent in what order they are written; thus a+b and ba give the same result, namely the sum of the numbers which are denoted by a and b. We may express this fact algebraically thus Similarly a+b=b+ a. a+b+c=a+c+b=b+a+c=b+c+a=c+a+b= c + b + a. 26. If an algebraical expression consist of some terms preceded by the sign + and some terms preceded by the sign we may write the former terms first in any order we please, and the latter terms after them in any order we please. This appears from the same considerations as before. Thus, for example, a+b-c-e=a+b-e-c=b+a-c-e=b+a-e-c. 27. In some cases it is obvious that we may vary the order of terms still further by mixing up the terms preceded by the sign with those preceded by the sign+. Thus, for example, if a denote 10, b denote 6, and c denote 5, then a+b-c=a-c+b=b-c+α. If however a denote 2, b denote 6, and c denote 5, then the expression a-c+b presents a difficulty because we are thus |