Thus 2a+36 is a binomial expression; a-2b+5c is a trinomial expression; and a-b+c-d-e may be called a multinomial expression or a polynomial expression. 22. Each of the letters which occur in a term is called a dimension of the term, and the number of the letters is called the degree of the term. Thus ab3c or a xa x b x b × bx c is said to be of six dimensions or of the sixth degree. A numerical coefficient is not counted; thus 9a3b4 and a3b4 are of the same dimensions, namely seven dimensions. Thus the word dimensions refers to the number of algebraical multiplications involved in the term; that is, the degree of a term, or the number of its dimensions, is the sum of the exponents of its algebraical factors, provided we remember that if no exponent be expressed the exponent 1 must be understood, as indicated in Art. 16. 23. An 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. We shall now give some more examples of finding the numerical values of algebraical expressions. Suppose a = 1, b=2, c=3, d=4, e=5, f=0. Then 362=3×4=12, 5b3=5×8=40, 965=9 × 32=288. e=51=5, e=52=25, e°=53 = 125. a2b3=1×8=8, 362c2=3×4 × 9=108. d3 +c2 — 7ab+ƒ2=64+9−14+0=59. EXAMPLES. II. If a=1, b=2, c=3, d=4, e=5, ƒ=0, find the numerical values of the following expressions. III. Remaining Signs. Brackets. 24. The difference of two numbers is sometimes denoted by the sign ~; thus a~b denotes the difference of the numbers represented by a and b; and is equal to a-b, or b-a, according as a is greater than b, or less than b. 25. The sign > denotes is greater than, and the sign < denotes is less than; thus a>b denotes that the number represented by a is greater that the number represented by b, and ba denotes that the number represented by b is less than the number represented by a. Thus in both cases the opening of the angle is turned towards the greater number. 26. The sign.. denotes then or therefore; the sign. denotes since or because. 27. The square root of any assigned number is that number which has the assigned number for its square or second power. The cube root of any assigned number is that number which has the assigned number for its cube or third power. The fourth root of any assigned number is that number which has the assigned number for its fourth power. And so on. Thus since 4972, the square root of 49 is 7; and so if a= b2, the square root of a is b. In like manner, since 125=53, the cube root of 125 is 5; and so if a=c3, the cube root of a is c. 28. The square root of a is denoted thus a, or simply thus a. The cube root of a is denoted thus a. The fourth root of a is denoted thus a. And so on. Thus 93; /8=2. The sign is said to be a corruption of the initial letter of the word radix. 29. When two or more numbers are to be treated as forming one number they are enclosed within brackets. Thus, suppose we have to denote that the sum of a and b is to be multiplied by c; we denote it thus (a+b)×c_or a+bxc, or simply (a+b) c or {a+b}c; here we mean that the whole of a+b is to be multiplied by c. Now if we omit the brackets we have a+bc, and this denotes that b only is to be multiplied by c and the result added to a. Similarly, (a+b-c)d denotes that the result expressed by a+b-c is to be multiplied by d, or that the whole of a+b-c is to be multiplied by d; but if we omit the brackets we have a+b-cd, and this denotes that c only is to be multiplied by d and the result subtracted from a+b. So also (a-b+c) x (d+e) denotes that the result expressed by a-b+c is to be multiplied by the result expressed by d+e. This may also be denoted simply thus, (a−b+c)(d+e); just as axb is shortened into ab. So also (a+b+c) denotes that we are to obtain the result expressed by a+b+c, and then take the square root of this result. So also (ab)2 denotes ab × ab; and (ab)3 denotes ab× ab × ab. So also (a+b-c)÷(d+e) denotes that the result expressed by a+b-c is to be divided by the result expressed by d+e. 30. Sometimes instead of using brackets a line is drawn over the numbers which are to be treated as forming one number. Thus a-b+cxd+e is used with the same meaning as (a−b+c) x (d+e). A line used for this purpose is called a vinculum. So also (a+b−c)÷(d+e) may a+b-c be denoted thus ; and here the line between d+e a+b-c and d+e is really a vinculum used in a particular sense. 31. We have now explained all the signs which are used in algebra. We may observe that in some cases the word sign is applied specially to the two signs + and -; thus in the Rule for Subtraction we shall speak of changing the signs, meaning the signs + and -; and in multiplication and division we shall speak of the Rule of Signs, meaning a rule relating to the signs + and 32. We shall now give some more examples of finding the numerical values of expressions. Suppose a=1, b=2, c=3, d=5, e=8. Then √(2b+4c) = √(4+12) = √√/(16) = 4. 3/4c-2b)/(12—4) = 3/(8) = 2. e (2b+4c)-(2d-b)/(4c-2b)=8x4-8x2-32-16=16. √{(e—b)(2e−5b)} = √{(8—2)(16—10)} = √√(6 × 6) = 6. Ke-dxb+c)-(d-c)(c+a)} (a+d)= {3×5-2×4}6=7×6=42. 3 /(c3+3c2b+3cb2 + b3) ÷ √(a2 + b2 - 2ab) =(27+54 +36 +8)/(1+4-4)=(125)+1=5. EXAMPLES. III. If a=1, b=2, c=3, d=5, e=8, find the numerical values of the following expressions. 10. (a+2b+3c+5e-4d)(6e-5d-4c-3b+2a). 11. (a+b+c)(e-d-c2). 12. (3d2-7c2). 13. e(d-3e)+d(d2+3e). 14. e-√e+1)+2}+(e− e) √(e-4). 15. √(a2+2ab+b2) × 3/(a3 +3a2b+3ab2+b3). 16. (c-3c2a+3ca2-a3)÷√(b+c2-2cb). IV. Change of the order of Terms. Like Terms. 33. When all the terms of an expression are connected by the sign it is indifferent in what order they are placed; thus 5+7 and 7+5 give the same result, namely 12; and so also a+b and b+a give the same result, namely, the sum of the numbers which are represented by a and b. We may express this fact algebraically thus, a+b=b+a. Similarly, a+b+c=a+c+b=b+c+a. 34. When an expression consists 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 is obvious from the common notions of arithmetic. Thus, for example, 7+8-2-3=8+7-2-3=7+8-3-2=8+7-3-2, 35. In some cases we may change the order of the terms further, by mixing up the terms which are preceded |