19. 16-2/20−2 √28+2√35. 20. 24+4/15-4/21-2/35. 21. 6+12-24-√8. 22. 5-10-15+√6. 23. a+3b+4+4a-4√3b-2√3ab. 24. 21+3/8-63-67-√24-56+2/21. 2 1 39. (28—10/3)3 — (7+4/3). 40. (26+15/3) - (26 + 15/3) ̄. 41. Given √5=2.23607, find the value of 10/2 √10+√18 √8+√3-√5 42. Divide 23+1+3x 2 by x-1+/2. 43. Find the cube root of 9ab2 + (b2 + 24a2) √√/b2 - 3a2. IMAGINARY QUANTITIES. 92. Although from the rule of signs it is evident that a negative quantity cannot have a real square root, yet imaginary quantities represented by symbols of the form -a, -1 are of frequent occurrence in mathematical investigations, and their use leads to valuable results. We therefore proceed to explain in what sense such roots are to be regarded. When the quantity under the radical sign is negative, we can no longer consider the symbol as indicating a possible arithmetical operation; but just as a may be defined as a symbol which obeys the relation √ax a = a, so we shall define a to be such that √-ax√-a-a, and we shall accept the meaning to which this assumption leads us. It will be found that this definition will enable us to bring imaginary quantities under the dominion of ordinary algebraical rules, and that through their use results may be obtained which can be relied on with as much certainty as others which depend solely on the use of real quantities. Thus the product Ja. -1 may be regarded as equivalent to the imaginary quantity √-a. 94. It will generally be found convenient to indicate the imaginary character of an expression by the presence of the symbol-1; thus 4 = √ √ 4 × ( − 1) = 2√−1. √−7a2 = √7a3 × ( − 1) = a √7 √−1. 95. We shall always consider that, in the absence of any statement to the contrary, of the signs which may be prefixed before a radical the positive sign is to be taken. But in the use of imaginary quantities there is one point of importance which deserves notice. by taking the square root, we have √= a × √− b = ± √ab. Thus in forming the product of a and b it would appear 96. It is usual to apply the term 'imaginary' to all expressions which are not wholly real. Thus a+b√-1 may be taken as the general type of all imaginary expressions. Here a and b are real quantities, but not necessarily rational. 97. In dealing with imaginary quantities we apply the laws of combination which have been proved in the case of other surd quantities. Example 1. a+b√1±(c+d√−1) = a + c + (b±d) √ -1. Example 2. The product of a+b-1 and c+d√-1 then = (a + b √√ − 1) (c + d / − 1) =ac− bd + (bc+ad) √ −1. 98. If a + b/− 1 = 0, then a = 0, and b=0. b√−1 = -a; .'. — b3 = a3 ; .'. a2 + b2 = 0. Now a and b are both positive, therefore their sum cannot be zero unless each of them is separately zero; that is, a = 0, and b = 0. 99. If a+b-1=c+d√-1, then a = c, and b = d. For, by transposition, a − c + (b − d) √− 1 = 0; therefore, by the last article, a c = 0, and b- d = 0; that is a = c, and b = d. Thus in order that two imaginary expressions may be equal it is necessary and sufficient that the real parts should be equal, and the imaginary parts should be equal. 100. DEFINITION. When two imaginary expressions differ only in the sign of the imaginary part they are said to be conjugate. Thus a-b-1 is conjugate to a + b √−1. Similarly √2+3/-1 is conjugate to √2-3√-1. 101. The sum and the product of two conjugate imaginary expressions are both real. 102. DEFINITION. The positive value of the square root of a+b2 is called the modulus of each of the conjugate expressions a+b√-1 and a-b√-1. 103. The modulus of the product of two imaginary expressions is equal to the product of their moduli. Let the two expressions be denoted by a+b√−1 and c+d√−1. Then their product = ac – bd + (ad + bc) √−1, which is an imaginary expression whose modulus = = · √(ac - bd)2 + (ad + bc)* · √(a2 + b2) (c2 + d3) √a2 + b2 × √c2 + d3 ; which proves the proposition. 104. If the denominator of a fraction is of the form a+b√-1, it may be rationalised by multiplying the numerator and the denominator by the conjugate expression a-b√-1. |