264. Sometimes equations are proposed in which the unknown quantity appears under the radical sign. Such equations are varied in character and often require special artifices for their solution. We shall consider a few of the simpler cases, which can generally be solved by the following method: Bring to one side of the equation a single radical term by itself: on squaring both sides this radical will disappear. By repeating this process any remaining radicals can in turn be removed. Square both sides; then 4x - 4√x + 1 = 4 x − 11, 9. √2x+11=√5. 10. √4x2-7x + 1 = 2 x − 1. 13. √x+3+ √x = 5. 14. 10√25+9x=3√x. 15. √x 4 + 3 = √x + 11. 265. When radicals appear in a fractional form in an equation, we must clear of fractions in the ordinary way, combining the irrational factors by the rules already explained in this chapter. CHAPTER XXIV. IMAGINARY QUANTITIES. 266. An imaginary quantity is an indicated even root of a negative quantity. In distinction from imaginary quantities all other quantities are spoken of as real quantities. Although from the rule of signs it is evident that a negative quantity cannot have a real square root, yet 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 va x vaa, so we shall define √ a to be such that αχν 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 algebraic 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. 267. Any imaginary expression not involving the operation of raising to a power indicated by an exponent that is an irrational or imaginary expression, can be reduced to the form a+b√1, which may be taken as the general type of all imaginary expressions. Here a and b are real quantities, but not necessarily rational. An imaginary expression in this form is called a complex number. If a = 0, the form becomes b-1, which is called a pure imaginary expression. 268. By definition, √1x-1=-1. Thus the product √a. √−1 may be regarded as equivalent to the imaginary quantity V-a. 269. It will generally be found convenient to indicate the imaginary character of an expression by the presence of the symbol V-1 which is called the imaginary unit; thus √−4=√4 x (− 1) = 2√ — 1. √-7 a2 = √7 a2 × (− 1)=a√√7√ −1. 270. 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 the following point deserves notice. Since (− a) × (—b) = ab, by taking the square root, we have √=ax√=b=±√ab. Thus in forming the product of V-a and V-b it would appear that either of the signs + or might be placed before Vab. This is not the case, for √=ax√-b=√a. √=1 × √b. √-1 =√ab(√-1)2=-√ab. 271. In dealing with imaginary quantities we apply the laws of combination which have been proved in the case of other surd quantities. Ex. 1. a + b√ −1 ±(c + dv − 1) = a + c + (b ± d) √ − 1. Ex. 2. The product of a + b√-1 and c + d√=1 =(a+b√ 1) (c+d√1) = ac - bd + (bc + ad) √ — 1. 272. The symbol V-1 is often represented by the letter i; but until the student has had a little practice in the use of imaginary quantities he will find it easier to retain the symbol √1. The successive powers of √1, or i, are as follows: and since each power is obtained by multiplying the one before it by √-1, or i, we see that the results must now recur. 273. If a + b√1= 0, then a = 0, and 60. Now a2 and b2 are both positive, hence their sum cannot be zero unless each is separately zero; that is, a = 0, and b = 0. 274. If a+b√-1=c+d√-1, then a = c, and b = d. For, by transposition, a − c + (bd) √−1=0; therefore, by the last article, ac that is, a = c and b = d. 0, and b-d=0; 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. The student should carefully note this article and make use of it as opportunity may offer in the solution of equa tions involving imaginary expressions. |