CHAPTER XXXIII. IMAGINARY QUANTITIES. 286. 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 V-a, V-1, called imaginary quantities, 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 v as indicating a possible arithmetical operation; but just as va may be defined as a symbol which obeys the relation Vax Vara, so we shall define V-a to be such that v-ax v-ar-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. 287. It is usual to apply the term imaginary to all expressions which are not wholly real. 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+bV-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 bv-1, which is called a pure imaginary expression. 288. By definition, V-1x-1=-1. .. va.V-1x va. V-1=a( -1); (va.V-1)=-a. that is, Thus the product va.V-1 may be regarded as equivalent to the imaginary quantity v-a. 289. It will generally be found convenient to indicate the imaginary character of an expression by the presence of the symbol V-1; thus V-4=V4x(-1)=2 V-1. V=7a’=V7a2 x (-1)=av7 V-1. 290. 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. Since (-a)x(-6)=ab, by taking the square root, we have ✓-ax-b=+Vab. Thus in forming the product of V-a and V-6 it would appear that either of the signs + or might be placed before Vab. This is not the case, for V-ax V-b=va.V-1x vb.V-1 = Vab(V-1) - Vab. 291. In dealing with imaginary quantities we apply the laws of combination which have been proved in the case of other surd quantities. Ecample 1. a+b7-1+c+d7-1)=a+c+b+d) V-1. Exumple 2. The product of a +6v-1 and c+dv - 1 =(a+bV-1) (c+dV1) =ac-bd+(bc+ad) V-1. 292. 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 V-1. It is useful to notice the successive powers of V-1 or i; thus For, if (V-1)=-1, i=i; (V-1)= -1, 12= -1; (V-1)3=-V-, 28= -1; (V-1)=1, 24=; and since each power is obtained by multiplying the one before it by V-1, or i, we see that the results must now recur. 293. If a+b7-1=0, then a=0, and b=0. a+bV-1=0, then 6-1=-a; .. – 62=a2; .:. a2+62=0. Now aand b2 are both positive, therefore their sum cannot be zero unless each of them is separately zero; that is, a=0, and b=0. 294. If a+b7-1=c+dV-1, then a=c, and b=d. For, by transposition, a-c+(6-d) V-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. The student should carefully note this article and make use of it as opportunity may offer in the solution of equations involving imaginary expressions. 295. DEFINITION. When two imaginary expressions differ only in the sign of the imaginary part, they are said to be conjugate. Thus a-ov-1 is conjugate to a+bV-1. 296. The sum and the product of two conjugate imaginary expressions are both real. For a+b7-1+a-bv-1=2a. =a2+%. + 297. If the denominator of a fraction is of the form a+bV-1, it may be rationalized by multiplying the numerator and the denominator by the conjugate expression a-6V-1. For instance, c+dV-1 _ (c+dV-1)(a-bv-1) ac+bd + (ad – bc) V-1 a? +62 ac+bd ad - bc ✓- -1. a? +62 a? +62 Thus, by reference to Art. 291, we see that the sum, difference, product, and quotient of two imaginary expressions is in each case an imaginary expression of the same form. 298. To find the square root of a+b V-1. Assume Va+bV-1=x+y=1, where x and y are real quantities. By squaring, a+6V-1=x2- y2 + 2xy V-1; therefore, by equating real and imaginary parts, 22 - y2=a... (1), 2cy=b. .(2); .. (22+ y)2 = (x2 - y2)2 + (2xy)? =a?+62; (3). From (1) and (3), we obtain Va+12+a va? +62 x2 = y2= 2 2 Va? +62 Va? +62 .'.x=+ y= == 2 2 Thus the required root is obtained. Since x and y are real quantities, x2 + y2 is positive, and therefore in (3) the positive sign must be prefixed before the quantity Va+62 Also from (2) we see that the product xy must have the same sign as b; hence x and y must have like signs if b is positive, and unlike signs if b is negative. -a ...(1), Example 1. Find the square root of - 724 V-1. Assume V-7-24 V-1=x+yV-1; then –7–24 V-1=x2 - y2 +2xy V-1; ... 22-y-=-7..... and 2xy=-24. =49+576 =625 ; ..22 + y2=25 From (1) and (2), 22=9 and y2=16 ; .. X=+3, y=+4. Since the product xy is negative, we must take x=3, y=--4; or x=-3, y=4. Thus the roots are 3–4V-1 and —3+4V-1; that is, V-7-24-1=+(3-4V-1), ...(2) . 299. The method of Art. 278 may also be used. 3. Simplify -- EXAMPLES XXXIII. 1. Add V-8 and ✓ – 18. 2. Add 4V-27 and 3V – 12. 9 27 3. V V-4+V-16+ V 4. Multiply 2-3+37-2 by 4-3-5V-2. Express with rational denominator : 1 3V-2+2V-5 5. 6. 3-V-2 3V-2-2-5 3+2V-1 3-2 V-1 a-XV7. a+xV-1 + 8. 2-5V-1 2+5 V-1 a-XV-1 a+#V-1 Find the square root of 9. -5+12V - 1. 10. – 11- 60 V-1. 11. -47+8V-3. Express in the form A +iB 3+5i 12. 1+i 14. 2-31 213-12 1-i 13. 13-172 |