EXERCISES. 1. Find a multiplier whose product by 3-2 shall be rational. The rules for the multiplication, division, involution, and evolution of quantities, are generally applicable, whether their exponents be integral or fractional, positive or negative. The only cases for which this proposition requires to be proved, are when the exponents are negative, and either integral or fractional. This may easily be proved by the reader, by considering that, by convention, am is the same as 1 am and the same as am. So =am-n, or = m 1 an-m 1 am an a". By this means, any quantity with a negative exponent, may be converted into one with a positive exponent; and the rules being then applied to the latter, the result may then be changed into one with negative exponents. IMAGINARY QUANTITIES. (218.) ' Imaginary quantities are not real quantities, for their values cannot be assigned either by terminate or interminate numbers.' These quantities, which are also called impossible quantities, occur when it is required to find an even root of a negative quantity, which is unassignable in terms of any real quantities, either rational or irrational; for any even power of any real quantity, whether positive or negative, is positive, and therefore no even root of a negative quantity can be found. Any root of a negative quantity is equal to the same root of that quantity taken positively, multiplied by the same root of For 1. aa (-1); therefore m a = m/ a my —1 When m is an odd number, "-11 (186.) When m is an even number, it will be equal to some power of 2, as 2", or to the product of some power of 2, as 2", by an odd number n. -1 is = 2 — 1, times in succession. Let m 2", then the mth root of equal to the square root taken r Thus, if r =3, m = 23 = 8, and m 1 = √ {√(√ — 1)}, which is imaginary, because √ — 1 is so. = Let m = 2′′n, then "— 1 = (~ — 1); and as n is odd, ~—1——1; therefore 1= 12—Ï; and, as in the preceding case, the imaginarity of this latter expression depends on 1, as is evident by giving r any particular value, as 3, as is shown above. (219.) All imaginary quantities therefore depend on the single imaginary√1. Also any quadratic imaginary is the product of a real quantity and 1. For -a =√a(−1)=√ a√ — I The product of two quadratic imaginaries leads to a para doxical result. Thus, a.√. a is the square of the square root of -a; and hence the result must be a. This But by the rule for multiplying radicals, √ a.√ -a =√(-a) (-a)=√a2, and this last quantity is usually expressed by a; so that if the upper sign be taken, the result is a, whereas the former product was — ɑ. apparent contradiction is, however, easily reconciled. In the present case a2 is the product (-—a) (—a), and its root is therefore -a; and being so, it cannot also be +a, for the double sign is properly used only when it is doubtful what the sign of the root is, and merely means that it is either positive or negative, but it does not imply that it has both these signs; and when the origin of the square known in any case, as in the present, no ambiguity exists, so that the sign of the root being known, the other sign is excluded. Thus, when a2 is the product, (-a) (a) =(-a); then a2 = √ (— a)2 = -a; and when a is the product, (+a) (+ a) = ( + a)2, √√ a2 = √(+ a)2 = + a aa, therefore √ √ Since == -1. a.. is The rules for addition and subtraction of imaginaries are the same as for surds. I. Multiplication of Quadratic Imaginaries. (220.) The product of two quadratic imaginaries is found by taking the square root of their product, considering them as real, and prefixing the negative or positive sign, according as their signs are like or unlike." b) For (+√=a) (+√ = 6) = √ a √√=1.√ √ =1 =√ ab√(-1)2=—√ ab, and (a)(√√ — b) = √ ab. √ (— 1)2 = — √ ab ; also (−√√—a) ( + √ —b) = −√ ab √ (− 1)2 = +√ ab If the imaginaries have coefficients, the products of the former must be multiplied by that of the latter. EXAMPLES. 1. Multiply 43 by 3-5 4√3 × 3√5=-12/15 or 4√3√1x3√5√-1=-12/15 2. Multiply 3+2√ — 5 by 2—3 √. 3+2√-5 2 6+4√-5-9√2+6/10 3. Multiply a b√ — 1 by a + b√ — 1 (a b√1) (a+b√ — 1) = a2+ab√—1—ab√1 +b2=a2+b2 EXERCISES. 1. Multiply 5-2 by 3√3 2. Multiply 3-5 by -2/3 3. Multiply 3+√2 by 2-3√1 N 4. Multiply ab √c by a + b √√ c 5. Find the square of ab√1 6. Find the product of a-3-band c-2 d 1... 15/6 2... 6/15 ANSWERS. 3... 6+2√2-9√−1+3√2 4... a2+b2c 5... a2-2ab1-12 6... ac-2ad-3 c√b-6√ bd II. Division of Quadratic Imaginaries. (221.) The quotient of two quadratic imaginaries is found by taking their quotient, considering them as real, and prefixing the positive or negative sign according as their signs are like or unlike." 3. Divide 2+3/1 by 2-1 The multiplier that will make the denominator rational is 2+1, which may be found from the general formula (217), by assuming c=2, and y=√1; or it may more easily be found by considering that (a+x) (a—x) = a2 —2. Hence |