EXAMPLES. 1. Reduce 5+ √3 tor. to a fraction having a rational numera Multiplying both numerator and denominator by 5-√3, Again, multiplying both numerator and denominator of this last fraction, by 6-2 v15, it becomes 4√30-4√15-6√10+10 √6-8√3-12√2 -24 or changing both signs of numerator and denominator, it becomes, after striking out the factor 2 from both, 6√2+4√3-5√6+3√10+2√15-2√30 1 4. Reduce V3+1 2+1 to an equivalent fraction having rational denominator. 2+12-16 Ans. 4 Vatic 5. Reduce first to a fraction having a rational Vb+ vx' denominator, and then to a fraction having a rational numerator. vat v x Vab- Vaxt vbx-x ✓btvc Vu+ x b-x a-T ах IMAGINARY QUANTITIES. (125.) We have already shown, that (see Note to the Rule under Art. 98) an even root of a negative quantity is impossible. Such expressions are called imaginary. a are all imaginary quantities. -a 2m) -a Surd quantities, though their values can not be accurately found, can, nevertheless be approximately obtained; but imaginary quantities can not have their values expressed by any means, either accurately or approximately. They must, therefore, be regarded merely as symbolical expressions. (126.) We will confine ourselves to the imaginary expressions arising from taking the square root of a negative quantity The general form of imaginaries of this kind, is v-ar Vax-1= Va.v-1, replacing the rational surd quantity Va, by b, we have V-a=bV-1, so that all imaginary quantities arising from extracting the square root of a minus quantity is of the form = c4 m, (127.) If we put V-1=c, we shall always have C2=-1 5 = V-1. And in general c4n=1 -1 C4m+3 =-v-1, m being any integer whatever. (128.) From which we easily deduce the following principles. (+V-a)x(+V-a)=-Va?=-a. -= . (+V-a)*(-V—b)=+ Vab. The above is in accordance with the usual rules for the multiplication of algebraic quantities, and must be considered as a definition of this symbol, and of the method of using it, and not as a demonstration of its properties. (129.) The student must not infer from what has been said, that imaginary quantities are useless. So far from being useless, they have lent their aid in the solution of questions, which required the most refined and delicate analysis. (130.) We will now, in order to become more familiar with the operations of imaginaries, perform some examples in MULTIPLICATION OF IMAGINARIES. 1. Multiply 4 V-1+V-2 by 2 V-1-V-3. OPERATION, 41-1+1-2 :-8-2V2+4V3+ 16. 2. Multiply 4+1-3 by. 2-V-2. OPERATION, 4+ 1-3 2- V-2 8+2 V-3-47-2+ V6. 3. Multiply 3-V-1 by 4+ V-1. OPERATION. 3- V-1 12-4-1 12-1-1+1=13-1-1. 4. Multiply 1-1V-3 into itself. OPERATION. 1-V-3 1-1V-3 -iv-3 1-1-3-*=-1-V-3. |