Page:Elementary algebra (1896).djvu/241

223 Rh IMAGINARY QUANTITIES. 223

this form is called a complex number. If a=0, the form becomes b\sqrt{} 1, which is called a pure imaginary expression.

268. By definition, \sqrt{}—1 xV—1=—1.

that is, G/a- V1 =—«4. Thus the product \sqrt{}a. V—1 may be regarded as equivalent to the imaginary quantity \sqrt{} -a.

269. It will generally be found convenient to indicate the imaginary character of an expression by the presence of the symbol —1 which is called the imaginary unit ; thus

\sqrt{}—4=V4Ex(—1)=2V-1. \sqrt{}-T@=Vid x(—1)=aV/7v— 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) x (—b)= ab, by taking the square root, we have

Va=axV—b=4 Vad.

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

Gy I Cay alo i eat = al

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+bV—-14+(e+dv—-1)=aic+(b+dv—-1.

Ex. 2. The product of a+ bV—1 and c+dV—1

=(a+bV—1)(¢+dv—1) =ac— bd + (be + ad)V =1.