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224 Rh 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 b=0.

For, if a+b—1=0, then b —1=—a; - b = -a a + b=0.

Now a and b 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 + (b-d)—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.