275. When two imaginary expressions differ only in the sign of the imaginary part, they are said to conjugate.
Thus a— b —1 is conjugate to a +b —
Similarly 2 + 3—1 is conjugate to 2—3—1.
276. The sum and the product of two conjugate imaginary expressions are both real.
For a+b —1+a—b —1=2a. Again (a +b—1)(a- b -1)=a—(—b)= a +b.
277. If the denominator of a fraction is of the form a+b -1, it may be rationalized by multiplying the numerator and the denominator by the conjugate expression a—b —1. For instance,
{c+d-1}{a+b-1} = {(c+d-1)(a-b-1)}{(a+b-1)(a-b-1}={ac+bd +(ad bc)-1}{a+b} = {}{}
Thus, by reference to Art. 271, we see that the sum, difference, product, and quotient of two imaginary expressions is in each case an imaginary expression of the same form.
278. Fundamental Algebraic Operations upon Imaginary Quantities.
Ex. 1. Find value of — a4 + 5 — 9a4 —2—4 a4.
V=¢ =Vaei(— 1) = eVv—1 5V—9a =5VIG(—1) = 1eV=1 2v—4at 2V4a(—1)=— 40V=1
= 12@V—-1
Ex 2. Multiply, 2-3 by 3 -2
2 -3=2 3 -1 3 -2=3 2 -1 2 3-1 3 2-1=6 6-1=-6 6
