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Complex Numbers 329
When you cube it, you get a vector at an angle of 900. If you subtract 360 from this
twice, you can see that it is in the same direction as 180 which lies exactly along the
negative real axis.
RECIPROCAL OF A COMPLEX QUANTITY
If the vector representing a complex number a ib has magnitude r that is greater
than 1, its reciprocal has magnitude 1/r, which is of course less than 1. An example of
this is shown in Fig. 22-7. The triangular portion of the shaded area above the real
number axis uses unit magnitude on the positive real axis for its base, and the magnitude
r of the quantity a ib for its top side. Scaling this area down to make the longest
side fit unit magnitude on the positive real axis, the side that was 1 in the bigger triangle
is now the reciprocal of the original complex quantity, in both magnitude and polar
angle. This is represented by the part of the shaded area below the real axis.
The algebra in the figure shows how to calculate these values. When you have the
quantity a ib in the denominator of an expression, it presents a problem that is rather
difficult to work out directly. However, if you multiply both the numerator and the
denominator of such a fraction by a ib, the denominator becomes the sum of two
squares, which is a real number. It’s always a lot easier to divide by a real number than
it is to divide by a complex number!
In the study and use of complex quantities, the quantity a ib is called the conjugate of
a ib, and vice versa. The product of two complex conjugates is always a pure real number.
Figure 22-7
Geometric representation
of the
reciprocal of a
complex quantity.