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Multiplying repeatedly by i works the same way no matter what number you start out
with. You can begin with a negative real, a positive imaginary, a negative imaginary, or
even the sum of a real number and an imaginary one. Each multiplication by i always
turns the complex number vector (a ray connecting the origin with the point in the coordinate
plane representing the number) counterclockwise through 90. The length of the
vector does not change unless you multiply a number by some quantity ib, where b is
a real number. In that case, the vector length is multiplied by b, and it rotates through
90 also. For example, if you multiply something by i2, the vector turns 90 counterclockwise
and becomes twice as long; if you multiply by i/12 or i(1/12), the vector turns
90 counterclockwise and becomes 1/12 as long.
Complex Numbers 325
THE COMPLEX NUMBER PLANE
The set of rectangular coordinates shown in Fig. 22-2A through D is known as the
complex number plane, in which you can plot quantities that are part real and part
imaginary. The real part is expressed toward the right for positive and toward the left
for negative. The imaginary part goes upward for positive and downward for negative.
Any point in the plane, representing a unique complex number, can be expressed as an
ordered pair (a,ib) or written algebraically as a ib, where a and b are real numbers
and i is the unit imaginary number.
If a 0, a complex number is called pure imaginary. If b 0, a complex number is
called pure real. If both parts are positive, the quantity is in the first quadrant of the
complex number plane. If the real part is negative and the imaginary part is positive, the
quantity is in the second quadrant. If both parts are negative, the quantity is in the third
quadrant. If the real part is positive but the imaginary part is negative, the quantity is in
the fourth quadrant.
MULTIPLYING COMPLEX QUANTITIES
Here is a geometric method by which you can multiply complex numbers. Suppose you
have two complex quantities that can be portrayed both in rectangular coordinates as
real and imaginary parts and in mathematician’s polar coordinates as a vector magnitude
and a direction angle. In Fig. 22-3, two complex quantities a ib and c id are
set out, each beginning at the positive real number axis. In polar form these complex
numbers are (,r) and (,s), respectively. Complete the triangles formed by these vectors,
the real axis, and vertical lines running from the plotted points down to the real axis.
These triangles are shaded in the diagram. Beginning at the magnitude of the first quantity
and multiplying each part of the second by the magnitude of the first, you can erect
a third, unshaded triangle. This triangle brings you to the product, in magnitude and
angle, or it can be read in rectangular coordinates as real and imaginary parts. Study the
calculations next to the diagram to see how the quantities that appear in the algebra are
reproduced in coordinate geometry.