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Unformatted text preview: complex number. He showed
t hat every e quation of the n th order has exactly n solutions (roots), no more and no
less. Gauss was also one of the first t o give a coherent account of complex numbers
and t o i nterpret t hem as points in a complex plane. I t is he who introduced t he
t erm c omplex n umbers a nd paved the way for general and systematic use of complex
numbers. T he n umber s ystem was once again broadened or generalized t o include
imaginary numbers. Ordinary (or real) numbers became a special case of generalized
(or complex) numbers.
T he u tility o f complex numbers can b e u nderstood readily by an analogy with
two neighboring countries X a nd Y , as illustrated in Fig. B .l. I f we w ant t o travel
from City a t o C ity b ( both in Country X ), t he s hortest route is t hrough Country Y,
although t he j ourney begins and ends in Country X . We may, if we desire, perform
this journey by a n a lternate route t hat lies exclusively in X , b ut this alternate route
is longer. In m athematics we have a similar situation with real numbers (Country
X ) a nd complex n umbers ( Country Y ). All realworld problems must s tart w ith real
numbers, and all the final results must also be in real numbers. B ut t he derivation
o f results is c onsiderably simplified by using complex numbers as an intermediary.
I t is also possible t o solve all realworld problems by an alternate method, using
real numbers exclusively, b ut such procedure would increase the work needlessly.
3 B.12 Algebra o f Complex Numbers
A complex n umber (a, b) or a + j b c an be represented graphically by a point
whose Cartesian coordinates are (a, b) in a complex plane (Fig. B.2). Let us denote
this complex n umber by z so t hat F ig. B .1 Use o f complex numbers can reduce t he work. z = a + jb (B.1) T he numbers a a nd b ( the abscissa and the ordinate) of z are the r eal p art and
the i maginary p art, respectively, of z. T hey are also expressed as
Re z = a 1m z = b
Note t hat in this plane all real numbers lie on t he horizontal axis, and all imaginary
numbers lie on the vertical axis.
Complex numbers may also be expressed in terms of polar coordinates. I f (r, 0)
are the polar coordinates of a point z = a + j b (see Fig. B.2), then
a =rcosO
b =rsinO and
z = a + j b = r cos 0 + j r sin 0
= r(cosO+jsinO) i
Imaginary b Real """ .................
 b .................... : :,. z· F ig. B .2 R epresentation of a number in t he complex plane. (B.2) 6 Background B.1 T he E uler f ormula s tates t hat
e i
1m j8 7 Complex N umbers = cos () + j sin () j re J6 T o prove t he E uler formula, we e xpand e j8 , cos (), a nd sin () using a Maclaurin series
e j8 1t (.())3
(.())4
(.())5 (.())6
( .())2
= 1 + j () + _ J_ + _ J_ + _ J_ + _ J_ + _ J_ + " .
2!
3!
4!
5!
6]
()2 = 1 + j () ()2 ()3 2T  j 3! + 4! + j 5!  ()4 ()6 ()4 ()5 Re ..... 05 j
(b) ()s F ig. B .3 Understanding some useful identities in terms of re j8 . ()7 Hence, it follows t hat
ej8 = cos 0 + j sin () (B.3) Usin...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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