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Unformatted text preview:  20 .
x a nd H =±j
T his notation allows us to determine t he s quare root of any negative number. For
example, N =V4xH=±2j
W hen i maginary numbers are included in t he number system, t he resulting
numbers are calle<l c omplex n umbers.
Origins o f Complex Numbers Ironically ( and c ontrary to popular belief), i t was n ot the solution of a quadratic
equation, such a s x 2 + 1 = 0, b ut a cubic equation with real roots t hat m ade = \ /10+ v'i08 + \/10  v'i08 = V20.392  V0.392 =2 We can readily verify t hat 2 is indeed a solution of x + 6 x  20 = O. B ut when
C ardano t ried t o solve t he e quation x 3  15x  4 = 0 by this formula, his solution
3 t This e quation is known as t he depressed cubic equation. A general cubic e quation
y3 + py2 + q y + r = 0
c an always b e r educed t o a d epressed cubic form b y s ubstituting y = x  ~. T herefore a ny g eneral
cubic e quation c an b e solved if we know t he s olution t o t he d epressed cubic. T he d epressed
cubic was independently solved, first by S cipione d el F erro (14651526) a nd t hen b y N iccolo
F ontana (14991557). T he l atter is b etter k nown in t he h istory o f m athematics a s T artaglia
( "Stammerer"). C ardano l earned t he s ecret o f t he d epressed cubic solution from Tartaglia. He
t hen showed t hat b y u sing t he s ubstitution y = x  ~, a g eneral cubic is reduced t o a d epressed
cubic. 4 Background was
x= ~2 + J 121 + ~2  B.1 5 Complex Numbers J121 W hat was C ardano t o make of this equation in the year 1545? I n those days
negative numbers were themselves suspect, and a square root of a negative number
was doubly preposterous! Today we know t hat
(2 ± N = 2 ± j 11 = 2 ± J 121
Therefore, C ardano's formula gives
x = (2 + j ) + (2  j) = Country 4 y We can readily verify t hat x = 4 is indeed a solution of x  15x  4 = O. C ardano
tried t o explain halfheartedly t he presence of J  121 b ut ultimately dismissed the
whole enterprise a s being "as s ubtle as i t is useless." A generation later, however,
R aphael B ombelli (15261573), after examining Cardano's results, proposed acceptance of imaginary numbers as a necessary vehicle t hat would t ransport t he
m athematician f rom t he real cubic equation to its real solution. In other words,
while we begin a nd e nd with real numbers, we seem compelled t o move into an
unfamiliar world o f imaginaries t o complete our journey. To m athematicians of t he
day, this proposal seemed incredibly strange. 7 Yet they could not dismiss the idea
of imaginary n umbers so easily because this concept yielded the real solution of a n
equation. I t t ook two more centuries for t he full importance of complex numbers t o
become evident i n t he works of Euler, Gauss, and Cauchy. Still, Bombelli deserves
credit for recognizing t hat such numbers have a role t o play in algebra. 7
I n 1799, t he G erman mathematician K arl F riedrich G auss, a t a ripe age
of 22, proved t he f undamental theorem of algebra, namely t hat every algebraic
equation in one unknown has a root in t he form of a...
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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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