Signal Processing and Linear Systems-B.P.Lathi copy

X a nd h j t his notation allows us to determine t he

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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 (1465-1526) a nd t hen b y N iccolo F ontana (1499-1557). 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 J-121 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 (1526-1573), 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.

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