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

Pythagoras a number mystic who regarded n umbers as t

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Unformatted text preview: umber or a fraction. Pythagoras, a number mystic, who regarded n umbers as t he essence and principle of all things in t he universe, was so appalled a t his discovery t hat he swore his followers to secrecy a nd imposed a d eath p enalty for divulging this s ecret'! T hese numbers, however, were included in t he n umber system b y t he t ime of Descartes, a nd t hey are now known as i rrational n umbers. U ntil recently, n egative n umbers were not a p art of the number system. T he c oncept of negative numbers must have appeared absurd to early man. However, t he medieval H indus h ad a clear understanding of the significance of positive a nd negative numbers. 2 ,3 T hey were also t he first to recognize the existence of absolute negative quantities. 4 T he works of B haskar (1114-1185) on arithmetic (L[lavatD a nd a lgebra ( B[jaganit) n ot only use t he decimal system b ut also give rules for dealing with negative quantities. Bhaskar recognized t hat positive numbers have two square roots. 5 M uch l ater, in Europe, t he b anking system t hat arose in Florence and Venice d uring t he l ate Renaissance (fifteenth century) is credited with developing a crude form of negative numbers. T he seemingly absurd subtraction of 7 from 5 seemed reasonable when bankers began t o allow their clients t o d raw seven gold ducats while t heir deposit stood a t five. All t hat was necessary for this purpose was t o w rite t he difference, 2, o n t he d ebit side of a ledger.6 T hus t he n umber system was once again broadened (generalized) t o include negative numbers. T he acceptance of negative numbers made i t possible t o solve equations such as x + 5 = 0, which h ad no solution before. Yet for equations such as x 2 + 1 = 0, l eading t o x 2 = - 1, t he solution could not be found in t he real number system. I t was therefore necessary t o define a completely new kind of number with its square e qual t o - 1. D uring t he t ime of Descartes and Newton, imaginary (or complex) n umbers came t o b e accepted as p art of the number system, b ut t hey were still r egarded as algebraic fiction. T he Swiss m athematician L eonhard E uler i ntroduced t he n otation i (for i maginary) a round 1777 t o represent H . Electrical engineers use t he n otation j i nstead of i t o avoid confusion with t he n otation i often u sed for electrical current. T hus j2 = -1 B.1 3 Complex N umbers G erolamo C ardano ( left) a nd K arl F riedrich G auss ( right). i maginary numbers plausible and acceptable t o early mathematicians. T hey could as pure nonsense when i t a ppeared as a solution t o x 2 + 1 = 0 dismiss H because this equation has no real solution. B ut in 1545, G erolamo C ardano of Milan published A rs M agna ( The G reat A rt), t he m ost i mportant algebraic work of t he Renaissance. In this book he gave a method of solving a general cubic equation in which a root of a negative number appeared in a n i ntermediate step. According t o his method, t he solution t o a t hird-order e quationt x3 + ax + b = 0 is given by x = \jb~\jb~ - 2+Y'4+2'7+ - 2-Y'4+2'7 For example, to find a solution of x 3 in t he above equation t o o btain + 6x - 20 = 0, we s ubstitute a = 6, b =...
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