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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 (11141185) 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 hirdorder e quationt
x3 + ax + b = 0 is given by x = \jb~\jb~
 2+Y'4+2'7+  2Y'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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 Spring '13
 Bayliss
 Signal Processing, The Land

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