BaalChatzaf Posted September 12, 2007 Share Posted September 12, 2007 (edited) If one is willing to consider ordered pairs of real numbers, the imaginary numbers are not at all mysterious.Consider pairs (a, b ) where a and b are real. Let us define arithmetical operations on such pairs.!!!!Damned emoticons!!!!Definition of addition: (a,b ) + (c,d ) = (a+c, b+d ) .Definition of multiplication: (a,b ) * (c,d ) = (a*c -b*d, b*c + a*d ). The "zero" of such pair arithmetic is (no surprise) (0,0 ) and has the property (a,b ) + (0,0 ) = (a,b ) [show this as an exercise].How about negation? Define negation: -(a,b ) = (-a, -b ) (that wasn't hard, was it?).Now here is the goody: Look at (0,1 ). This is no more mysterious than any other pair of real numbers. Now multiply it by itself (0,1 ) * (0,1 ) = (0*0 - 1*1, 1*0 + 0*1 ) = (-1,0 ) = -(1,0 ).If one identifies the pair (a, 0 ) with the number a (show that this is arithmetically sensible as an exercise) we get (0,1 )*(0,1 ) = -1. In short, (0,1 ) is a number, which when squared is -1. Voila! It is none other than mysterious, imaginary, crazy i. Ba'al Chatzaf Edited September 12, 2007 by BaalChatzaf Link to comment Share on other sites More sharing options...
Michael Stuart Kelly Posted September 12, 2007 Share Posted September 12, 2007 !!!!Damned emoticons!!!!Bob,Before you hit "Add Reply" when you post, drift your gaze up the screen a bit. There is a section called "Post Options" and a little box is checked by default beside "Enable emoticons?". Uncheck it and you will be fine. No emoticons for that post.Michael Link to comment Share on other sites More sharing options...
tjohnson Posted September 12, 2007 Share Posted September 12, 2007 If one is willing to consider ordered pairs of real numbers, the imaginary numbers are not at all mysterious.Ba'al ChatzafYes, it is unfortunate that the term 'imaginary' was used in this respect. Similarly for 'rational' and 'irrational' numbers, there are connotations from natural language that probably discourage people from learning more about the evolution of number systems. Link to comment Share on other sites More sharing options...
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