Imaginary Numbers are not so imaginary or mysterious


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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 by BaalChatzaf
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If one is willing to consider ordered pairs of real numbers, the imaginary numbers are not at all mysterious.

Ba'al Chatzaf

Yes, 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.

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