Rational and complex numbers

[Richard E. Zippel <RZ @ MIT-MC>]

    Date: Sunday, 12 June 1983  19:54-EDT
    From: MOON at SCRC-TENEX
    To:   Richard E. Zippel <RZ>
    cc:   common-lisp at SU-AI
    Re:   Rational and complex numbers

    Is there really more than one reasonable injection of Z into Q?
Any map that sends 1 to an element of Q (other than 0) and all the elements
accordingly is reasonable.  Basically, any time you are counting things by
halves, thirds, fourths or some other fraction you are presuming a different
injection.  (otherwise known as scaling)  It might be nice to be able to
have integers with these scaling factors built in.  (multiples of pi or
pi/4 might be nice for trig calculations.  Certainly radians would be a more
palatable measure of angles if the scaling factor of pi were built in.
acos(0) would be 1/2(radians).  When converted to floating point (the reals)
it would of course give 1.57 ...)  Why should these conversions be
any different from the standard one?  (The answer is: because it is more
common) 

I'm not suggesting that the natural coercion from Z to Q be thrown away as
much as suggesting that there is a lot more power and flexibility lurking
thee than w currently acknowledge.