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I_SQUARE

                                        July 21, 2010

      This is how one gets
      to irrationality (of
      the mathematical
      variety, at least):

      If you square a number,
      that is, multiply it by
      itself, you always get       positive X positive = positive
      a positive number.           negative X negative = positive

         And when you take
         a square root of
         a number there's
         an ambiguity in
         the solution:
         either a positive
         or a negative         That in itself is
         result works.         interesting, though
                               not the point at hand.
         Is it at all
         reasonable to
         consider a                            If you have some picture
         square root                           in your head of equations
         of a negative                         as some sort of ideal
         number?                               machines-- turn the crank
                                               and get a dependable
         There's no possible                   result-- that falls apart
         solution. There's no                  quickly.
         number you can square
         that will get you a
         negative.

         Rather than just say
         "you can't go there"
         mathematicians decided
         to declare these
         "irrational" numbers,
         and they found things
         you can do with them.

         They factored out the insanity                         BIG_WALTER
         as radical negative one: a hard
         knot of irrationality that             A friend of mine once
         doesn't really make any sense          wondered if there was a
         whatsoever, but nevertheless           similar trick you could
         *can* be used in equations to          do with "division by
         determine practically useful           zero": can you factor
         results.                               out 1/0 as Chi, and
                                                keep going with your
         They labeled it "i", and               calculations even
         observed that "i squared"              though you have no idea
         is not irrational, but                 what this Chi might mean?
         rather an ordinary negative
         integer: minus one.                       The analogy breaks
                                                   down because there's
           This means that it's entirely           no mathmatical
           possible to perform calculations        operation that can
           with irrational numbers that            turn Chi back into a
           yield rational results.                 meaningful number.

           In fact, such calculations are                 1/0 and 4/0 are
           routinely done by electrical                   indistinguishable:
           engineers working with analog                  the singularity eats
           circuitry.                                     the identity of the
                                                          numerator.

                                                             (Just like
               The modern era saw an obsession                squaring a
               with the weirdness of quantum                  number eats
               mechanics and relativity, and                  it's sign?)
               yet the oddity of irrational
               numbers is shrugged off and
               ignored.

               Reality has some incomprehensibly
               weird aspects to it, and they don't
               just exist down on the quantum level.

                     Some of them come straight
                     out of mathematics...

                                                       [ref]

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