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restore as before, because complex numbers are not simply ordered pairs; complex numbers do depend on a unique square root of -1
== Significance of Elementary Proofs ==
== Significance of Elementary Proofs ==
Elementary proofs are preferred over non-elementary proofs for at least one reason:
Elementary proofs are preferred over non-elementary proofs for at least two reasons:
*elementary proofs minimize the underlying assumptions, as in avoiding the assumption that there is a unique, algebraically manipulable square root of negative one
*elementary proofs are typically impossible to simplify further in a logically significant manner
*elementary proofs are typically impossible to simplify further in a logically significant manner