Z for the set of integers is apparently due to Bourbaki. The letter stands
for _Zahl_ (_Zahlen_).
In a message to a mathematics history mailing list, Julio Gonzalez Cabillon
wrote that he believed Bourbaki was responsible for both of the above symbols,
quoting Weil, who wrote, "...it was high time to fix these notations once
and for all, and indeed the ones we proposed, which introduced a number of
modifications to the notations previously in use, met with general approval."
In Landau's Grundlagen der Analysis (1930, p. 64), he denotes the set of
integers by a fraktur Z with a bar over it. He does not seem to introduce
symbols for the sets of rationals, reals, or complex numbers. [Stacy Langton]
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Reference: http://members.aol.com/jeff570/nth.html