Possibly because x and y are used to denote the real and imaginary parts, respectively.
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It is Z from the German for "to count". The counting, or natural numbers are denoted by N.
There are no real reason why it is denoted by z, but that the real number axis is denoted by x, imaginary number is denoted by y, the real part of a complex number is denoted by a, the imaginary part of a complex number is denoted by b, so there is z left.
why an set of integer denoted by z
The set of integers consists of zero, the natural numbers and their inverse (negatives). This is denoted by a boldface Z standing for the German word Zahlen. It means that Z is a subset of the sets of rational and real numbers and is countably infinite.
It is the set of integers, denoted by Z.