No, zero is an integer but it is neither a postiive nor a negative integer.
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the positive integer
There are just three possible cases: Positive integer: positive x positive = positive. Negative integer: negative x negative = positive. Zero: zero x zero = zero.
a positive integer
A positive integer is an integer to the right of zero on the number line. It is more then zero
The sum of zero and a negative integer can never be zero - it will always be negative and nonzero. Although zero is also an integer, it is neither negative nor positive and cannot be the other integer used.