Given real numbers: define integers? I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following:

Integer numbers are just special cases (a subset) of real numbers. Imagine a world where you know only real numbers. How are integers defined using mathematical operations?

Knowing only the set of complex numbers $a + bi$, I could define real numbers as complex numbers where $b = 0$. Knowing only the set of real numbers, I would have no idea how to define the set of integer numbers.
While searching for an answer, most definitions of integer numbers talk about real numbers that don't have a fractional part in their notation. Although correct, this talks about notation, assumes that we know about integers already (the part left of the decimal separator), and it does not use any mathematical operations for the definition. Do we even know what integer numbers are, mathematically speaking?

Under the representation of infinite decimals, the integers consist of those numbers whose numerals only have 0s, in any base system, after their decimal point or only have 9s after their decimal point.
