An integer is any whole number. For example, 4, 16, -8 are all integers. There is no decimal part. Obviously, there is no "maximum number" as these numbers continue up to infinity.
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an integer can be represented as any letter of the alphabet
A natural number is an integer equal to 1 or greater. A rational number can be an integer or a fraction represented as x/y whereby x and y are both integers.
The set of real numbers is represented by a fancy script "R". However, real numbers are just represented by a number (be it integer, rational, irrational, or otherwise). The key here is that it is not a complex number (number containing the imaginary number "i").
No. An irrational number is one that is not a rational number. A rational number is once that equals one integer divided by another. So an irrational number cannot be represented by one integer divided by another integer, so it cannot be an integer!