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Yes integers are whole numbers
Always. By definition the Integers are the whole numbers; and the whole numbers are the integers.
All integers are rational numbers. There are integers with an i behind them that are imaginary numbers. They are not real numbers but they are rational. The square root of 2 is irrational. It is real but irrational.
What you need to know about integers is that integers is the name for the group of numbers that include whole numbers and negative numbers. But integers DO NOT include fractions.
No- not exactly. Negative integers are not counting numbers. Positive integers are identified with counting numbers. Many authors like to start with zero as a counting number.