Page numbers in a book, your phone number and (most) addresses. Any time you count people, places or things. When someone is on the 40 yard line in football, that is an integer. Ever since you learned how to count you used integers, you just didn't use the word "integer".
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You have it backwards. Integers are a subset of real numbers.
Counting, checking your bank account what has been deducted, reading a weather report, or even when you check your weight. The integers are positive and negative numbers we use in life.
I thinl the term you want is integers.
Example: Question: How many toes have you? Answer: At the last count, nine.