Integers are WHOLE number (not numbers with decimals or anything like that.) They are used for many things like equation, calculating, and more.
By the way if you are going to "fling out the exact change" the number is not considered an integer if the number of the amount has a decimal in it like 6.25 dollars that is wrong.
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The theory of numbers is based on the theory of integers. It is important to understand the property of integers, and the extension of the system that they form, to understand the properties of the extensions: rational numbers, real numbers, complex numbers and so on.
Extending the set of all integers to included rational numbers give closure under division by non-zero integers. This allows equations such as 2x = 3 to be solved.
Non-positive integers are zero and the negative integers.
The set of integers represents the integers.
The boundary points of the integers is simply the integers.
consecutive integers